Question

Researchers collected two different samples, X and Y, of temperatures, in degrees Celsius, of the habitat for Florida scrub lizards. The confidence interval 36±1.66 was constructed from sample X, and the confidence interval 36±1.08 was constructed from sample Y. Assume both samples had the same standard deviation.
Which of the following statements could explain why the width of the confidence interval constructed from X is greater than the width of the confidence interval constructed from Y?

A. The sample size of X is greater than the sample size of Y, and the confidence level is the same for both intervals.
B. The sample size of X is greater than the sample size of Y, and the confidence level used for the interval constructed from X is less than the confidence level used for the interval constructed from Y.
C. The sample size is the same for X and Y, and the confidence level used for the interval constructed from X is less than the confidence level used for the interval constructed from Y.
D. The sample size is the same for X and Y, and the confidence level is the same for both intervals.
E. The sample size is the same for X and Y, and the confidence level used for the interval constructed from X is greater than the confidence level used for the interval constructed from Y

Answer 1

Answer:

E. The sample size is the same for X and Y, and the confidence level used for the interval constructed from X is greater than the confidence level used for the interval constructed from Y

Step-by-step explanation:

Confidence interval formula is given as

CI = \bar{x} \pm z*(\sigma/\sqrt{n})

Here, we can see that the width of confidence interval is z*(\sigma/\sqrt{n}) , which is dependent only on z critical value, standard deviation (sigma) and sample size (n)

This simply entails that either z or n or both are greater for x as compared to y.

Option E is correct answer because x and y can have sample sizes, but if the z critical is greater for x, then the width will be larger.