Homework

Due Thursday, September 8

(Note: None of these problems are to be turned in to be graded, and some of them may be better understood after further class meetings. But I think it will be good if you can try to read the pertinent sections of the S&W (Samuels and Witmer) text and try to do these relatively simple exercises prior to the next class meeting.)

Problem 1 (0 points)

Consider Exercise 3.2 on p. 77 of S&W. Draw 10 random samples of size 5 using the last two digits in each of the first two columns (the ones labeled 01 and 06) on p. 670. Give the frequency of each of the six outcomes indicated at the bottom of p. 77. (I realize that I didn't cover the use of the table of random digits in class, but I think that you should be able to read over Sec. 3.2 and work this problem anyway. I may extend this exercise, and make further use of it, after I cover sampling distributions in class.)

Problem 2 (0 points)

Do Exercise 3.5 on p. 83 of S&W. (Note that the answers are in the back of the book.)

Problem 3 (0 points)

Do Exercise 3.13 on p. 92 of S&W. The wording of the parts should be What is the probability that a randomly selected person in this study is classified as being ... instead of "What is the probability that someone in this study is ..." since the wording in the text is a bit vague --- are we to take it to be that "someone in the study" means at least one person in the study? (Also, since the same person can feel stressed at one time and not stressed at another time, the wording isn't precise enough. My alternative wording indicates that we are to randomly select one of the 6549 subjects and examine the classifications obtained from that person's reported information.) In addition, the word either does not belong in the statement of part (d).

Due Thursday, September 22

(Note: For each of the problems to be turned in on Sep. 22, there is a similar problem in this set among the ones not to be turned in. Some of the answers to the problems not to be turned in (and I really don't want you to turn them in --- no need for me to sort through the extra pages) are in the back of the book, and I will post answers (or even more complete solutions) for the ones not to be turned in for which answers are not given. Because of all of this, I will only discuss with you the ones not to be turned in (since if you understand them, you should be able to do the others, and I don't want to give you hints on the others or wind up doing them for you).)

Problem 4 (0 points)

Do part (a) of Exercise 3.9 on p. 88 of S&W. (Note that the answer is in the back of the book.)

Problem 5 (3 points)

Do part (a) of Exercise 3.11 on p. 88 of S&W.

Problem 6 (0 points)

Do Exercise 3.12 on p. 92 of S&W. The wording of the parts should be What is the probability that a randomly selected person in this study ... instead of "What is the probability that someone in this study is ..." since the wording in the text is a bit vague --- are we to take it to be that "someone in the study" means at least one person in the study? (Note that the answers are in the back of the book.)

Problem 7 (2 points)

Do Exercise 3.14 on p. 92 of S&W.

Problem 8 (2 points)

Do part (b) of Exercise 3.18 on p. 101 of S&W.

Problem 9 (0 points)

Do part (a) of Exercise 3.19 on p. 102 of S&W.

Problem 10 (3 points)

Do Exercise 3.20 on p. 102 of S&W.

Problem 11 (0 points)

Do Exercise 3.22 on p. 102 of S&W. (Note that the answer is in the back of the book. Also, I'll point out that the clever ones among you should figure out that the mean can be easily obtained using the np formula for the mean of a binomial random variable (see the middle of p. 109). But it may be good to also get the mean by computing the weighted average of the possible outcomes, since this is the procedure that should be used for Problem 10 above.)

Due Thursday, September 29

Problem 12 (1 point)

Do part (c) of Exercise 3.15 on p. 95 of S&W.

Problem 13 (0 points)

Do Exercise 3.16 on p. 96 of S&W. (Note: For each of the problems to be turned in on Oct. 2, there is a similar problem in this set among the ones not to be turned in. Some of the answers to the problems not to be turned in (and I really don't want you to turn them in --- no need for me to sort through the extra pages) are in the back of the book, and I hope to post answers (or even more complete solutions) for the ones not to be turned in for which answers are not given. Because of all of this, I will only discuss with you the ones not to be turned in (since if you understand them, you should be able to do the others, and I don't want to give you hints on the others or wind up doing them for you).)

Problem 14 (2 points)

Do part (a) of Exercise 3.29 on p. 111 of S&W.

Problem 15 (0 points)

Do part (a) of Exercise 3.31 on p. 111 of S&W. (Note that the answer is in the back of the book.)

Problem 16 (3 points)

Do part (b) of Exercise 3.33 on p. 111 of S&W.

Problem 17 (0 points)

Do Exercise 3.43 on p. 117 of S&W.

Problem 18 (2 extra credit points)

Do part (b) of Exercise 3.44 on p. 117 of S&W, indicating all values of n for which the probability exceeds 0.95. (Note: Extra credit points contribute to the numerator, but not the denominator, when the HW component of your grade is determined. If, at the end of the semester, you've earned some extra credit points, then in a way they can make up for some of the points that you've missed throughout the semester.)

Problem 19 (0 points)

Do Exercise 4.4 on p. 131 of S&W. (Note that the answer is in the back of the book.)

Problem 20 (0 point)

Do part (a) of Exercise 4.7 on p. 132 of S&W. (Note: The desired value can be easily obtained from Table 4 on p. 677 of S&W.)

Problem 21 (1 points)

Do part (d) of Exercise 4.7 on p. 132 of S&W.

Problem 22 (2 points)

Do part (a) of Exercise 4.10 on p. 132 of S&W.

Due Thursday, October 6

Problem 23 (0 points)

Enter the 11 ewe milk yield values from Exercise 2.32 on p. 39 of S&W into SPSS and create a normal scores plot. (Note: If you click on Variable View at the bottom of the SPSS Data Editor, you can change the name of VAR00001 to yield.) You should get a plot that is fairly consistent with an assumption of approximate normality --- a close look might suggest mild negative skewness, but with a small sample size, the apparent mild deviation from approximate normality shouldn't be taken too seriously.

Problem 24 (0 points)

Read the 39 bull average daily gain values from Example 2.34 on p. 45 of S&W into SPSS, getting the data from the CD included with the book (the data set is called cattlewt). Assume that the sample is a random sample obtained from some distribution that we want to make inferences about (the distribution of average daily weight gains for the hypothetical population of all similar bulls given the same diet and living under the same conditions).

• (a) Give an estimate, based on the sample mean, of the distribution mean.
• (b) Give an estimate, based on the sample median, of the distribution median.
• (c) Give an estimate, based on the sample 75th percentile, of the distribution 75th percentile.
• (d) Give an estimate, based on the sample standard deviation, of the distribution standard deviation.
• (e) Give an estimate, based on the sample skewness, of the distribution skewness.
• (f) Give an estimate, based on the sample kurtosis, of the distribution kurtosis.
• (g) Create a normal scores plot. (You should get a plot that exhibits a clear pattern of positive skewness.)
• (h) Create a histogram.

Problem 25 (2 points)

Read the 36 serum CK concentrations from Example 2.6 on pp. 14-16 of S&W into SPSS, getting the data from the CD included with the book (the data set is called serum-CK). Assume that the sample is a random sample obtained from some distribution that we want to make inferences about.

• (a) Give an estimate, based on the sample mean, of the distribution mean.
• (b) Give an estimate, based on the sample median, of the distribution median.
• (c) Give an estimate, based on the sample 25th percentile, of the distribution 25th percentile.
• (d) Give an estimate, based on the sample standard deviation, of the distribution standard deviation.
• (e) Give an estimate, based on the sample skewness, of the distribution skewness.

Give the first three estimates by rounding to the nearest tenth, and give the last two estimates by rounding to the nearest hundredth. (This may be expressing more precision than is warranted.) It is not necessary to print out any computer output or show any work --- for this problem, just give me the five values.

Problem 26 (5 points)

For parts (a) through (e) below, match the description [A], [B], [C], [D]. or [E] (using each description exactly once --- thus it is your task to produce the best one-to-one matching of data sets and descriptions of underlying distributions), which best describes the distribution underlying the observed data. Here are the 5 descriptions:

• [A] the distribution has (clear) negative skewness;
• [B] the distribution has (clear) positive skewness;
• [C] the distribution has (not severe) light tails;
• [D] the distribution has (perhaps slightly) heavy tails;
• [E] the distribution is approximately normal (having no strong signs of appreciable deviations from normality).

Here are the 5 data sets (all available on the CD that came with S&W book):

• (a) the 15 pepper stem lengths from p. 64 (peppers data set from Ch. 2 section of S&W CD);
• (b) the 31 dog glucose measurements from p. 25 (glucose data set from Ch. 2 section of S&W CD --- but note that the 4 values greater than 100 don't read in correctly, and so you need to correct them in the Data Editor);
• (c) the 28 lamb birthweights from p. 181 (lamb-wt data set from Ch. 6 section of S&W CD);
• (d) the 14 radish growth measurements from p. 19 (darkness variable of radish data set from Ch. 2 section of S&W CD);
• (e) the 83 moisture content values (moisture data set from Ch. 4 section of S&W CD --- the data values don't seem to be printed in the book).

You don't have to turn in any work to support your answers (so you can save printer ink). (Note: The skewed distributions should be easy to identify. The others are a bit more subtle. But look for a consistent pattern suggesting the light-tailed or heavy-tailed shape. Don't let the straight line segments put on the plots by SPSS make you think that three of the plots are close to straight --- if you ignore the straight line segments and focus on the curvature of the pattern of the points, you ought to see shapes like the ones I drew on the board suggested by the plotted points. Again, look for a consistent pattern of curvature in the points --- the one for the (approximately) normal distribution isn't necessarily real straight, but the points jiggle about somewhat of a straight line pattern with no consistent trends in their departure from straightness.) (Note: For a lot of the problems to be turned in on Oct. 6, there is a similar problem in this set among the ones not to be turned in. Some of the answers to the problems not to be turned in (and I really don't want you to turn them in --- no need for me to sort through the extra pages) are in the back of the book, and I hope to post answers (or even more complete solutions) for the ones not to be turned in for which answers are not given. Because of all of this, I will only discuss with you the ones not to be turned in (since if you understand them, you should be able to do most of the others, and I don't want to give you hints on the others or wind up doing them for you). Since there isn't a problem similar to this one, I've prepared this web page giving example data sets for you.

