Question

Assuming that the population is normally​ distributed, construct a 90​% confidence interval for the population mean for each of the samples below. Explain why these two samples produce different confidence intervals even though they have the same mean and range.

Sample​ A:	1    1    4    4    5    5    8    8
Sample​ B:	1    2    3    4    5    6   7   8

a. Construct a 90​% confidence interval for the population mean for sample A

b. Construct a 90​% confidence interval for the population mean for sample B

Explain why these two samples produce different confidence intervals even though they have the same mean and range.

A.The samples produce different confidence intervals because their sample sizes are different.

B.The samples produce different confidence intervals because their critical values are different.

C.The samples produce different confidence intervals because their standard deviations are different.

D.The samples produce different confidence intervals because their medians are different.

Answer 1

For calculating 90​% confidence interval for the population mean for sample A and B, first we need to identify mean and standard deviation. We used MS Excel for calculating mean and average.

Answer a)

90​% confidence interval for the population mean for sample A

Answer b)

90​% confidence interval for the population mean for sample A

90% CI Sample A = (2.71, 6.29) and 90% CI Sample B = (2.859, 6.141)

Thus, we can see that there is difference in 90% CI for Sample A and Sample B

These two samples produce different confidence intervals even though they have the same mean and range because their standard deviations are different. (Option C is correct)

Explanation

Confidence Interval Formula is as follows:

From the formula of confidence interval, we can see that CI depends on 4 variables namely mean, critical t value, sample size, and standard deviation.

Now, in our case mean and sample size are same. Also, critical t value is same as we are finding 90% CI. So the only factor differentiating sample A from sample B is standard deviation.