Question

Direct mail advertisers send solicitations (a.k.a. "junk mail") to thousands of potential customers in the hope that some will buy the company's product. Suppose a company wants to test the response to a new flyer, and sends it to 1000 people randomly selected from their mailing list of over 200,000 people. They get orders from 123 of the recipients.

a) Is it plausible that 30% of all recipients order the new flyer? Is it plausible that 12% of all recipients ordered the new flyer?

b) The company must decide whether to now to a mass mailing. The mailing won't be cost-effective unless it produces at least a 10% return. What does your confidence interv% with a margin of error 1%?

Answer 1

Part a)

Total sample size n = 1000

Number of responses = x = 123

proportion ( p) = x/n

P = 123/1000 = 0.123

.

For 30% , P^ = 0.30

.
SE = sqrt(p*(1-p)/n)

SE = sqrt(0.123*0.877/1000)

SE = 0.0104

.

Formula of Z score is:

Z = (p^-p)/SE

Z = (0.30-0.123)/0.0104

Z = 17.01923

This is an exceptionally large value of Z obtained

Hence yes it is plausible to obtain a proportion of 30%

.

For 12%, p^ = 0.12

Z = (0.12-0.123)/0.0104

Z = -0.29

Yes this value of Z is within the casual limits of -2 and +2

Hence this value of Z is not plausible

.

Part b)

Confidence interval for p = 0.123 with margin of error 0.01 ( or 1%) is as follows:

(p -margin of error, p + margin of error)

(0.123 -0.01, 0.123 + 0.01)

(0.113 , 0.133)

That is the confidence interval is 11.3% to 13.3% , which is completely greater than 10% and hence yes the company must decide for mass mailing