Problem 27 (0 points)

Do Exercise 5.2 on p. 156 of S&W. (Note that the answers are in the back of the book.)

Problem 28 (2 points)

Do item (iii) of part (a) of Exercise 5.3 on p. 156 of S&W.

Problem 29 (0 points)

Do part (c) of Exercise 5.15 on p. 164 of S&W. (Note that the answer is in the back of the book.)

Problem 30 (0 points)

Do part (a) of Exercise 5.16 on p. 164 of S&W. (Note that the answer is in the back of the book.)

Problem 31 (3 points)

Do part (b) of Exercise 5.16 on p. 164 of S&W.

Problem 32 (0 points)

Do Exercise 6.1 on p. 184 of S&W. (Note that the answers are in the back of the book.)

Problem 33 (1 point)

Do part (a) of Exercise 6.2 on p. 184 of S&W.

Problem 34 (0 points)

Do Exercise 6.6 on p. 185 of S&W.

Problem 35 (0 points)

Do Exercise 6.7 on p. 185 of S&W.

Due Thursday, October 13

Problem 36 (0 points)

Do Exercise 6.10 on p. 194 of S&W. (Note that the answers are in the back of the book. Also, even though one could easily do this problem without using SPSS, I recommend that you do use SPSS in order to confirm that you can properly use the software to obtain a confidence interval for the distribution mean.)

Problem 37 (2 points)

Do part (a) of Exercise 6.11 on p. 194 of S&W.

Problem 38 (0 points)

Do Exercise 6.13 on p. 194 of S&W.

Problem 39 (0 points)

Do Exercise 6.14 on p. 194 of S&W.

Problem 40 (0 points)

Do Exercise 6.15 on pp. 194-195 of S&W.

Problem 41 (0 points)

Do Exercise 6.17 on p. 195 of S&W.

Problem 42 (0 points)

Do Exercise 6.19 on p. 196 of S&W.

Problem 43 (2 points)

Do Exercise 6.22 on p. 196 of S&W.

Problem 44 (2 extra credit points)

Consider Exercise 6.22 on p. 196 of S&W. Use the given values to form a 99% confidence interval for the mean, only take the sample size to be 34 (as opposed to 101). Round the confidence bounds to the nearest hundreth (even though such implied accuracy may not be warranted). (Hint: Use SPSS, as described here, to obtain the necessary critical value.)

Problem 45 (0 points)

Do Exercise 6.25 on p. 196 of S&W.

Problem 46 (3 points)

Consider the sample of 28 observations given in Table 6.2 on p. 181 of S&W (and available on the S&W CD).

• (a) Give a 90% confidence interval for the mean of the parent distribution of the data. (Note: By parent distribution, I just mean the distribution of birthweights of all single birth Rambouillet lambs born to parents living under basically the same conditions as those which led to the observed sample.) Round the confidence bounds using the guidelines described in class --- don't express more accuracy than is warranted.
• (b) Give a 95% confidence interval for the mean of the parent distribution of the data (and note that it is wider).
• (c) Give a 99% confidence interval for the mean of the parent distribution of the data (and note that it is the widest of the three).

Problem 47 (0 points)

Do part (a) of Exercise 6.40 on p. 212 of S&W. (Note that the answer is in the back of the book.)

Problem 48 (3 points)

Do part (a) of Exercise 6.41 on p. 212 of S&W.

Problem 49 (0 points)

Do Exercise 7.1 on p. 225 of S&W. (Note that the answer is in the back of the book. Also, the request should be for the estimated standard error of the estimator (as opposed to the estimate --- so it should be Y instead of y in the sample means.)

Problem 50 (2 points)

Do Exercise 7.2 on p. 225 of S&W. (The request should be for the estimated standard error of the estimator (as opposed to the estimate --- so it should be Y instead of y in the sample means.)

Due Thursday, October 20

(Note: For most of the problems to be turned in on Oct. 20, there is a similar problem in this set among the ones not to be turned in. Some of the answers to the problems not to be turned in (and I really don't want you to turn them in --- no need for me to sort through the extra pages) are in the back of the book, and I hope to post answers (or even more complete solutions) for the ones not to be turned in for which answers are not given. Because of all of this, I will only discuss with you the ones not to be turned in (since if you understand them, you should be able to do the others, and I don't want to give you hints on the others or wind up doing them for you).)

Problem 51 (6 points)

For parts (a) and (b), use Welch's method. (Note: I recommend that you use SPSS for this problem. Also, note that the data can be read in from the CD that came with S&W.)

• (a) Do Exercise 7.19 on pp. 233-234 of S&W. (Be sure to note that a 90% confidence interval is requested, as opposed to a 95% interval.)
• (b) Give the p-value which results from a test to determine whether or not there is statistically significant evidence to support the claim that caffeine has an effect on heart rate. (Note: Since the previous statement doesn't indicate that the test should assess whether caffeine increases heart rate, a two-tailed test is called for. (Perhaps a one-tailed test would be preferred by some, but the rationale behind a two-tailed test might be that if many substances are considered, some might raise heart rate, some might lower it, and some may not affect it, and in a study to determine which substances affect heart rate, two-tailed tests are desirable because one generally wants to detect changes in either direction.))
• (c) Comment on the validity of the method used in parts (a) and (b).

Problem 52 (0 points)

For parts (a) and (b), use Welch's method. (Note: I recommend that you use SPSS for this problem. Also, note that the data can be read in from the CD that came with S&W.)

• (a) Do Exercise 7.21 on p. 234 of S&W.
• (b) Give the p-value which results from a test to determine whether or not there is statistically significant evidence to support the claim that the color of light affects the mean size of two week old plants.
• (c) Comment on the validity of the method used in parts (a) and (b).

Problem 53 (0 points)

Do part (a) of Exercise 7.29 on pp. 244-245 of S&W. (Note that the answer is in the back of the book.)

Problem 54 (2 points)

Do part (a) of Exercise 7.30 on p. 245 of S&W. Give the value of the test statistic, and state whether or not you can reject if testing to determine if there is statistically significant evidence to support the claim that the two distribution means differ.

Problem 55 (2 points)

Use the summary statistics of Exercise 7.30 on p. 245 of S&W to make a statement about the p-value which results from a test to determine if there is statistically significant evidence to support the claim that the mean of the male distribution is greater than the mean of the female distribution. Be as specific as possible using Table 4 on p. 677.

Problem 56 (2 extra credit points)

Use the summary statistics of Exercise 7.30 on p. 245 of S&W to make a statement about the p-value which results from a test to determine if there is statistically significant evidence to support the claim that the mean of the female distribution is greater than the mean of the male distribution. But instead of using a table, as described above, use SPSS to get a more precise p-value (rounded to 2 significant digits). (Hint:Use CDF.T.)

Problem 57 (0 points)

Do part (c) of Exercise 7.30 on p. 245 of S&W.

Problem 58 (0 points)

Do Exercise 7.35 on p. 246 of S&W. (Assume that the test done was a two-tailed test.)

Problem 59 (3 points)

Do Exercise 7.36 on p. 246 of S&W. (Assume that the test done was a two-tailed test.)

Problem 60 (3 points)

Do Exercise 7.56 on p. 266 of S&W, except instead of stating whether or not there is statistically significant evidence to support the theory at the 0.1 level, report an appropriate p-value instead. (Note that the data is on the CD (under the name settler) that comes with S&W. After I got the data into SPSS, I clicked on Variable View on the Data Editor to change the Width to 6 and the Decimals to 3 for the density variable.)

Due Thursday, October 27

Problem 61 (0 points)

Consider the data of Exercise 7.21 on p. 234 of S&W. Test the null hypothesis that the choice of light color doesn't affect plant growth against the general alternative.

• (a) Report the (approximate) p-value which results from using the Mann-Whitney test.
• (b) Report the p-value which results from using Student's two-sample t test. (One should find that the p-value for part (a) is considerably smaller than the p-value for part (b), although neither is highly statistically significant. Typically skewness hurts the power of Student's t test, even though a cancellation effect leads to (approximate) validity.)

Problem 62 (0 points)

Consider the data of Exercise 7.31 on p. 245 of S&W. Test the null hypothesis that the thymus weight distribution for the 15th day is the same as the thymus weight distribution for the 14th day against the general alternative that the distributions differ.

• (a) Report the (exact) p-value which results from using the Mann-Whitney test.
• (b) Report the p-value which results from using Student's two-sample t test.

Problem 63 (4 points (3 for (a), 1 for (b))

Consider the data of Exercise 7.51 on pp. 264-265 of S&W. Test the null hypothesis that selection for the hypnosis group does not affect total ventilation against the general alternative.

• (a) Report the (exact) p-value which results from using the Mann-Whitney test.
• (b) Report the p-value which results from using Student's two-sample t test.

Problem 64 (3 points (2 for (a), 1 for (b))

Consider the data of Exercise 7.56 on pp. 266 of S&W. Test the null hypothesis that the two density distributions are the same against the general alternative that the distributions differ.

• (a) Report the (exact) p-value which results from using the Mann-Whitney test.
• (b) Report the p-value which results from using Student's two-sample t test. (One should find that the p-value for part (b) is just a tad smaller than the p-value for part (a), despite the fact that skewness typically hurts the performance of Student's t test. (In this case, the skewed distributions didn't ptoduce a lot of extreme outliers. It's the extreme outliers that really hurt the power of Student's t test.))

