Question

Direct mail advertisers send solicitations​ ("junk mail") to thousands of potential customers in the hope that some will buy the​ company's product. The response rate is usually quite low. Suppose a company wants to test the response to a new flyer and sends it to

1020

people randomly selected from their mailing list of over​ 200,000 people. They get orders from

100

of the recipients. Use this information to complete parts a through d.

​a) Create a

90%

confidence interval for the percentage of people the company contacts who may buy something.

Answer 1

Here we need to find out 90% confidence interval for the percentage of people the company contacts who may buy something.

So in this problem we have given that,

sample size (n) = 1020

number of successes (x) = people get orders = 100

So we will calculate percentage value associated with a population i.e Estimate of population proportion (p^{^})

(denoted by p^hat)

Formula for p^

p^ =

p^ = 0.0980

So we need to find out confidence interval for population proportion,

So formula for confidence interval for population proportion is

When constructing a confidence interval for a population proportion, we need to check if the observed number of successes and failures are at least 10, means
npˆ ≥ 10 and  n(1 − pˆ) ≥ 10

npˆ = 1020*0.0980 = 100

n(1 − pˆ)= 1020*(1-0.0980) = 920

So both are greater than 10, then we can calculate 90% confidence interval   

Now we will find out

confidence level (c) = 0.90

=1-0.90 = 0.1

= 0.05

= 1-0.05 = 0.95

See above value in Z table to get or use below command in Excel to get

=NORMINV(0.95,0,1) = 1.645

Now we have all value to put in formula, putting them in given formula for confidence interval,

0.0980 +/- 1.645 * Sqrt ( 0.0980* (1-0.0980)/1020) ( sqrt = Square root)

0.0980 +/- 1.645 * 0.093

0.0980 +/- 0.01531

(0.0980 - 0.01531 , 0.0980 + 0.01531 )

(0.08269, 0.11331 )

Convert above confidence interval in to percentage multiplying by 100

(8.30%, 11.30% )

Interpretation :

We are 95% confident that the percentage of people the company contacts who may buy something lies between 8.30% and 11.30%