Question

As part of a project about response bias, Ellery surveyed a random sample of 25 students from her school. One of the questions in the survey required students to state their GPA aloud. Based on the responses, Ellery said she was 90% confident that the interval from 3.14 to 3.52 captures the mean GPA for all students at her school.
Required:

a. Interpret the confidence level.

b. Explain what would happen to the length of the interval if the confidence level were increased to 99%.

c. How would a 90% confidence interval based on a sample of size 200 compare to the original 90% interval?

d. Describe one potential source of bias in Ellery's study that is not accounted for by the margin of error.

Answer 1

Answer:

a. We are 90% sure that the mean GPA for all students at her school is between 3.14 and 3.52.

b. The length would increase.

c. The interval would be narrower.

d. The biggest potential source of bias is making the students state their grades aloud, which makes it possible that students with low grades could lie their grades, making it seem higher, and the interval biased.

Step-by-step explanation:

x% confidence interval:

A confidence interval is built from a sample, has bounds a and b, and has a confidence level of x%. It means that we are x% confident that the population mean is between a and b.

Margin of error of a confidence interval:

$M = z\frac{\sigma}{\sqrt{n}}$

In which z is related to the confidence level(higher the confidence level, higher z) $\sigma$ is the standard deviation of the population and n is the size of the sample.

This shows that if we increase the confidence level, the margin of error gets larger and the interval gets wider, while if we increase the sample size the margin of error gets smaller and the interval gets narrower.

a. Interpret the confidence level.

We are 90% sure that the mean GPA for all students at her school is between 3.14 and 3.52.

b. Explain what would happen to the length of the interval if the confidence level were increased to 99%.

We would increase the confidence level, and the interval would get wider, that is, the length would increase.

c. How would a 90% confidence interval based on a sample of size 200 compare to the original 90% interval?

Larger sample size means that the interval would be narrower.

d. Describe one potential source of bias in Ellery's study that is not accounted for by the margin of error.

The biggest potential source of bias is making the students state their grades aloud, which makes it possible that students with low grades could lie their grades, making it seem higher, and the interval biased.