Problem 65 (1 extra credit point)

Construct an empirical Q-Q plot using the data from Exercise 7.56 on p. 266 of S&W. There doesn't seem to be a real slick way to do this using SPSS. However, if you get the two ordered samples into two separate columns on the SPSS Data Editor, you can use Graphs > Scatter. The Simple scatter plot is the one preselected by SPSS's defaults, and so just click Define to specify the details of the plot. Click the variable name corresponding to the ordered 800 m values into the Y Axis box, and click the variable name corresponding to the ordered 250 m values into the X Axis box. Finally, click OK. (The part I haven't found a good way to do using SPSS is creating the two ordered samples.)

Problem 66 (0 points)

Consider the data of Exercise 7.79 on p. 296 of S&W. Test the null hypothesis that toluene does not affect rat brain dopamine concentration against the general alternative.

• (a) Report the (exact) p-value which results from using the Mann-Whitney test.
• (b) Report the p-value which results from using Student's two-sample t test.

Problem 67 (0 points)

Do Exercise 8.1 on p. 316 of S&W.

Problem 68 (0 points)

Do Exercise 8.4 on p. 316 of S&W.

Problem 69 (0 points)

Do Exercise 8.9 on p. 325 of S&W.

Problem 70 (0 points)

Do Exercise 8.19 on p. 333 of S&W.

Problem 71 (0 points)

Do Exercise 8.20 on p. 333 of S&W.

Problem 72 (0 points)

Do Exercise 8.21 on p. 333 of S&W. (Note that the answer is in the back of the book.)

Problem 73 (0 points)

Do Exercise 8.24 on p. 334 of S&W.

Problem 74 (0 points)

Do Exercise 8.25 on p. 338 of S&W.

Problem 75 (0 points)

Do Exercise 8.35 on p. 344 of S&W.

Due Thursday, November 3

Problem 76 (0 points)

Consider the data from Exercise 2.9 on p. 25 of S&W. (If you bring the data into SPSS from the S&W CD, make sure that the values are read in correctly. (I think that for this data set there shouldn't be a problem.)) This problem deals with doing tests about the mean of the distribution underlying the data (which is the distribution of the hypothetical population of all 2 year old Holsteins maintained under the same conditions were those corresponding to the sample).

• (a) Use a t test to do a one-sided test about the distribution mean. Specifically, report the p-value that results from the one-sided test which is appropriate if one wants to determine if there is statistically significant evidence that the distribution mean exceeds 450.
• (b) Use a t test to do a one-sided test about the distribution mean. Specifically, report the p-value that results from the one-sided test which is appropriate if one wants to determine if there is statistically significant evidence that the distribution mean exceeds 500.
• (c) Comment on the quality of the result from part (a). Can the test be considered to be valid, or is the result suspect? Are there concerns about possible conservativeness?

Problem 77 (0 points)

Consider the data from Exercise 2.10 on p. 25 of S&W. (If you bring the data into SPSS from the S&W CD, make sure that the values are read in correctly. (I think that for this data set some of the values are incorrect, and you should correct them prior to doing the tests which are requested below.)) This problem deals with doing tests about the mean of the distribution underlying the data.

• (a) Use a t test to do a one-sided test about the distribution mean. Specifically, report the p-value that results from the one-sided test which is appropriate if one wants to determine if there is statistically significant evidence that the distribution mean is less than 90.
• (b) Use a t test to do a one-sided test about the distribution mean. Specifically, report the p-value that results from the one-sided test which is appropriate if one wants to determine if there is statistically significant evidence that the distribution mean is less than 95.
• (c) Comment on the quality of the result from part (b). Can the test be considered to be valid, or is the result suspect? Are there concerns about possible conservativeness?

Problem 78 (6 points)

Consider the data from Example 6.3 on p. 181 of S&W. (If you bring the data into SPSS from the S&W CD, make sure that the values are read in correctly. (I think that for this data set there shouldn't be a problem.)) This problem deals with doing tests about the mean of the distribution underlying the data.

• (a) Use a t test to do a one-sided test about the distribution mean. Specifically, report the p-value that results from the one-sided test which is appropriate if one wants to determine if there is statistically significant evidence that the distribution mean is greater than 5.0.
• (b) Use a t test to do a one-sided test about the distribution mean. Specifically, report the p-value that results from the one-sided test which is appropriate if one wants to determine if there is statistically significant evidence that the distribution mean is less than 5.0.
• (c) Comment on the quality of the result from part (a). Can the test be considered to be valid, or is the result suspect? Are there concerns about possible conservativeness?

Problem 79 (4 points)

Consider the data from Exercise 9.2 on p. 356 of S&W. (If you bring the data into SPSS from the S&W CD, make sure that the values are read in correctly.) This problem deals with doing a test about the mean of the distribution underlying the observed differences.

• (a) Report the p-value which results from doing a t test of the null hypothesis that the distribution mean is 0 against the alternative that the mean is not equal to 0.
• (b) Comment on the quality of the result from part (a). Can the test be considered to be valid, or is the result suspect? Are there concerns about possible conservativeness?

Problem 80 (0 points)

Consider the data from Exercise 9.3 on p. 356 of S&W. (If you bring the data into SPSS from the S&W CD, make sure that the values are read in correctly.) This problem deals with doing a test about the mean of the distribution underlying the observed differences.

• (a) Report the p-value which results from doing a t test of the null hypothesis that the distribution mean is 0 against the alternative that the mean is not equal to 0.
• (b) Comment on the quality of the result from part (a). Can the test be considered to be valid, or is the result suspect? Are there concerns about possible conservativeness?

Problem 81 (1 extra credit point)

Do Exercise 9.9 on p. 358 of S&W. (Note: Extra credit points contribute to the numerator, but not the denominator, when the HW component of your grade is determined. If, at the end of the semester, you've earned some extra credit points, then in a way they can make up for some of the points that you've missed throughout the semester.)

Problem 82 (0 points)

• (a) Do part (a) of Exercise 9.1 on p. 355 of S&W.
• (b) Using the same data, give the p-value which results from using Student's t test to determine whether or not there is statistically significant evidence that the two varieties of wheat differ with respect to yield.
• (c) Comment on the validity of the p-value produced in part (b).
• (d) Using the same data, give the p-value which results from (incorrectly) using Student's two-sample t test to determine whether or not there is statistically significant evidence that the two varieties of wheat differ with respect to yield.
• (e) Using the same data, give the p-value which results from using the signed-rank test to determine whether or not there is statistically significant evidence that the two varieties of wheat differ with respect to yield.

Problem 83 (10 points)

Consider the data of Exercise 9.4 on pp. 356-357 of S&W.

• (a) Give a point estimate for the median change in shrinkage temperature due to the electrical stimulation.
• (b) Give a point estimate for the mean change in shrinkage temperature due to the electrical stimulation.
• (c) Give a point estimate for the standard error associated with the mean change in shrinkage temperature due to the electrical stimulation.
• (d) Give a 99% confidence interval for the mean change in shrinkage temperature due to the electrical stimulation.
• (e) Comment on the validity of the interval produced in part (d).
• (f) Give the p-value which results from using Student's t test to determine whether or not there is statistically significant evidence that electrical stimulation affects the collagen shrinkage temperature.
• (g) Comment on the validity of the p-value produced in part (f).
• (h) Give the p-value which results from (incorrectly) using Student's two-sample t test to determine whether or not there is statistically significant evidence that electrical stimulation affects the collagen shrinkage temperature.
• (i) Give the p-value which results from using the signed-rank test to determine whether or not there is statistically significant evidence that electrical stimulation affects the collagen shrinkage temperature.
• (j) Comment on the validity of the p-value produced in part (i).

Problem 84 (2 extra credit points)

Do Exercise 9.36 on p. 384 of S&W. (Answer Yes or No, and provide an explanation for your answer (not being too brief, nor too rambling). To get full credit, you should use some sort of statistical measure/procedure to support your written explanation.) (Note: Extra credit points contribute to the numerator, but not the denominator, when the HW component of your grade is determined. If, at the end of the semester, you've earned some extra credit points, then in a way they can make up for some of the points that you've missed throughout the semester.)

Problem 85 (6 points)

Consider the data of Exercise 9.49 on pp. 386-387 of S&W. Test the null hypothesis that caffeine has no effect on RER against the general alternative and report the resulting p-value, using each of the tests indicated below. Due to the limitations of the software and tables that we're working with, for this problem it's okay to report the p-values using only one accurate significant digit. (Take this to be a general policy for the rest of the semester --- if there is no way to get two accurate significant digits for a p-value using the methods that I've covered in class, it's fine to report a p-value using just one (hopefully accurate) significant digit.)

• (a) Student's t test
• (b) Wilcoxon signed-rank test
• (c) sign test

Problem 86 (1 extra credit point)

Find a way to get two accurate significant digits for the p-values for parts (b) and (c) of the immediately preceding problem. (For the sign test, you can make use of the fact that the null sampling distribution of the test statistic is a binomial distribution.)

Problem 87 (6 points)

Consider the data of Exercise 9.52 on p. 387 of S&W. Test the null hypothesis that nerve regeneration does not affect CP level against the general alternative and report the resulting p-value, using each of the tests indicated below.

• (a) Student's t test
• (b) Wilcoxon signed-rank test
• (c) sign test

Problem 88 (0 points)

Consider the data of Exercise 9.53 on pp. 387-388 of S&W. Test the null hypothesis that benzamil does not affect healing against the general alternative and report the resulting p-value using the sign test.

Problem 89 (0 points)

Consider the data for Exercise 2.20 on p. 31 of S&W. Do a test to determine if there is statistically significant evidence that the median of the parent distribution is less than 3.50, and report the p-value.

