Q 8.6RP. - Elementary Statistics

Question

Suppose that you intend to find a $95\%$ confidence interval for a population mean by applying the one-mean $z-$interval procedure to a sample of size $100$

a. What would happen to the accuracy of the estimate if you used a sample of size $50$ instead but kept the same confidence level of $0.95$ ?

b. What would happen to the accuracy of the estimate if you changed the confidence level to $0.90$ but kept the same sample size of $100$?

Step-by-Step Solution

Answer

Part (a) The precision of the estimate decreases as the margin of error increases.

Part (b) The precision of the estimate increases as the length of the confidence interval decreases.

Part (a) Step 1: Given information

Applying the one-mean z-interval technique to a sample of size $100$yields a $95\%$ confidence interval for a population mean.

Part (a) Step 2: Explanation

The precision of the estimate would be reduced if we utilized a sample of size $50$ instead of size $100$ and kept the same confidence level.

Because the margin of error increases as the sample size drops for a certain degree of confidence, we know that the precision of the estimate reduces as the margin of error increases.

Part (b) Step 1: Explanation

The precision of the estimate might improve if we increased the confidence level to $0.90$ while keeping the sample size at $100$

Because the margin of error and hence the length of the confidence interval falls as the level of confidence lowers for a certain sample size, As the length of the confidence interval reduces, the precision of the estimate increases.