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A large clothing manufacturer plans to introduce a new line of sports clothes for women if preliminary market research shows that at least 50% of the population are favorably impressed by the new line. Five hundred women were surveyed; 265 of the women were favorably impressed. The research manager wants to test the hypothesis at the .05 significance level. The z-value associated with this significance level is 1.645.
-How many women from a sample of 500 must be favorably impressed with the new line in order to reject the null hypothesis at the specified significance level?
A) 250
B) 251
C) 249
D) 269
E) 268

Question

Quizplus · Accepted Answer

To reject the null hypothesis at the .05 significance level, we calculate the minimum number of women who need to be favorably impressed using the formula for the test of proportion: $Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$, where $\hat{p}$ is the sample proportion, $p_0$ is the population proportion (0.5 in this case), and $n$ is the sample size (500). Rearranging the formula to solve for $\hat{p}$ when $Z = 1.645$, we find that more than 268 women need to be favorably impressed to reject the null hypothesis. Therefore, the minimum number is 269.

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