Problem 90 (3 points)

Consider the data for Example 2.34 on p. 45 of S&W. Do a test to determine if there is statistically significant evidence that the median of the parent distribution exceeds 1.30, and report the p-value. (Note: I had thought that SPSS always gave an exact p-value for the the sign test, but upon doing this problem, I now know that if the sample size is deemed to be too large, SPSS uses a normal approximation. (It's incredible that 39 is too large for an exact p-value --- I don't know of any other statistical software package that does the sign test and uses an approximation if n is 39.))

Due Thursday, November 10

Problem 91 (2 points)

Consider the data of Example 11.1 on pp. 463-464 of S&W. Use the Kruskal-Wallis test to test the null hypothesis of no difference between treatments against the general alternative and report the resulting p-value. (Note: You'll find a description of how to get SPSS to do the Kruskal-Wallis test on my web page for Chapter 11 of S&W. There, you will also see the p-value obtained from applying the K-W test to another data set in S&W, and you can try to duplicate that result in order to check that you're doing it correctly.)

Problem 92 (2 extra credit points)

Do parts (a) and (b) of Exercise 11.7 on p. 476 of S&W.

Due Thursday, November 17

Note: If SPSS reports 0.000 as a p-value, write down p-value < 0.0005 in response to my request for a p-value. (Do this for the rest of the semester, with the exception of cases for which a more precise p-value can be obtained from a table which I have given you.)

Problem 93 (0 points)

Use the anorexia treatment data that was given with the midterm exam to do the following items. (Note: Here is a link to a listing of the 72 values of the response variable, which is the weight change. The first 29 values are the cognitive behavioral sample, the next 26 values are the control sample, and the last 17 values are the family therapy sample. Perhaps you can paste these into SPSS. Then you can create a column having 29 1s, followed by 26 2s, followed by 17 3s, to indicate the three groups. To use SPSS to generate output helpful for parts (a), (b), (c), (f), (g), and (h), use Analyze > Compare Means > One-Way ANOVA. Click the response variable into the Dependent List box and click the variable giving the group indicators into the Factor box. Next, click Post Hoc and click to check the boxes for Tukey (notTukey's-b) and Dunnett's T3, and change the Significance Level from 0.05 to 0.1 (because 90% simultaneous confidence intervals are requested for this problem). Click Continue. Then, click Options and click to check the box for Welch. Click Continue, and then finally click OK.)

• (a) Give the p-value which results from a one-way ANOVA F test of the null hypothesis of equal means against the general alternative.
• (b) Give the p-value which results from a Tukey studentized range test of the null hypothesis of equal means against the general alternative. (The easy way to get this using SPSS is to generate Tukey studentized range confidence intervals (aka Tukey HSD intervals), and report the smallest value found in the column labeled sig.)
• (c) Generate 90% Tukey studentized range (aka Tukey HSD) confidence intervals and use them to determine which pairs of means are significantly different. Identify all pairs for which one mean can be determined to be larger than another mean, stating which mean is determined to be the larger one for each pair.
• (d) Produce a probit plot of the pooled residuals in order to check on the approximate normality assumption associated with parts (a) through (c). (Note: Unfortunately, it seems like the residuals cannot be produced using Analyze > Compare Means > One-Way ANOVA. However, you can use Analyze > General Linear Model > Univariate. In the initial dialog box, click the column with the 72 weight change values into the Dependent Variable box, and the column with the group indicators in it into the Fixed Factor(s) box.) Under Save, elect to save Unstandardized Predicted Values and Unstandardized Residuals. Then click Continue, followed by OK. The desired residuals will appear in a new column (of length 72) in the Data Editor. Then one just needs to use Graphs > Q-Q to generate the desired probit plot.) Also, comment on how well the assumption of approximate normality is met, stating how you would characterize the error term distribution if approximately normal is not a good description. (Note: For this part, give an answer based on the assumption that the error term distribution is the same for all three treatments, even though an examination of individual probit plots suggests that this may not be a good assumption.)
• (e) Plot the pooled residuals against the predicted values in order to check on the assumption of homoscedasticity associated with parts (a) through (c). (Note: Unfortunately, it seems like the residuals and predicted values (which are just the sample means) cannot be produced using Analyze > Compare Means > One-Way ANOVA. However, you can use Analyze > General Linear Model > Univariate. In the initial dialog box, click the column with the 72 weight change values into the Dependent Variable box, and the column with the group indicators in it into the Fixed Factor(s) box.) Under Save, elect to save Unstandardized Predicted Values and Unstandardized Residuals. Then click Continue, followed by OK. The desired residuals and predicted values will appear in two new columns (of length 72) in the Data Editor. Then one just needs to use Graphs > Scatter to generate the desired plot. (Select Simple, and then click Define. Put the residuals on the Y Axis, and the predicted values on the X Axis. Finally, click OK to create the plot.))
• (f) Give the p-value which results from a Welch test of the null hypothesis of equal means against the general alternative.
• (g) Give the p-value which results from using Dunnett's T3 procedure to do a test of the null hypothesis of equal means against the general alternative. (The easy way to get this using SPSS is to generate Dunnett T3 confidence intervals, and report the smallest value found in the column labeled sig.)
• (h) Generate 90% Dunnett T3 (simultaneous) confidence intervals and use them to determine which pairs of means are significantly different. Identify all pairs for which one mean can be determined to be larger than another mean, stating which mean is determined to be the larger one for each pair.
• (i) Give the p-value which results from a Kruskal-Wallis test of the null hypothesis of identical distributions against the general alternative.

Problem 94 (27 points (3 points for each part))

Use the vehicle head injury data that will be distributed in class on Nov. 3 to do the following items. (Note: Here is a link to a listing of the 70 values of the response variable. The first 10 values are the Compact sample, the next 10 values are the Heavy sample, ..., and the last 10 values are the Van sample. Perhaps you can paste these into SPSS. Then you can create a column having 10 1s, followed by 10 2s, ..., followed by 10 7s, to indicate the seven groups.)

• (a) Give the p-value which results from a one-way ANOVA F test of the null hypothesis of equal means against the general alternative.
• (b) Give the p-value which results from a Tukey studentized range test of the null hypothesis of equal means against the general alternative. (The easy way to get this using SPSS is to generate Tukey studentized range confidence intervals (aka Tukey HSD intervals), and report the smallest value found in the column labeled sig.)
• (c) Generate 95% Tukey studentized range (aka Tukey HSD) confidence intervals and use them to determine which pairs of means are significantly different. (You should find that 2 of the 21 possible pairs provide statistically significant differences.) Identify all pairs for which one mean can be determined to be larger than another mean, stating which mean is determined to be the larger one for each pair. (Note: On the answer sheet you don't have to give all of the intervals. Just indicate which means are different. For example, if there is strong evidence that the mean of the 3rd class (Light) is greater than the mean of the 4th class (Medium), put μ₃ > μ₄.)
• (d) Produce a probit plot of the pooled residuals (be sure to turn in the plot) in order to check on the approximate normality assumption associated with parts (a) through (c). (Note: Unfortunately, it seems like the residuals cannot be produced using Analyze > Compare Means > One-Way ANOVA. However, you can use Analyze > General Linear Model > Univariate. In the initial dialog box, click the column with the 70 head injury readings into the Dependent Variable box, and the column with the group indicators in it into the Fixed Factor(s) box.) Under Save, elect to save Unstandardized Predicted Values and Unstandardized Residuals. Then click Continue, followed by OK. The desired residuals will appear in a new column (of length 70) in the Data Editor. Then one just needs to use Graphs > Q-Q to generate the desired probit plot.) Also, comment on how well the assumption of approximate normality is met, stating how you would characterize the error term distribution if approximately normal is not a good description.
• (e) Plot the pooled residuals against the predicted values (be sure to turn in the plot) in order to check on the assumption of homoscedasticity associated with parts (a) through (c). (Note: Unfortunately, it seems like the residuals and predicted values (which are just the sample means) cannot be produced using Analyze > Compare Means > One-Way ANOVA. However, you can use Analyze > General Linear Model > Univariate. In the initial dialog box, click the column with the 70 head injury readings into the Dependent Variable box, and the column with the group indicators in it into the Fixed Factor(s) box.) Under Save, elect to save Unstandardized Predicted Values and Unstandardized Residuals. Then click Continue, followed by OK. (Note: Before clicking OK, I suggest that you click to check the Spread vs. level plot box under Options. This is a convenient way to examine the collection of the sample standard deviations.) The desired residuals and predicted values will appear in two new columns (of length 70) in the Data Editor. Then one just needs to use Graphs > Scatter to generate the desired plot. (Select Simple, and then click Define. Put the residuals on the Y Axis, and the predicted values on the X Axis. Finally, click OK to create the plot.)) You should not see any strong pattern indicating that variablity varies systematically with the mean response, but at the same time it can be noted that there may be heteroscedasticity (but not occuring in the typical pattern of increasing variance with increasing mean). Although formal tests for heteroscedasticity do not produce statistically significant evidence of heteroscedasticity (Levene's test results in a p-value of about 0.70), it may be interesting to see if allowing for unequal variances leads to different conclusions than those suggested by the results of parts (a) through (c).)
• (f) Give the p-value which results from a Welch test of the null hypothesis of equal means against the general alternative.
• (g) Give the p-value which results from using Dunnett's T3 procedure to do a test of the null hypothesis of equal means against the general alternative. (The easy way to get this using SPSS is to generate Dunnett T3 confidence intervals, and report the smallest value found in the column labeled sig.)
• (h) Generate 95% Dunnett T3 (simultaneous) confidence intervals and use them to determine which pairs of means are significantly different. (You should find that 1 or 2 of the 21 possible pairs provide statistically significant differences.) Identify all pairs for which one mean can be determined to be larger than another mean, stating which mean is determined to be the larger one for each pair. (Note: Although all of the p-values provide strong evidence of some differences, when trying to determine the nature of the differences, parts (c) and (h) do not give results in perfect agreement.)
• (i) Give the p-value which results from a Kruskal-Wallis test of the null hypothesis of identical distributions against the general alternative.

