Explanation of answers to Quiz #14
9 out of 14 students correctly identified D as the correct answer
to the first problem. (The other 5 students answered with B. B
would only be good if it was known that the distributions
underlying the data were normal (and even in that case, given that the
means are known to be 0, the variances should not be estimated with the
usual sample variance estimator which uses the sample mean to estimate
the distribution mean).) The test statistics indicated by D is a
modification of Levene's statistic that adjusts for the fact that the
means are known to be 0.
3 out of 14 students correctly identified D as the correct answer
to the second problem. (The other 11 students answered with C.
C is the null distribution of the usual F statistic (the
statistic indicated by B), that is based on the usual sample variances,
which use sample means to estimate the distribution means.) The sum of
the Xi2 divided by the true variance of the
Xi has a chi-square distribution with n degrees
of freedom, and
the sum of
the Yj2 divided by the true variance of the
Yj has a chi-square distribution with n degrees
of freedom, and these two sums are independent since all of the
Xi are independent of all of the
Yj.
Since the variances are equal under the null hypothesis, we can view the
statistic as being a ratio of two chi-square random variables (since
the unknown variances needed to get the chi-square distributions cancel
one another) if the null hypothesis is true. If each chi-square random
variable is divided by its df, n, then we have an
Fn,n distribution as the null sampling distribution.
Since the sample sizes are equal, we can indeed view the chi-square
random variables as being divided by their df, since the "missing"
values of n simply cancel one another out.