You would like to know if a majority of students at Emory support the university-wide vaccine mandate. You conduct a hypothesis test with the hypotheses:
𝐻
0
:
𝑝
=
0.5
, and
𝐻
𝑎
:
𝑝
≠
0.5
. You collect data from 50 students, calculate a positive test statistic, and find a p-value of 0.06. What would be true regarding a 95% confidence interval for the same sample data?

Question

You would like to know if a majority of students at Emory support the university-wide vaccine mandate. You conduct a hypothesis test with the hypotheses: 
𝐻
0
: 
𝑝
=
0.5
, and 
𝐻
𝑎
: 
𝑝
≠
0.5
. You collect data from 50 students, calculate a positive test statistic, and find a p-value of 0.06. What would be true regarding a 95% confidence interval for the same sample data?

BlackTom AI · Accepted Answer

We need to analyze what a 95% confidence interval (CI) for the population proportion p would look like given the study details and relate each answer option to those facts.
Option 1: "It would include 0.5". Since the p-value for testing H0: p = 0.5 against Ha: p ≠ 0.5 is 0.06 (which is greater than the 0.05 significance level), we fail to reject the null at alpha = 0.05. A fundamental property of confidence intervals is that if the null value p0 is not rejected by a two-sided test at level alpha, then p0 lies inside the corresponding (1−alpha) confidence interval. Therefore, a 95% CI for p would indeed contain 0.5. This makes this option consistent with the observed p-value and standard CI behavior.
Option 2: "If we increase the sample size, the confidence interval would be wider." This is incorrect. Increasing the sample size reduces the standard error, which shrinks the width of the CI (assuming the same confidence level). So larger samples yield narrower, not wider, intervals. The statement contradicts the known effect of sample size on CI width.
Option 3: "It would not include 0.5" This would contradict the conclusion from the p-value above. Since the 95% CI should include the null value when the two-sided test is not significant at alpha = 0.05, asserting that the interval would exclude 0.5 is inconsistent with the given p = 0.06. Therefore, this option is incorrect.
Option 4: "It would include 0.06" We must distinguish between a p-value and a confidence interval for p. The 95% CI is an interval for the true population proportion p, while 0.06 is a p-value from a hypothesis test, not a plausible bound for p itself. Unless the observed p-hat and standard error happen to place 0.06 inside the CI by coincidence (which is not a general rule and cannot be guaranteed from the information given), this statement is not a guaranteed property of the 95% CI. More importantly, the CI is not defined to include the p-value as a parameter value, so this option is misleading and generally incorrect.
In summary, option 1 is consistent with the standard link between hypothesis tests and confidence intervals; options 2, 3, and 4 describe scenarios that either contradict fundamental CI behavior or misinterpret the relationship between p-values and CI endpoints.

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