Problem 95 (0 points)

Consider the data of Exercise 11.57 on pp. 522-523 of S&W. (The data can be read into SPSS, in a convenient form, from the S&W CD --- it's the alkaline data set from Ch. 11.)

• (a) Give the p-value which results from an ANOVA F test to determine if there is statistically significant evidence that dose effects alkaline phosphatase level in dogs. (To produce the results needed to obtain the p-values requested in parts (a), (b), and (c), you can use Analyze > General Linear Model > Univariate. Click level into the Dependent Variable box, and click both sex and dose into the Fixed Factor(s) box. Upon clicking OK, you should get the ANOVA table from which all three p-values can be obtained. (Just read them from the Sig. column.))
• (b) Give the p-value which results from an ANOVA F test to determine if there is statistically significant evidence that the mean alkaline phosphatase level in dogs depends on sex.
• (c) Give the p-value which results from an ANOVA F test to determine if there is statistically significant evidence for rejecting the null hypothesis of an additive model in favor of the alternative that not all of the interaction terms are 0.

Problem 96 (7.5 points)

Consider the data of Exercise 11.17 on p. 497 of S&W.

• (a) Give the p-value which results from an ANOVA F test to determine if there is statistically significant evidence that flooding effects ATP concentration in birch trees.
• (b) Give the p-value which results from an ANOVA F test to determine if there is statistically significant evidence that the mean ATP cencentration is not the same for the two species of birch trees.
• (c) Give the p-value which results from an ANOVA F test to determine if there is statistically significant evidence for rejecting the null hypothesis of an additive model in favor of the alternative that not all of the interaction terms are 0.

Problem 97 (0 points)

Use the CPK data which will be distributed in class on Nov. 3 to do the following items. (Note: Here is a link to a listing of the 56 values of the response variable. After pasting in these response values, and creating two more variables, I recommend that you go to Variable View and give the names CPK, period, and subject to the variables, since the output may be easier to digest if the variables are all given names.)

• (a) Give the p-value which results from an ANOVA F test of the null hypothesis of no treatment effects against the general alternative. (Note: Since there is only one observation per cell, one has to assume an additive model. Also, in this case the blocking variable (subject) is a random effects variable (although when there is only one observation per cell, the same results are obtained whether the blocking variable is fixed or random). To get started, use Analyze > General Linear Model > Univariate. Click CPK in as the Dependent Variable, period as a Fixed Factor, and subject as a Random Factor. Click Model, and then click the Custom button. Highlight period in the Factors and Covariates box, and click the right arrow underneath Build Terms to put it into the Model box. Do the same thing to put subject into the Model box. (Do not highlight period and subject together --- the two factors have to be put into the Model box one at a time.) Then, underneath the arrow, use the scroll-down menu to select Main effects, and then click Continue. To set yourself up for some of the parts below, click Post Hoc, highlight period and click it into the Post Hoc Tests for box, and click to check Tukey and Dunnett's T3 before clicking Continue. Under Save, click to save the unstandardized residuals and predicted values. Finally click OK to produce the output needed for parts (a) through (d). (Note: If I don't specify that you change some aspect about SPSS, then stay with the default settings. That is, sometimes you may encounter that a button has already been checked. If so, leave it that way --- don't click it to uncheck it. You should also be aware that if you run the same procedure more than once with the same data set in a given SPSS session, then the settings that you used the last time will still be in effect. This is usually what you want, for the most part, but there might be some things that you will want to alter before running the procedure again.))
• (b) Give the p-value which results from a Tukey studentized range test of the null hypothesis of no treatment effects against the general alternative. (The easy way to get this using SPSS is to generate Tukey studentized range confidence intervals (aka Tukey HSD intervals), and report the smallest value found in the column labeled sig.)
• (c) Give the p-value which results from a Dunnett T3 test of the null hypothesis of no treatment effects against the general alternative. (The easy way to get this using SPSS is to generate Dunnett T3 confidence intervals, and report the smallest value found in the column labeled sig.) It can be noted that this p-value is very much different from the p-values obtained using the procedures based on an assumption of homoscedasticity, which casts doubt on the truth of that assumption, and makes the smaller p-values obtained quite suspect. With the Dunnett T3 intervals, none of the treatments are found to be different!
• (d) Plot the pooled residuals against the predicted values (be sure to turn in the plot) in order to check on the assumption of homoscedasticity associated with parts (a) and (b). You should see a funnel-like shape, suggesting severe heteroscedasticity. If you look at a probit plot of the pooled residuals, it should appear to be rather odd, and not at all close to a straight line pattern. Due to the severe heteroscedasticity, the residuals cannot be viewed as iid observations from any distribution --- so the plot is hard to interpret ... the odd pattern is largely due to the heteroscedasticity, and we cannot use the plot to check the assumption of approximate normality. At this point, we have no good evidence of treatment effects, since Dunnett's T3 procedure should be considered to be the most reliable of the tests considered so far.
• (e) Transform the response variable (CPK activity) by taking logs to the base 10. (This can be done in SPSS using Transform > Compute. Type log10CPK into the Target Variable box, enter information into the Numeric Expression box to create the desired trasformed variable, and click OK.) Examine a plot of the residuals against the predicted values for this fitted model, and if there is still evidence of heteroscedasticity, try a stronger transformation, using Transform > Compute to make -1/SQRT(CPK) a new variable which can be used as the response. (-1/SQRT(CPK) is what goes in the Numeric Expression box. You could name this new variable invsrCPK.) More severe choices are -1/CPK and -1/( CPK**2 ). (Note: Once we go past log to negative powers (inverse square root, inverse, and inverse square), we should multiply by -1 to prevent the response values from being reversed ... e.g., if 1/CPK is used, the largest CPK value becomes the smallest transformed value. But if -1/CPK is used, the largest CPK value yields the largest transformed variable. Doing it this way, you'll know that you've gone to too powerful a transformation when the funnel-like pattern reverses direction, with the variability generally getting smaller as the response values get larger. (In addition to having a funnel-like pattern when a transformation is needed to tame heteroscedasticity, it's often the case that the pointy part of the funnel is not at 0, and this indicates that the additive model doesn't fit well in addition to it needing a nonconstant variance term. If we can believe that the additive model fits decently, and could check on the distributions of the error terms, then there would be less need to transform to reduce heteroscedasticity, since in theory Dunnett's T3 procedure can be used. But with only one observation per cell, several things can work against having good accuracy with Dunnett's T3 procedure, and it is often better to work with a transformed response variable, since it may result in a better fitting model and greater overall accuracy.) Choose the best transformation from among log (base 10), negative inverse square root, negative inverse, and negative inverse square, and then repeat part (a) using the transformed response variable.
• (f) Repeat part (b) using the transformed response variable.
• (g) Repeat part (c) using the transformed response variable.
• (h) Produce a probit plot of the pooled residuals (be sure to turn in the plot) in order to check on the approximate normality assumption associated with parts (e) through (g). You should see a pattern indicating heavy tails and perhaps slight negative skewness, and the estimated skewness and kurtosis (obtained using Analyze > Explore) of about -0.3 and 3.0 are in agreement with this assessment. All in all, the test results should be a bit conservative, which doesn't bother us since none of the p-values are boarderline --- all of the significance values associated with the ANOVA F tests, Tukey's procedure, and Dunnett's T3 procedure are either less than 0.01 or greater than 0.1. All three procedures give consistent results, with both sets of simultaneous confidence intervals suggesting that the 4th treatment (period) differs from the other three, and that there are no statistically significant differences between the other three treatments (periods). While it's a bit awkward reporting results based on a transformed response, overall it seems best to transform with this data --- the raw data exhibits severe heteroscedasticity, and combined with possible severe nonnormality (something that can't be adequately checked, given the hetroscedasticity) and indications that the additive model for the nontransformed data doesn't fit well, it may well be that Dunnett's T3 procedure applied to the untransformed data didn't perform accurately. Using the transformed data, we can conclude that exercise does not seem to strongly affect transformed CPK activity (since the control period and the two postexercise periods are the group of three that don't exhibit significant differences), but that psychosis does affect transformed CPK activity (since the psychosis period is the 4th period --- the one that is significantly different from the others). If psychosis doesn't affect CPK activity, it wouldn't affect transformed CPK activity, and so the presence on an effect on transformed CPK can be taken to mean that CPK activity is affected by psychosis, even though due to the transformation, we shouldn't say anything about the mean CPK activity (but we can model the transformed CPK activity and make statements about the mean transformed CPK activity).

Problem 98 (16 points (no points for parts (a)-(d), and 4 points each for parts (e)-(h))

Use the survival time data, described here, to do the following items. (Note: Here is a link to a listing of the 48 values of the response variable. After pasting in these response values, and creating two more variables, I recommend that you go to Variable View and give the names time, poison, and treatment to the variables, since the output may be easier to digest if the variables are all given names.) You can do this problem similarly to the way I suggest that Problem 97 be done, only in this case both poison and treatment should be treated as fixed effects factors (there are no random factors), and also there is no need to go to Model and build a Custom model, since the default selection of a full factorial model (which includes interactions) is fine. (So, one can start out like in Problem 95, only since the data is not on the S&W CD you'll have to create the three variables yourself. (Do this carefully. As a check, I'll give you that the p-value for part(c) should be about 0.11. If you get something else, then I suggest that you check to make sure that you coded the two factor variables, poison and treatment, correctly.))

