# Finding confidence interval given z-score and null hypothesis

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

1. 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.

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.