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 1180 people randomly selected from their mailing list of over 200,000 people. They get orders from 148 of the recipients. Use this information to complete parts a through d.
) 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 based on the given information, we can use the following formula:
CI = p̂ ± Z * √(p̂ * (1-p̂) / n)
where:
- p̂ is the sample proportion (the number of orders divided by the total number of people contacted)
- Z is the Z-score corresponding to the desired confidence level (90% in this case)
- n is the sample size (the number of people contacted)
First, we calculate the sample proportion:
p̂ = (number of orders) / (total number of people contacted)
= 148 / 1180
≈ 0.1254
Next, we need to find the Z-score corresponding to a 90% confidence level. We can use a standard normal distribution table or a statistical calculator. For a 90% confidence level, the Z-score is approximately 1.645.
Now we can plug the values into the formula to calculate the confidence interval:
CI = 0.1254 ± 1.645 * √(0.1254 * (1-0.1254) / 1180)
Calculating the values inside the square root:
√(0.1254 * (1-0.1254) / 1180) ≈ 0.0139
Plugging back into the formula:
CI = 0.1254 ± 1.645 * 0.0139
Calculating the values:
CI = 0.1254 ± 0.0229
Therefore, the 90% confidence interval for the percentage of people the company contacts who may buy something is approximately:
0.1025 ≤ p ≤ 0.1483
This means that we can be 90% confident that the true percentage of people who may buy something lies within the range of 10.25% to 14.83%.