Question

In marketing, response modeling is a method for identifying customers most likely to respond to an advertisement. Suppose that in past campaigns 76.2% of customers identified as likely respondents responded to a nationwide direct marketing campaign. After making improvements to their model, a team of marketing analysts hoped that the proportion of customers identified as likely respondents who responded to a new campaign would increase. The analysts selected a random sample of 1500 customers and found that 1185 responded to the marketing campaign.

The marketing analysts want to use a one‑sample ?‑test to see if the proportion of customers who responded to the advertising campaign, ?, has increased since they updated their model. They decide to use a significance level of ?=0.01.

Select the correct null (?0) and alternative (?1) hypotheses.

?0:?=0.762 and ?1:?≠0.762

?0:?̂=0.762 and ?1:?̂>0.762

?0:?=0.762 and ?1:?>0.762

?0:?̂=0.790 and ?1:?̂>0.790

?0:?=0.790 and ?1:?>0.790

Determine the ?‑value for this test. Give your answer precise to at least three decimal places.

?=

Determine the value of the ?‑statistic. Give your answer precise to at least two decimal places.

?-value:

Select the correct decision and conclusion.

A. The analysts should reject the null hypothesis. There is insufficient evidence (?>0.01) that the proportion of customers who responded to the marketing campaign is greater than 0.762.

B. The analysts cannot make any valid decision or conclusion because the requirements for using a one-sample ?‑test for a proportion have not been met.

C. The analysts should not reject the null hypothesis. There is insufficient evidence (?>0.01) that the proportion of customers who responded to the marketing campaign is greater than 0.762.

D. The analysts should reject the null hypothesis. There is sufficient evidence (?<0.01) that the proportion of customers who responded to the marketing campaign is greater than 0.762.

E.The analysts should not reject the null hypothesis. There is sufficient evidence (?<0.01) that the proportion of customers who responded to the marketing campaign is greater than 0.762.

Answer 1

The null and alternative hypothesis is

?0:?=0.762 and ?1:?>0.762

Sample size = n = 1500

x = 1185

Test statistic is

$z=\frac{\hat{p}-P_{0}}{\sqrt{P_{0}*(1-P_{0})/n}}$

$z=\frac{0.79-0.762}{\sqrt{0.762*(1-0.762)/1500}}=\frac{0.028}{\sqrt{0.000121}}=2.55$

P-value = P(Z >2.55) = 0.0054

P-value < 0.01 we reject null hypothesis.

Conclusion: D. The analysts should reject the null hypothesis. There is sufficient evidence (?<0.01) that the proportion of customers who responded to the marketing campaign is greater than 0.762.