Finding confidence interval given z-score and null hypothesis

Question

I am working on a homework question and got stumped on this question.

9. Suppose researchers perform a large-sample test of a population proportion where the null hypothesis is that the population proportion is 0.4, and the alternative hypothesis is that the population proportion is not equal to 0.4 (two-sided). They obtain a z-statistic of -1.5 under the null hypothesis.

a) (*1 point) What is the (two-sided) p-value? Choose the best answer: i) Two-sided p-value < 0.05 ii) 0.05 < Two-sided p-value < 0.32 iii) Two-sided p-value > 0.32 iv) Two-sided p-value > 0.68 v) Two-sided p-value > 0.95 vi) Two-sided p-value = -1.5

b) (*1 point) Would the 68% confidence interval include 0.4? Explain your answer.

For part a I said that the two-sided p value is in between 0.05 and 0.32 since the z-score is -1.5. However, I am not sure on how to solve the second part with this limited information. I was thinking since the p-value is not enough to reject the null hypothesis, 0.4 would be included, but I want to prove it mathematically. Thanks for any assistance.

Answer 1

Minitab statistical software does one sample tests for binomial proportion with confidence intervals of specified levels.

It may be worthwhile to use formulas from your text or class notes for $Z$ statistics and confidence intervals to see how the following Minitab output was computed.

Test and CI for One Proportion 

Test of p = 0.4 vs p ≠ 0.4


Sample  X   N  Sample p         68% CI         Z-Value  P-Value
1       5  21  0.238095  (0.145667, 0.330523)    -1.51    0.130

Using the normal approximation.