• (a) Give the p-value which results from an ANOVA F test of the null hypothesis that the type of poison has no effect on survival time against the general alternative.
• (b) Give the p-value which results from an ANOVA F test of the null hypothesis that the type of treatment has no effect on survival time against the general alternative.
• (c) Give the p-value which results from an ANOVA F test of the null hypothesis that there are no interactions between poison and treatment against the general alternative that not all of the interaction terms are equal to 0.
• (d) Examine a plot the pooled residuals against the predicted values and note that there is a classic funnel pattern. (If you look at a probit plot of the pooled residuals, you should see a strong heavy tail pattern. Of course, due to the severe heteroscedasticity, the residuals should not be viewed as iid observations from any distribution, and so the plot is hard to interpret ... the apparent heavy tailedness could be due to the heteroscedasticity.)
• (e) Considering all of the transformations described in Problem 97 (and no other transformations), transform the response variable (time), using the transformation that results in the best compatibility with an assumption of homoscedasticity. (Do this carefully --- if you carefully consider the appropriate plots, it should be fairly easy to pick the best transformation.) Repeat part (a) using the transformed response variable.
• (f) Repeat part (b) using the transformed response variable.
• (g) Repeat part (c) using the transformed response variable.
• (h) Since there is not strong evidence to indicate that an additive model is not sufficient, fit an additive model (i.e., use only main effects --- no interactions) using the same transformed response variable used in parts (e), (f), and (g), and examine the pooled residuals, with a probit plot and also plotting them against the predicted values. You should note that removing the possibility of interactions did not create appreciably greater heteroscedasticity, and it can also be seen that the probit plot of the residuals looks nicer than one based on the full model (which includes the interaction terms). Using the resisuals from the fitted additive model, give estimates of the skewness and kurtotsis of the error term distribution.

Problem 99 (3.5 points)

Using the whiteness data which will be distributed in class on Nov. 3, use Friedman's test to test the null hypothesis of no differences between detergents against the general alternative, and report the p-value.

Problem 100 (0 points)

Using the whiteness data which will be distributed in class on Nov. 3, use Friedman's test to test the null hypothesis of no differences between washing machines against the general alternative, and report the p-value. (This can be done is SPSS by entering the data for each washing machine into a separate column of the Data Editor (making sure that the values are ordered the same way in each column with respect to the detergents (e.g., by typing in the values in the exact arrangement as they are on the handout, all of the Detergent A values are in the 1st row, all of the Detergent B values are in the 2nd row, and so on)), and then using Analyze > Nonparametric Tests > K Related Samples. Then one just has to put all 3 of the variables (each corresponding to a different washing machine) into the Test Variables box and click OK. In the output, the value of the test statistic is given beside of Chi-Square, and the approximate p-value is given beside of Asymp. Sig. (but for a small number of blocks, it may be appreciably better to use a table of the exact null distribution).) (Note: For this test, detergent is a blocking variable. I'd view the blocking variable as a fixed effects variable, thinking that 4 specific detergents are used, and not a random selection of 4 detergents from a large collection of possible detergents. In the previous problem, the washers served as a blocking variable. Again, I'd treat the blocks as fixed effects since 3 different models were used. If three different washers of the same model were used, I'd treat the blocking variable as a random effect. With only one observation per cell, it makes no difference whether the blocking variable is treated as a fixed effects or a random effects variable. Finally, I think that in this situation, assuming an additive model isn't so bad. While there may be differences between detergents and differences between washers, I wouldn't think that strong interactions exist --- I wouldn't think that the detergent that works best with Machine 1 is different from the detergent that works best with Machine 2. Rather, the issue is whether or not there are any differences at all. (Is the cheaper store brand just as good as the more expensive name brand? Given some people's loyalty to certain brands, clearly the judging of whiteness should be done blinded. (Not by putting people's eyes out before they judge the whiteness, but rather by not letting the judges know which brands they are rating until after the task is done.)))

Problem 101 (1 extra credit point)

Use an ANOVA F test to address the situation of Problem 100 (instead of Friedman's test), and report the p-value.

Due Thursday, December 1

Problem 102 (0 points)

Consider the data of Exercise 12.7 on pp. 538-539 of S&W (which can be brought into SPSS from the CD included with S&W (fatfree data set)).

• (a) Give the value of Pearson's sample correlation coefficient.
• (b) Give the p-value which results from a test of the null hypothesis that the two variables are uncorrelated against the general alternative that their correlation is not 0.
• (c) Give the value of Spearman's rank correlation coefficient. (See my web page comments about Section 12.7 of S&W for pertinent SPSS information, and additional comments. Because Pearson's correlation is stronger than Spearman's correlation, it is sensible to emphasize Pearson's correlation, which measures the strength of the linear relationship between the two variables. If Spearman's correlation was the greater of the two by more than just a little, then there would be evidence that the relationship is nonlinear, and it would be good to also report, and even emphasize, Spearman's rank correlation coefficient. If you look at a scatter plot of energy plotted against fatfree (put fatfree on the horizontal axis), you will see a fairly strong linear trend.)

Problem 103 (6 points)

Consider the data of Exercise 12.27 on pp. 563-564 of S&W (which can be brought into SPSS from the CD included with S&W).

• (a) Give a scatter plot of plant weight plotted against leaf area. (So put weight on the y axis, and area of the x axis.)
• (b) Give the value of Pearson's sample correlation coefficient.
• (c) Give the p-value which results from a test of the null hypothesis that the two variables are uncorrelated against the general alternative that their correlation is not 0. (Since SPSS just gives one significant digit, it's okay for you to do so too.)
• (d) Give the value of Spearman's rank correlation coefficient. (See my web page comments about Section 12.7 of S&W for pertinent SPSS information, and additional comments. Because Pearson's correlation is stronger than Spearman's correlation, it is sensible to emphasize Pearson's correlation, which measures the strength of the linear relationship between the two variables. If Spearman's correlation was the greater of the two by more than just a little, then there would be evidence that the relationship is nonlinear, and it would be good to also report, and even emphasize, Spearman's rank correlation coefficient. Note that a linear trend between the two variables indicates that plant weight increases more or less linearly with leaf area.)

Problem 104 (0 points)

Consider the data of Exercise 12.3 of S&W, and fit a simple least squares regression model.

• (a) Give the value of R².
• (b) Give the least squares estimate of the intercept parameter.
• (c) Give a 95% confidence interval for the intercept parameter.
• (d) Since there isn't statistically significant evidence that an intercept is needed, and having E(Y | x = 0) = 0 seems sensible in this situation, fit a no intercept model and give an estimate of E(Y | x).

Problem 105 (6 points)

Consider the data of Exercise 12.5 of S&W, and fit a simple least squares regression model.

• (a) Give the value of R².
• (b) Give the least squares estimate of the slope parameter.
• (c) Give a 95% confidence interval for the slope parameter.
• (d) Give a plot of the (unstandardized) residuals against the (unstandardized) predicted values.

Due Thursday, December 8

Problem 106 (2 extra credit points)

Consider the data of Exercise 12.43 on pp. 575-576 of S&W. Give a fairly simple regression model for log age as a function of diameter. (Use base ten logs.) Be sure to estimate any parameter(s) in your model. (Hint: I would alter the model

log age = b₀ + b₁ diameter + error

in two ways.) (Note: One might think it would be more natural to model diameter as a function of age, but if the purpose is to use diameters to estimate ages as opposed to going to the trouble of using ¹⁴C-dating, one would want to model age (or perhaps some function of age in order to get a better model for a least squares regression fit) as a function of diameter.)

Problem 107 (3 points)

Consider the data of Exercise 12.45 of S&W, and fit a simple least squares regression model. (Note: The data on the S&W CD isn't right. The values of the response variable are okay, but for the predictor variable you should create a new variable in the Data Editor having three values of 0, three values of 0.06, three values of 0.12, and three values of 0.30.) (Don't do any transformations, even though some sort of transformation seems to be appropriate.)

• (a) Give a scatter plot of yield against sulfer dioxide concentration, and draw the fitted regression line onto the plot. (To do this using SPSS, first make the scatter plot, and then double-click the plot to open up the Chart Editor. By clicking on the plotted points on the graph in the Chart Editor in just the right way (it's somewhat sensitive to how you go about it) you can color the points blue. Once that is done, click the 10th icon on the bar near the top of the editor to add the fitted regression line. Close the Properties window that opened up (by clicking Close in the Properties window), and then use File > Close in the Chart Editor to close it down and cause the plot with the fitted line to appear in the main Output window.) (Note that the fitted line doesn't miss the tightly packed set of points for x = 0.30. Sometimes when there is apparent heteroscedasticity, the fitted line can miss a tightly packed set of points for an x value at the end of the range of x values.)
• (b) Give a plot of the studentized deleted residuals against the (unstandardized) predicted values. (Note that there is one really large studentized deleted residual.)

Problem 108 (4 points)

Consider the obesity data described on p. 75 of G&H, and available on the G&H web site.

• (a) Without using any transformations, and including a constant term, use multiple regression to develop an estimate for E(Y | h, w), where h is height and w is weight.
• (b) Consider the observation which results in the studentized deleted residual having the largest magnitude. Do you think that this observation is such that it has great influence on the fit of the regression model? Answer yes or no, and give a not too lengthy one sentence explantion in support of your answer.

Due Thursday, December 15

Problem 109 (18 points)

Consider the caterpillar data described on pp. 581-582 of S&W, and available on the S&W CD. Transform the data as is indicated in S&W.

