Question

1. a publication reports that 42% of adults believe that radio stations should wait until after thanksgiving to begin playing holiday music. believing the true percentage of adults who feel this way is not as high as 42%, a researcher surveys a random sample of 400 adults, and 156 say that radio stations should wait until after thanksgiving to begin playing holiday music. from this information, which of the following statements is false?

a. the alternative hypothesis should be “ha: p < 0.42” because the researcher believes the true proportion is smaller.
b. the test statistic will be equal to approximately -1.2, leading to a p - value close to 0.12.
c. if the significance level is set at 0.10, the decision should be to fail to reject the null hypothesis.
d. the difference between 0.42 and 0.39 is small, and this means the results are statistically significant but not practically significant.
e. because the researcher has a theory that the true percentage is not as high as 42%, a one - sided test would provide stronger evidence against the null hypothesis than a two - sided test.

Explanation:

Step1: Calculate sample proportion

The sample proportion $\hat{p}=\frac{156}{400} = 0.39$.

Step2: Analyze option A

The researcher believes the true proportion is smaller, so the alternative hypothesis $H_{a}:p < 0.42$ is correct.

Step3: Calculate test statistic

The formula for the test statistic in a one - sample proportion test is $z=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}$. Substituting $p = 0.42$, $\hat{p}=0.39$, and $n = 400$:

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Using the standard normal table, the $P$-value for $z=-1.2$ (one - sided) is $P(Z < - 1.2)=0.1151\approx0.12$. So option B is correct.

Step4: Make a decision for option C

Since the $P$-value ($0.12$) is greater than the significance level $\alpha = 0.10$, we fail to reject the null hypothesis. Option C is correct.

Step5: Analyze option D

Statistical significance is determined by the $P$-value. Here, $P - value=0.12>0.05$ (common significance level), so the results are not statistically significant. Option D is false.

Step6: Analyze option E

A one - sided test uses the information about the direction (the researcher believes $p<0.42$) and will have a smaller $P$-value (stronger evidence against the null hypothesis) compared to a two - sided test when the direction is correct. Option E is correct.

Answer:

D. The difference between 0.42 and 0.39 is small, and this means the results are statistically significant but not practically significant.