Question

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?
the alternative hypothesis should be \hₐ: p < 0.42\ because the researcher believes the true proportion is smaller.
the test statistic will be equal to approximately -1.2, leading to a p - value close to 0.12.
if the significance level is set at 0.10, the decision should be to fail to reject the null hypothesis.
the difference between 0.42 and 0.39 is small, and this means the results are statistically significant but not practically significant.
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: Calculate test statistic

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

$$
LATEXBLOCK0
$$

The $P$-value for $z=-1.2$ (one - sided test) is $P(Z\lt - 1.2)=0.1151\approx0.12$.

Step3: Analyze significance level

If $\alpha = 0.10$, since $P - value=0.12\gt0.10$, we fail to reject the null hypothesis.

Step4: Analyze practical significance

A small difference between the hypothesized proportion ($p = 0.42$) and the sample proportion ($\hat{p}=0.39$) means the results are not statistically significant (because $P - value\gt\alpha$ when $\alpha = 0.10$). Statistical significance is determined by the $P$-value and significance level, not just the difference in proportions.

Step5: Analyze one - sided vs two - sided test

When the researcher has a specific direction (the true proportion is less than $0.42$), a one - sided test is more appropriate and provides stronger evidence against the null hypothesis in the hypothesized direction than a two - sided test.

Answer:

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