Direct mail advertisers send solicitations (a.k.a. "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 1000 people randomly selected from their mailing list of over 200,000 people. They get orders from 123 of the recipients.
Create a 90% confidence interval for the percentage of people the company contacts who may buy something.My answer is (0.106,0.14)
Explain what this interval means. Need help with this part.
Explain what "90% confidence" means.Need help with this part.
The company must decide whether to now do a mass mailing. The mailing won't be cost-effective unless it produces at least a 5% return. What does your confidence interval suggest? Explain. My explanation is Our confidence interval suggests that the company should do mass mailings. The entire interval is well above the cutoff of 5%.
90% confidence interval indicates that 90% of the confidence intervals that you form using the same procedure will contain the true mean of the population.
You had a sample and you are trying to predict the mean response from the total population. That is why we create a confidence interval.
If I was going to answer the question what does this interval mean... I cannot say that there is a 90% chance that the interval contains the true population number. I can say that I am 90% confidence that the interval contains the population number. I can also say that 90% of the intervals formed in this manner will contain the true popoulation number.
I know it is confusing. This is one of the hardest concepts for students.
To create a 90% confidence interval for the percentage of people who may buy something, we can use the formula for confidence intervals for proportions.
First, we calculate the sample proportion, which is the number of successes divided by the total sample size:
Sample Proportion (p̂) = 123/1000 = 0.123
Next, we calculate the standard error, which measures the variability of our estimate:
Standard Error (SE) = √((p̂(1 - p̂))/n)
SE = √((0.123(1 - 0.123))/1000)
SE ≈ 0.0106
To calculate the confidence interval, we can use the formula:
Confidence Interval = p̂ ± (z * SE)
Given that we want a 90% confidence interval, we need to find the z-value that corresponds to a 90% confidence level. In the case of a two-tailed test, the z-value is approximately 1.645.
Confidence Interval = 0.123 ± (1.645 * 0.0106)
Confidence Interval ≈ 0.106 to 0.140
Now let's explain what this interval means. The confidence interval (0.106 to 0.140) suggests that, based on the sample data, we can be 90% confident that the true percentage of people in the population who may buy something lies within this range. In other words, if we were to repeat this experiment multiple times and construct 90% confidence intervals from each sample, approximately 90% of those intervals would contain the true population proportion.
Next, let's explain what "90% confidence" means. A 90% confidence level means that if we were to repeat this experiment multiple times and construct 90% confidence intervals from each sample, approximately 90% of those intervals would contain the true population proportion. It does not mean that there is a 90% chance that the true proportion falls within a specific interval, but rather that this method of constructing intervals provides a level of confidence in capturing the true proportion.
Based on the confidence interval of 0.106 to 0.140, we can see that the entire interval is well above the cutoff of 5%. This suggests that there is a high probability that the percentage of people who may buy something is above the 5% threshold. Therefore, our confidence interval suggests that the company should proceed with mass mailings because there is a good chance of achieving at least a 5% return.
The confidence interval you provided, (0.106, 0.14), represents the range of possible values for the true percentage of people who may buy something based on the sample data.
In this case, we are 90% confident that the true percentage lies between 10.6% and 14%. This means that if we were to repeat the sampling process many times, 90% of the resulting confidence intervals would contain the true percentage of people who may buy something.
A 90% confidence level means that, out of all the possible samples we could have taken, 90% of the intervals constructed using these samples would contain the true percentage of people who may buy something.
With a confidence interval that ranges from 10.6% to 14%, it suggests that there is a high probability that the percentage of people who may buy something is above the 5% threshold that the company requires for the mass mailing to be cost-effective. Therefore, based on the confidence interval, the suggestion is to proceed with the mass mailing as the entire interval is comfortably above the 5% cutoff.