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 11501150 people randomly selected from their mailing list of over​ 200,000 people. They get orders from 114114 of the recipients.

Create a 90​% confidence interval for the percentage of people the company contacts who may buy something.

To create a confidence interval for the percentage of people who may buy something, we can use the formula for estimating a population proportion.

The formula for calculating the confidence interval for a proportion is:

CI = p̂ ± Z * √(p̂(1 - p̂) / n)

Where:
CI is the confidence interval
p̂ is the sample proportion (in this case, the proportion of recipients who placed an order)
Z is the Z-score (corresponding to the desired confidence level)
n is the sample size (the number of people who received the flyer)

In this scenario, the sample proportion p̂ is calculated as the number of recipients who placed an order divided by the total number of recipients: p̂ = 114/1150 = 0.0991.

To determine the Z-score, we need to specify the desired confidence level. Since the question asks for a 90% confidence interval, the corresponding Z-score for a two-tailed test is 1.645. This value can be obtained from a standard normal distribution table or calculated using statistical software.

Now, we can plug in the values into the formula to calculate the confidence interval:

CI = 0.0991 ± 1.645 * √((0.0991 * (1 - 0.0991)) / 1150)

Calculating the expression inside the square root:

√((0.0991 * (1 - 0.0991)) / 1150) ≈ 0.0098

Substituting this value into the formula:

CI = 0.0991 ± 1.645 * 0.0098

Calculating the values:

CI = 0.0991 ± 0.0161

This gives us the confidence interval:

CI ≈ (0.0829, 0.1153)

Therefore, at a 90% confidence level, we can estimate that the percentage of people the company contacts who may buy something is between approximately 8.29% and 11.53%.