Question

A company is deciding whether to renew its ad buy with a local TV station. It will renew the ad if a 2-sided hypothesis test at the 10% level concludes that more than 19% of the local residents remember the ad. They decide to test this by contacting 300 randomly selected local residents. Sixty-seven (67) of the 300 remember the ad.
a. Conduct the requested hypothesis test based on this sample using the normal approximation. (Use Minitab.) Report your conclusions, stated in terms of the data and the sampling scenario.

b. The station advocates that the decision be made based on a 1-sided hypothesis test using the normal approximation at the 10% level using this sample. Why? (Hint: If you conduct the appropriate 1-sided test based on this sample, what will the p-value be? Confirm your answer with Minitab by conducting this 1-sided test.)

Answer 1

Answer:

a) If we perform a two-sided test, we can not reject the null hypothesis, so we have no enough evidence that the proportion of people who remember the ad is different from 0.19.

b) In the case of a one-side test, the null hypothesis is rejected. We have evidence to claim that more than 19% of the people remember the ad.

Step-by-step explanation:

The null and alternative hypothesis are:

$H_0: \pi=0.19\\\\H_1:\pi\neq0.19$

To be a two-sided test, it has to be a not-equal sign, allowing π to be significantly less or more than0.19.

The significance level is 10%.

The sample size is 300.

The sample proportion is p=67/300=0.2233.

The standard deviation is:

$\sigma_M=\sqrt{\frac{\pi(1-\pi)}{N} }=\sqrt{\frac{0.19*0.81}{300} }=0.02265$

The statistic z is:

$z=\frac{p-\pi-0.5/N}{\sigma_M} =\frac{0.2233-0.19+0.5/300}{0.02265}=\frac{0.03163}{0.02265}=1.397\\\\P(|z|>1.397)=0.16242$

The P-value is bigger than the significance level, so this two-sided test can not reject the null hypothesis.

b) In this case, the null and alternative hypothesis are:

$H_0: \pi\leq0.19\\\\H_1:\pi>0.19$

The P-value in this condition is:

$P(z>1.397)=0.08121$

In this case, the P-value is lower than the significance level, so the effect is significant. The null hypothesis is rejected.