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 1150 people randomly selected from their mailing list of over 200,000 people. They get orders from 114 of the recipients.

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 1150 people randomly selected from their mailing list of over 200,000 people. They get orders from 114 of the recipients.

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

Answer 1

n = 1150

x = 114
p = x/n
p = 114/1150 = 0.099
z critical value = ____

(Then, I got this formula from the question in the 'Related Questions' below. I hope this helps.)
E = za/2 *sqrt((p (1-p)/)/n))

Answer 2

To create a confidence interval for the percentage of people who may buy something, we can use the formula:

Confidence Interval = Sample Proportion ± (Critical Value) * (Standard Error)

First, let's calculate the sample proportion (p̂) using the number of orders received (114) and the sample size (1150):

p̂ = Number of orders / Sample Size = 114 / 1150 = 0.0991

Next, we need to determine the critical value, which depends on the level of confidence desired. For a 90% confidence level, we need to find the z-score corresponding to a confidence level of 0.90. Consulting a z-table, we find that the z-score for a 90% confidence level is approximately 1.645.

Now, we need to calculate the standard error (SE) using the sample proportion:

SE = sqrt((p̂ * (1 - p̂)) / n)

SE = sqrt((0.0991 * (1 - 0.0991)) / 1150) ≈ 0.0091

Finally, we can calculate the confidence interval by plugging in the values:

Confidence Interval = 0.0991 ± (1.645 * 0.0091)

Confidence Interval = 0.0991 ± 0.015

The 90% confidence interval for the percentage of people who may buy something is approximately 0.0841 to 0.1141, or 8.41% to 11.41%.

Answer 3

To create a confidence interval for the percentage of people who may buy something, we can use the following formula:

Confidence Interval = Sample Proportion ± (Z * √((Sample Proportion * (1 - Sample Proportion)) / Sample Size))

In this case, the sample proportion is the number of people who bought something divided by the sample size. Let's calculate the confidence interval step-by-step:

Step 1: Calculate the sample proportion
Sample Proportion = Number of people who bought something / Sample size
Sample Proportion = 114 / 1150
Sample Proportion ≈ 0.0991

Step 2: Find the Z-value for a 90% confidence interval
Since we want a 90% confidence interval, we need to find the Z-value for a 95% confidence level (1 - (1 - 0.90) / 2 = 0.95).
Using a standard normal distribution table or calculator, the Z-value for a 95% confidence level is approximately 1.645.

Step 3: Calculate the standard error
Standard Error = √((Sample Proportion * (1 - Sample Proportion)) / Sample Size)
Standard Error = √((0.0991 * (1 - 0.0991)) / 1150)
Standard Error ≈ 0.00934

Step 4: Calculate the margin of error
Margin of Error = Z * Standard Error
Margin of Error = 1.645 * 0.00934
Margin of Error ≈ 0.01532

Step 5: Calculate the lower and upper bounds of the confidence interval
Lower Bound = Sample Proportion - Margin of Error
Lower Bound = 0.0991 - 0.01532
Lower Bound ≈ 0.0838

Upper Bound = Sample Proportion + Margin of Error
Upper Bound = 0.0991 + 0.01532
Upper Bound ≈ 0.1144

Therefore, the 90% confidence interval for the percentage of people the company contacts who may buy something is approximately 8.38% to 11.44%.