# Thread: Stats: Creating confidence interval

1. ## Stats: Creating confidence interval

mm

2. Originally Posted by chillerbros17
Direct mail advertisers send solicitations to thousands of potential customers in the hope that some will buy the company's product. The rsponse rate is usually quite low. Suppose a company wants to test the response to a new flyer, and sends it to 1000 people randomly selected fromtheir mailing list of over 200,000 people. THey get orders from 123 of the recipients.

a) Create a 90% confidence interval for the percentage of people the company contacts who may buy something.
The number of respondents to test mailing of 1000 is 123, both these
numbers are large enough that we may assume that the Normal
approximation to the Binomial distribution is appropriate.

Then (N is the sample size in this case 1000, and p is the proportion of
recipients that will respond):

z = [123 - pN]/sqrt[N p (1-p)]

has a standard normal distribution, and so in 90% of cases lies between
+/-1.645. So:

-1.645 < [123 - pN]/sqrt[N p (1-p)] < 1.645

90% of the time, rearranging these inequalities:

-1.645 sqrt[N p (1-p)] < 123 -pN < 1.645 sqrt[N p (1-p)]

or:

[123-1.645 sqrt[N p (1-p)]]/N < p < [123+1.645 sqrt[N p (1-p)]]/N

90% of the time.

The problem now is that we don't know the value of sqrt[N p (1-p)], so
we have to estimate it from out test sample, for which N=1000, and our
estimate for p is pest=123/1000=0.123. In which case our estimate of
sqrt[N p (1-p)] is 10.39, and our inequality is now:

[123-1.645*10.39]/1000 < p < [123+1.645*10.39]/1000,

or:

0.106 < p < 0.140.

Which gives a 90% confidence interval for p of approximately (0.106, 0.140).

b) Explain what this interval means.
This interval is a random variable which 90% of the time contains the
actual proportion of respondents.

c) Explain what "90% confidence" means.
see above

d) The company must decide whether to now do a mass mailing. The mailing won't be cost-effective unless it produces at least 5% return. What does your confidence interval suggest? Explain.
The confidence interval suggests that the results obtained from the test
shot are unlikely to have been the result of a test on a population with a
response rate as low as 5%, but only suggests this as this is the wrong
procedure to apply for this last part.

RonL