Question

A statistics instructor would like to ask "clicker" questions that about \(80 \%\) of her students in a large lecture class will get correct. A higher proportion would be too easy and a lower proportion might discourage students. Suppose that she tries a sample of questions and receives 76 correct answers and 24 incorrect answers among 100 responses. The hypotheses of interest are \(H_{0}: p=0.80\) vs \(H_{a}: p \neq 0.80 .\) Discuss whether or not the methods described below would be appropriate ways to generate randomization samples in this setting. Explain your reasoning in each case. (a) Sample 100 answers (with replacement) from the original student responses. Count the number of correct responses. (b) Sample 100 answers (with replacement) from a set consisting of 8 correct responses and 2 incorrect responses. Count the number of correct mses.

Short answer

Method a would not be appropriate for hypothesis testing as it does not reflect the proportions of the null hypothesis. However, method b would be appropriate as it simulates the proportions of the null hypothesis.

Step-by-step solution

Understanding method a

In method a, we sample 100 answers (with replacement) from the original student responses and count the number of correct responses. This would reflect the true proportions in the observed data, namely, that 76% of responses were correct and 24% were incorrect. However, this does not match the null hypothesis that 80% of students answer correctly. Therefore, this method would not generate an appropriate randomization sample under the null hypothesis.

Understanding method b

In method b, we sample 100 answers (with replacement) from a set consisting of 8 correct responses and 2 incorrect responses. This method would indeed simulate results under the null hypothesis that 80% of responses are correct. Therefore, method b is an appropriate way to generate a randomization sample for testing this hypothesis.