Question

A company is willing to renew its advertising campaign with a local radio station only if the station can demonstrate that there is evidence that more than 20% of the residents of the city have heard the ad and recognize the company’s product. The station contacts 500 residents, and finds that 114 remember the ad.

(a) The station plans to conduct a test at ? = 0.10, but the company wants ? lowered to 0.05. Why?

(b) Use the p-value method to conduct the test at ? = 0.10. Would the conclusion differ if ? = 0.05?

(c) Use the critical value method to conduct the test at ? = 0.10. It is not necessary to write a conclusion.



Answer 1

Null hypothesis: H0: customers who remember the ad <= 0.20

Alternate hypothesis: Ha: customers who remember the ad> 0.20

a.

The station wants the null hypothesis to be rejected (while the company wants to make sure there is enough evidence before rejecting the null hypothesis). In other words, the company wants to make sure there is a strong evidence to reject the null hypothesis.

At at 10% significance level, there is more chance that the null hypothesis will be rejected. Thus, for the station, a higher significance level is better. Whereas, the lower the significance level, the stronger the evidence will be (if the null is rejected) that the ratio of customers who remember the ad is greater than 20%.

b.

We know that:

population proportion, P= 20%= 0.20

sample proportion, p= 114/500= 0.228

sample size, n= 500Test statistic= (p-P) / sqrt(P*(1-P)/n)

= (0.228-0.20) / sqrt(0.20*0.80/500)

= 0.028 / 0.01788854

= 1.565248

The p-value of this score= area to the right of this z score= 1- area to the left of 1.57

= 1- 0.94169

= 0.05821

Since the p-value is less than 0.10, we reject the null hypothesis and conclude that the proportion of residents who remember the ad is indeed greater than 20%, at 10% significance.

If we were to use a significance level of 5%, since the p-value would be greater than 0.05, we would fail to reject the null hypothesis and conclude that there is not enough evidence to conclude that the proportion of residents who remember the ad is greater than 20%, at 5% significance.

c.

Critical Z value at 0.10 significance = 1.28 (this value can be found from a z distribution table).

Since the z statistic is greater than 1.28, we reject the null hypothesis.

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