Chap 6 DQ: An instructor claims that more than $75 \%$ of MATH 1340 students dislike discussion questions. A large random sample of students in MATH 1340 is obtained, and it yields a test statistic of $z=1.15$. Test the claim at the 0.06 significance level using both the critical value and $P$-value methods, and state your conclusion in nontechnical terms. Your DQ Response must list all 7 of the following items: 1. The null and alternative hypotheses in symbolic form, with the claim identified as one of them 2. The value of $\alpha$ 3. The test statistic 4. The critical value(s) 5. The P-value 6. The decision of your test, including brief rationale using both the $\mathrm{CV}$ and $P$-value methods* 7. The conclusion of your test, stated in nontechnical language
Solution
Step 1 ：State the null and alternative hypotheses. The null hypothesis is that the proportion of students who dislike discussion questions is 75% or less, and the alternative hypothesis is that the proportion is more than 75%. The claim is identified as the alternative hypothesis.
Step 2 ：The significance level, \( \alpha \), is 0.06.
Step 3 ：The test statistic is \( z = 1.15 \).
Step 4 ：Calculate the critical value. The critical value is the z-score that corresponds to the 94th percentile (1 - 0.06 = 0.94) in the standard normal distribution. The critical value is approximately \( z = 1.55 \).
Step 5 ：Calculate the P-value. The P-value is the probability of obtaining a result as extreme as, or more extreme than, the observed data, assuming the null hypothesis is true. The P-value is approximately 0.125.
Step 6 ：Make a decision. The test statistic (1.15) is less than the critical value (1.55), and the P-value (0.125) is greater than the significance level (0.06). Therefore, using both the critical value and P-value methods, we fail to reject the null hypothesis.
Step 7 ：State the conclusion in nontechnical terms. There is not enough evidence to support the claim that more than 75% of MATH 1340 students dislike discussion questions. Therefore, the final answer is \(\boxed{\text{There is not enough evidence to support the claim that more than 75% of MATH 1340 students dislike discussion questions.}}\)
