IDENTIFYING REJECTION REGION INVOLVING POPULATION PROPORTION

1. A pharmaceutical company claims that a new vaccine is 95% effective in preventing a disease. A clinical trial with 1,000 participants finds that only 920 people were successfully protected. Should we reject the company’s claim at α = 0.05?

• A. Reject the claim

• B. Fail to reject the claim

• C. More data is needed

• D. It depends on public opinion

Answer explanation

Correct Answer: A. Reject the claim

Explanation: If 920 out of 1,000 people were protected, the sample proportion is 0.92 (92%). A hypothesis test comparing this to the claimed 95% effectiveness would likely result in rejecting the null hypothesis if the difference is statistically significant at α = 0.05.

2. A research study claims that a new herbal remedy reduces cold symptoms faster than traditional medicine. A clinical trial is conducted with 500 patients, and the average recovery time is 6.2 days compared to 6.5 days with traditional medicine. Should we reject the null hypothesis at

α = 0.01?

• A. Reject the claim, the remedy is not effective

• B. Fail to reject the claim, the remedy is effective

• C. Fail to reject the null hypothesis, the difference is not significant

• D. Accept the null hypothesis completely

Answer explanation

Correct Answer: C. Fail to reject the null hypothesis, the difference is not significant

Explanation: With a small difference in recovery time (6.2 vs. 6.5 days) and a strict α = 0.01, the remedy may not show a statistically significant effect in reducing symptoms.

3. A fair coin is expected to land on heads 50% of the time. A student flips a coin 100 times and records 38 heads. Should the student reject the assumption of fairness at α = 0.05?

• A. Yes, the coin is biased

• B. No, the result is within an expected range

• C. The result proves the coin always lands on tails

• D. More flips are needed for an accurate conclusion

Answer explanation

Correct Answer: B. No, the result is within an expected range

Explanation: Using a hypothesis test, the observed result of 38 heads out of 100 flips may not be extreme enough to reject the assumption of a fair coin at α = 0.05.

4. A university study claims that students who study in the morning score higher on exams than those who study at night. A sample of 200 students finds that morning students score an average of 82, while night students score an average of 79, with a p-value of 0.03. Should the university reject the null hypothesis at α = 0.05?

• A. Yes, because the p-value is less than 0.05

• B. No, because the difference is too small

• C. No, because a larger sample size is needed

• D. It depends on the students’ preferences

Answer explanation

Correct Answer: A. Yes, because the p-value is less than 0.05

Explanation: Since the p-value (0.03) is below the significance level (α = 0.05), we reject the null hypothesis and conclude that morning students perform better.

5. A company claims that a new marketing strategy increases sales by 20%. After implementing the strategy in 50 stores, analysts find an average increase of 15% with a standard deviation of 5%. Should the company reject its own claim at α = 0.05?

• A. Yes, the data does not support the claim

• B. No, the increase is still close to 20%

• C. More data is needed to make a conclusion

• D. Rejecting the claim depends on customer feedback

Answer explanation

Correct Answer: A. Yes, the data does not support the claim

Explanation: If the sample mean (15%) is significantly lower than the claimed 20%, hypothesis testing at α = 0.05 would likely lead to rejecting the claim.