Question

In marketing, response modeling is a method for identifying customers most likely to respond to an advertisement. Suppose that in past campaigns 31.6% of customers identified as likely respondents did not respond 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 did not respond to a new campaign would decrease. The analysts selected a random sample of 1500 customers and found that 441 did not respond to the marketing campaign.

The marketing analysts want to use a one‑sample ?‑test to see if the proportion of customers who did not respond to the advertising campaign, ?, has decreased since they updated their model. They decide to use a significance level of ?=0.05.

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.

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

?0:?=0.294 and ?1:?<0.294

?0:?=0.316 and ?1:?<0.316

?0:?=0.316 and ?1:?≠0.316

?0:?̂=0.316 and ?1:?̂<0.316

?0:?̂=0.294 and ?1:?̂<0.294

?=

?-value:

Select the correct decision and conclusion.

The analysts should not reject the null hypothesis. There is insufficient evidence (?>0.05) that the proportion of customers who did not respond to the marketing campaign is less than 0.316.

The analysts should not reject the null hypothesis. There is sufficient evidence (?<0.05) that the proportion of customers who did not respond to the marketing campaign is less than 0.316.

The analysts should reject the null hypothesis. There is insufficient evidence (?>0.05) that the proportion of customers who did not respond to the marketing campaign is less than 0.316.

The analysts should reject the null hypothesis. There is sufficient evidence (?<0.05) that the proportion of customers who did not respond to the marketing campaign is less than 0.316.

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.

Answer 1

Here claim is that the proportion of customers who did not respond to the advertising campaign, ?, has decreased since they updated their model.

So hypothesis is

?0:?=0.316 and ?1:?<0.316

Now

SE is

Hence test statistics is

So P value is

As P value is less than alpha=0.05, we reject the null hypothesis

So answer is

The analysts should reject the null hypothesis. There is sufficient evidence (?<0.05) that the proportion of customers who did not respond to the marketing campaign is less than 0.316.