• (a) Fit an additive model, and give the value of the F statistic which is used to test the null hypothesis that diet does not affect head weight against the general alternative.
• (b) Based on the additive model fit in part (a), give an estimate of the difference in mean log head weight for caterpillars on Diet 1 and caterpillars on Diet 2, considering caterpillars having the same body weight.
• (c) Fit a model which allows different slopes and different intercepts and give an estimate of the difference in the slope corresponding to Diet 1 and the slope corresponding to Diet 3.
• (d) Give the p-value which results from a test of the null hypothesis that the change in mean log head weight per unit change in log body weight is the same for all of the diets against the general alternative.
• (e) Give a plot of the (unstandardized) residuals against the (unstandardized) predicted values for a fitted additive model based on the log transformed data (which is the model considered in parts (a) and (b)).
• (f) Repeat part (e), only this time use the untransformed head weight and body weight values.

Problem 110 (0 points)

Consider the caterpillar data described on pp. 581-582 of S&W, and available on the S&W CD. Transform the data as is indicated in S&W.

• (a) Using an additive model, give an estimate of the difference in mean log head weight for caterpillars on Diet 2 and caterpillars on Diet 3, considering caterpillars having the same body weight.
• (b) Fit a model which allows different slopes and different intercepts and give the value of the F statistic which is used to test the null hypothesis that the intercepts corresponding to the three diets are all equal against the general alternative.
• (c) Using the fitted model from part (b), give an estimate of the difference in the slope corresponding to Diet 1 and the slope corresponding to Diet 2.

Problem 111 (27 points & 3 extra credit points)

Consider the snow geese data (distributed in class on 11/17). (Here is a link to the 36 TIME values. Here is a link to the 36 TEMP values. Here is a link to the 36 HUM values. Here is a link to the 36 LIGHT values. Here is a link to the 36 CLOUD values.)

• (a) Use Graphs > Scatter > Matrix to create a scatterplot matrix of the dependent variable and the four predictor variables. (You don't have to submit this graphic display.)
  • (i) Which of the predictor variable appears to have a strongest correlation with the dependent variable?
  • (ii) Which of the predictor variable appears to have a weakest correlation with the dependent variable?
  It can be noted that none of the predictor variables are highly correlated with one another. (Some pairs of predictors have statistically significant correlations, but none of them are close to 0.9 or - 0.9.)
• (b) Fit a multiple regression model using all four predictor variables (linearly --- i.e., don't transform them or add any higher order terms). You can do this using Analyze > Regression > Linear, clicking the dependent variable (TIME) into the Dependent box, the four predictor variables into the Independent box, and using the default Method of Enter.
  • (i) Give the value of R².
  • (ii) Look at the four p-values corresponding to the t tests about the coefficients of the four predictor variables. Which of the predictor variables has the largest p-value? (The p-value is greater than 0.2. It can be noted that this variable is not the one which should have been identified for item (ii) of part (a) --- thus it is the case that a variable which by itself is not strongly correlated with the dependent variable, contributes appreciably to a better fit when added to a model based on the other three variables, while a variable which is more strongly correlated with the dependent variable doesn't contribute much when added to a model containing the other three predictor variables. This illustrates that one should not simply use the collection of sample correlations between each of the predictor variables and the dependent variable to do variable selction (although if there are a very large number of potential predictors, some like to use the sample correlations to obtain an initial set of variables to work with, but in such a case, the variables initially screened out should be eventually considered, since the value of some variables may not become apparent unless they are used in a model containing certain other variables).)
  (Note: I wouldn't bother with saving residuals and predicted values yet. Once a model is firmed up, I'll request that you produce some plots pertaining to its residuals.)
• (c) Remove the weak predictor identified in part (b) and fit a multiple regression model using just the other predictor variables. Give the value of R². (It will be lower than the value of R² requested in part (b), but not a lot lower. (If a weak predictor is added to a model, it will tend to increase the value of R² a little, even if the predictor is not really related to the dependent variable.) Notice that all three t tests yield small p-values, and so one should not be compelled to remove any other predictors from the model. This three variable model can also be obtained by putting all four predictor variables into the Independent box and using Backward as the Method, and also by putting all four predictor variables into the Independent box and using Stepwise as the Method. So, ignoring possible transformations and higher order polynomial models, one can feel pretty good about a model based on using just the three predictor variables identified so far.
• (d) Now compare the first-order model based on the three predictors already identified (i.e., the model fit in part (c)) to the second-order polynomial model based on the same three predictors. Give the p-value which results from a F test done to determine of there is statistically significant evidence that any of the coefficients for the second-order terms is nonzero. (To do this you can use the output generated when doing part (c) to obtain one of the needed SSR values. To obtain the other SSR value which is needed, and the MSE value which is needed, one needs to create new variables for the second-order terms (which can be done using Transform > Compute), and then fit a regression model using the three first-order terms (the three predictor variables used in part (c)) and all of the second-order terms corresponding to the three first-order terms. The SSR value and MSE value needed can be obtained from the resulting ANOVA table (which is part of the regression output). Note that by SSR I mean the SS (sum of squares) value in the Regression row of the ANOVA table, and by MSE I mean the MS (mean squares) value in the Residual row of the ANOVA table. (The mean squares due to error is the same thing as the mean squares corresponding to the residuals.) Once the desired SSR values and the MSE value are obtained, one can use Transform > Compute to obtain the value of the desired F statistic and the p-value. (To do this, type in Fstat for the name of the Target Variable, and in the Numeric Expression box put (SSRfull - SSRred)/(df*MSEfull), where SSRfull is the SSR value for the 2nd-order model, SSRred is the SSR value for the 1st-order model, MSEfull is the MSE value for the 2nd-order model, and df is the difference between the number of coefficients estimated for the 2nd-order model and the number of coefficients estimated for the 1st-order model. (Note: In this problem, I'm using SSR for the sum of the squares due to regression, whereas in part (c) of the next problem, I let SSR be the sum of squared residuals. Either way of obtaining the F statistic works, since SSRred - SSRfull, where SSR is the sum of the squared residuals, is equal to SSRfull - SSRred, where SSR is the sum of squares due to regression. You can see it given differently in different books --- but either way you do it you get the same value. So in the numerator of the F statistic used for determing whether or not there is significant evidence that at least some of the 2nd-order terms are useful, you can take the difference of the SS (sum of squares) values on the regression row of the ANOVA table in the output (subtracting result for 1st-order model from result for 2nd-order model) or you can take the difference of the SS values in the residual row of the ANOVA table in the output (subtracting result for 2nd-order model from result for 1st-order model). In the denominator of the F statistic, you can use the MS (mean square) value from the residual row of the ANOVA table in the output, or you can use SSRfull/(n - p - 1), where here SSR needs to be the sum of the squared residuals (the SS value from the residual row of the ANOVA table in the output).) Clicking OK produces the value of the F statistic (in 36 different places down a column in the Data Editor (but this works out okay)). To obtain the p-value, this, type in pvalue for the name of the Target Variable, and in the Numeric Expression box put 1 - CDF.F(Fstat,df1,df2), where df1 is the same as the df value used in the F statistic, and df2 is the df value associated with the MSEfull value used in the F statistic. Clicking OK puts the desired p-value in 36 different places down a column in the Data Editor.)
• (e) At this point it would be good to check on the residuals. (If we go further with fine-tuning the current model, and much later look at the residuals only to find they indicate that a transformation of the dependent variable is called for, it's a bit like spending time polishing a scat specimen (or more crudely, polishing a turd).) So give a plot of the unstandardized residuals against the unstandardized predicted values. (Turn in this plot.) It can be noted that there is a severe funnel shape. With such a shape, often a log transformation (of y) would be a good next step. But with this data we have a problem in that some of the y values are negative. Although it will lead to a somewhat screwy dependent variable, add 30 to each y value, and then take the log (base 10). (I think that in this analysis, the main thing of interest is to determine which of the variables possibly influences the time that the geese fly off to breakie, and if we determine their transformed take-off times are related to some of the variables included in the study, we would have that their untransformed take-off times are related to these variables.)
• (f) Repeat item (i) of part (b) using transformed time (as prescribed in part (e)) as the dependent variable.
• (g) Since the output generated in doing part (f) indicates that not all of the variables (the four original predictor variables) are statistically significant, do some variables selection (starting with the four original predictor variables) automatically by using both the Backward and Stepwise methods of SPSS's regression routine. You should find that both methods result in the same two (predictor) variable model. (It is interesting to note that the stepwise procedure, as a last step, removed the predictor variable that it entered on the first step --- so the best predictor, if one had to select only use and use a simple regression model, doesn't make the final cut! (I hope that you're gaining an appreciation that good regression analysis can be a bit tricky --- it doesn't always work out to determine the best single predictor, put it in the model, and leave it in the model.)) Give the two predictor variables selected by both methods.
• (h) Repeat part (d) using transformed time (as prescribed in part (e)) as the dependent variable. (The 1st-order model should be the one determined in part (g), and the 2nd-order model should be the model obtained by expanding the 1st-order model to include the 2nd-order terms for the two predictors.) You should find that the collection of 2nd-order terms is not statistically significant. Also, if you were do start with the set of variables used for the 2nd-order model (the two 1st-order terms and the three 2nd-order terms), and do variable selction using both the Backward method and the Stepwise method, in each case you will wind up with the two (predictor) variable 1st-order model as your final model. (Note: This is perhaps a bit rare --- for a lot of data sets for which a 2nd-order model isn't a statistically significant improvement over the 1st-order model, applying such variable selection methods to the set of 1st-order and 2nd-order terms will result in models different from the 1st-order model and different from each other.) At this point, unless residuals indicate a problem, we might be tempted to take the relatively simple (except for the somewhat screwy dependent variable) model using only two untransformed predictor variables as out final model. (Note that the variables in the model seem to indicate that the timing of the geese depends much more on how the morning looks than it does on how the morning air feels.)
• (i) Repeat part (e) using the model selected in part (g). You should not see a clear funnel pattern. (Before fitting the model in order to save the needed residuals, click on Statistics and check the box for the Durbin-Watson statistic.)
• (j) Give the value of the Durbin-Watson statistic (based on the model fit for parts (g) and (i)). (Note: It only makes sense to compute the Durbin-Watson statistic if the data in the Data Editor is in some sort of a natural order. In this case, we have time-ordered data.) Since the value of the Durbin-Watson statistic is greater than 2, we don't have strong evidence of a problem with positive autocorrelation. (Positive autocorrelation is a condition in which successive error term values tend to be on the whole closer in value to one another than what occurs with independent error terms. Positive autocorrelation leads to values of the Durbin-Watson statistic less than 2 (although values not too much less than 2 shouldn't be taken as strong evidence of positive autocorrelation). Ideally, the value of the dependent variable should be observed at equally-spaced points in time, and we don't have that with this data. Because of this problem, one can also check on the autocorrelation issue by plotting the residuals against time.)
• (k) Plot the unstandardized residuals against a variable created from the dates, assinging the value of 1 to 11/10/87, the value 4 to 11/13/87, the value 5 to 11/14/87, ..., the value 75 to 01/23/88, and the value 76 to 01/24/88. (Note that 01/24/88 is 75 days after 11/10/87, so if we say that 11/10/87 is Day 1, then 01/24/88 is Day 76.)
• (l) As a further check of the fitted model, plot the deleted studentized residuals against the variable (prescribed in part (k)) created from the dates. Here I am having you plot the deleted studentized residuals against the date variable in order to make it easier to determine which observations lead to unusual deleted studentized residuals. Also, one can compare this plot which the plot called for in part (k). If an observation results in a relatively large (in magnitude) deleted studentized residual, but not a relatively large (in magnitude) unstandardized residual, it indicates that the observation had a rather large influence on the fit (the determination of the estimates of the coefficients). For this fitted model, it doesn't appear to be the case that any of the unusual observations have high influence. However, there is one observation which leads to a rather large residual. But if 33, instead of 30, is added to each of the original dependent variable (TIME) values before taking the log of this variable to obtain a transformed variable to model, the problem of the large residual disappears (and so I would prefer to use this second transformation of TIME). (Some refer to regression modeling as an art. I definitely think that it's a skill that's not easy to master. There's a lot more to it than I have time to teach you this semester. But if you have a good understanding of this HW problem, I would say that you've had a good first lesson.)

