Quiz #1

Answers are given in the answer boxes below. Scroll down for explantions for the answers (given in red).

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answer
Question 1 A
Question 2 D

Question 1

Consider two experiments very similar to the one of Example 1.1 on p. 2 of Samuels and Witmer, pertaining to Pasteur's vaccine (which hopefully protects sheep against anthrax). In one experiment, Drug A is given to 3 sick people who were randomly selected from a group of 6 sick people, with the remaining 3 people serving as a control group. In the other experiment, Drug B is given to 24 sick people who were randomly selected from a group of 48 sick people, with the remaining 24 people serving as a control group. In both experiments, all of the people who took a drug got better and all of the control group members who did not take a drug remained sick. The tables below display the experimental results.

Drug A control
got better 3 0
remained sick 0 3

Drug B control
got better 24 0
remained sick 0 24

Which one of the following statements is false? Neither outcome is impossible even if both drugs are uneffective. If half of the people in both experiments would have gotten better anyway (without an effective drug), random assignment could have placed the ones who would have gotten better into the groups receiving the drugs. While the chance that this would happen is very small for the Drug B experiment, it's not really small for the Drug A experiment. Since it cannot be ruled out that random assignment of subjects who got better without benefit of an effective drug to the group getting Drug A resulted in the observed outcome of the experiment, one cannot be sure that Drug A is actually effective. Technically, we cannot be sure about Drug B, even though the random assignment explanation is rather far-fetched. In any case, we cannot be sure about the effectiveness of Drug A, and so [ C ] is true. Since the outcome for Experiment B is very very small if the drug in not effective, it seems reasonable to believe that there is some other explanation for the outcome: the explanation being that Drug B is effective. The strength of the evidence against the random assignment explantion isn't as great for Drug A. Because of all of this, Drug B should be favored over Drug A, and so [ B ] is true and [ A ] is false.

Question 2

A and B are events for which P(A) = 0.7 and P(B) = 0.5. All but one of the values below is an impossible value for the probability that A or B (or both) occur.

Which one of the following values cannot be ruled out as the value of the probability that A or B occur? [ A] and [ E ] can be ruled out, because a probability cannot be less than 0 or greater than 1. It can be noted that all of the outcomes in in A are certainly in A or B, and so the probability of A or B has to be at least as large as the probability of A, which rules out [ B ] and [ C ]. So only [ D ] cannot be ruled out. (Another way to get the answer is to note that the desired probability is equal to P(A) + P(B) - P(A and B) = 0.7 + 0.5 - P(A and B) = 1.2 - P(A and B). The probability subtracted off cannot be larger than 1, and so the desired probability has to be at least as large as 0.2, which rules out [ B ]. But actually it can be noted that the probability subtracted off cannot be larger than P(B) = 0.5, since the intersection of A and B cannot be larger than B, and this results in the desired probability having to be at least as large as 0.7, which rules out [ C ].)