Problem 112 (0 points)

Consider the chemical engineering data described here. (Here is a link to the 24 Yield values. Here is a link to the 24 Time values. Here is a link to the 24 Temp values.)

• (a) Use Graphs > Scatter > Matrix to create a scatterplot matrix of the dependent variable and the two predictor variables. Which of the predictor variable appears to have a strongest correlation with the dependent variable?
• (b) Fit a multiple regression model using both predictor variables (linearly --- i.e., don't transform them or add any higher order terms). You can do this using Analyze > Regression > Linear, clicking the dependent variable (Yield) into the Dependent box, the two predictor variables (Time and Temp) into the Independent box, and using the default Method of Enter. For now, just save the Unstandardized Predicted Values and the Studentized Residuals.
  • (i) Give the value of R².
  • (ii) Give the p-value which results from the t test of the null hypothesis that the coefficient of Time is 0 against the alternative that the coefficient of Time is not 0.
  • (iii) Give the p-value which results from the t test of the null hypothesis that the coefficient of Temp is 0 against the alternative that the coefficient of Temp is not 0. (You should find that both p-values are less than 0.05, indicating that both predictors should be used in a good model of the phenomenon.)
  • (iv) Examine a plot of the studentized residuals against the unstandardized predicted values (so residuals on vertical axis and predicted values on horizontal axis). (You should see a pattern indicating nonlinearity --- the residuals corresponding to the lowest 4 predicted values are all negative, the residuals corresponding to the highest 3 predicted values are all negative, and the majority of the residuals corresponding to the middle group of predicted values are positive ... there is a curved pattern as opposed to a "band" centered on 0.)
• (c) The apparent nonlinearity makes it worthwhile to try fitting a 2nd-order model. To do this, use Transform > Compute to create three new variables (perhaps calling them Time2 (for the square of Time), Temp2 (for the square of Temp), and TimeTemp (for the product of Time and Temp)). Fit the 2nd-order model (using 5 predictors), and this time save the Unstandardized Predicted Values, the Studentized Residuals, the Studentized deleted Residuals.
  • (i) Give the value of R².
  • (ii) Upon looking at the results from the t tests about the coefficients of the variables, you should find that 4 of them indicate that the p-value is less than 0.0005. Give the variable that has the largest p-value associated with it, and give the p-value.
  • (iii) Since all of the p-values for the 2nd-order terms are less than 0.05, we know that these terms are usful in the model, and there is no real need to do an F test. Nevetheless, do an F test of the null hypothesis that all of the coefficients for the 2nd-order variables are 0 against the alternative that these coefficients are not all equal to 0. Give the value of the test statistic, and the resulting p-value. (To do this you can use the output generated when fitting the 2nd-order model to obtain one of the needed SSR (sum of squared residuals) values. The other one can be obtained from the output generated when the 1st-order model was initially fit. One can use Transform > Compute to obtain the value of the desired F statistic and the p-value. (To do this, type in Fstat for the name of the Target Variable, and in the Numeric Expression box put (SSRred - SSRfull)/(df*SSRfull/(n-p-1)), where SSRfull is the SSR value for the 2nd-order model, SSRred is the SSR value for the 1st-order model, df is the difference between the number of coefficients estimated for the 2nd-order model and the number of coefficients estimated for the 1st-order model, n is the sample size, and p is the number of predictor variables used in the full (in this case, 2nd-order) model. (Note: In this problem, I'm using SSR for the sum of the squared residuals, whereas in part (d) of the previous problem, I let SSR be the sum of squares due to regression. Either way of obtaining the F statistic works, since SSRred - SSRfull, where SSR is the sum of the squared residuals, is equal to SSRfull - SSRred, where SSR is the sum of squares due to regression. You can see it given differently in different books --- but either way you do it you get the same value. So in the numerator of the F statistic used for determing whether or not there is significant evidence that at least some of the 2nd-order terms are useful, you can take the difference of the SS (sum of squares) values on the regression row of the ANOVA table in the output (subtracting result for 1st-order model from result for 2nd-order model) or you can take the difference of the SS values in the residual row of the ANOVA table in the output (subtracting result for 2nd-order model from result for 1st-order model). In the denominator of the F statistic, you can use the MS (mean square) value from the residual row of the ANOVA table in the output, or you can use SSRfull/(n - p - 1), where here SSR needs to be the sum of the squared residuals (the SS value from the residual row of the ANOVA table in the output).) Clicking OK produces the value of the F statistic (in 24 different places down a column in the Data Editor (but this works out okay). To obtain the p-value, type in pvalue for the name of the Target Variable, and in the Numeric Expression box put 1 - CDF.F(Fstat,df1,df2), where df1 is the same as the df value used in the F statistic, and df2 is the df value associated with the MSE of the 2nd-order model (which is n - p - 1). Clicking OK puts the desired p-value in 36 different places down a column in the Data Editor. (Note: SSRfull/(n - p - 1) is the same as the MSE for the 2nd-order model. But rather than take the MSE value reported in the output, which has been rounded to 3 significant digits, it's better to use the SSR value reported in the output, since it is reported with 4 significant digits, and so we won't be subjected to as much rounding error.)))
  • (iv) At this point it would be good to check on the residuals. A plot of the studentized residuals (from the 2nd-order model) against the predicted vlaues (from the 2nd-order model) doesn't strongly suggest nonlinearity, and there is no clear funnel pattern indicative of heteroscedasticity. Further checks can be done by plotting the residuals against both Time and Temp, and noting that curved patterns are not found. Given that the 2nd-order model seems to be a good stopping point (there is not enough data to fit a 3rd-order model, and the lack of apparent nonlinearity and the high R² value suggest that the 2nd-order model may be adequate), one can check to see if any of the 24 observations is a bothersome outlier by comparing the studentized deleted residuals to the studentized residuals. (An observation that has a large studentized deleted residual but a relatively small studentized residual is a point that had high influence on the fit.) A good way to compare these residuals graphically is to plot the studentized deleted residuals against the studentized residuals. (Upon doing this, one should see a strong diagonal pattern, and upon closer inspection, one can note that for each of the 24 observations, the studentized residual and studentized deleted residual values are quite similar.) Finally, one can create a probit plot of the residuals to see that we don't have drastic nonnormality. (Often the problit plot is made form the standardized residuals (or the unstandardized residuals), but it'll be okay to just use the studentized residuals.) I suggest that you examine all of the plots indicated in this part.

Problem 113 (1.5 extra credit points)

Do Exercise 10.5 on p. 400 of S&W, only respond to the question by reporting an appropriate p-value.

Problem 114 (1.5 extra credit points)

Consider Exercise 10.13 on pp. 408-409 of S&W. Instead of giving a fictitious data set, simply fill in the 2 by 2 table of counts in such a way so that the value of the usual (uncorrected) chi-square statistic is 0, in addition to being in agreement with the information already on p. 408.

Problem 115 (3 extra credit points)

Do part (c) Exercise 10.17 on p. 409 of S&W. (Be sure to address both items.)

Problem 116 (3 extra credit points)

Do Exercise 10.53 on p. 434 of S&W, only respond to the question by reporting an appropriate p-value.

Problem 117 (3 extra credit points)

Do Exercise 10.62 on p. 441 of S&W. (Note: S&W provides answers for 10.59 and 10.61, which are similar.)