Asoccer ball manufacturer wants to estimate the mean circumf

Asoccer ball manufacturer wants to estimate the mean circumference of soccer bals within o 05 in (b) Repeat the size required to construc a 90% confidence interval or the populason mean Assume the population standard deviation is 0.25 in part (a) using a standard deviation of 0.35 in. Which standard deviation requires alarger sample size? Explain (a) The minimum sample size required to construct a 90% confidence interval using a population standard deviation of 0.25 in is bals (Round up to the nearest integer.) (b) The minimum sample size required to construct a 90% confidence interval using a standard deviation of 0.35 in is bals (Round up to the nearest integer) population standard deviation of in requires larger sample size Due to the increased variability in the population, a v sample size is needed to ensure the desied accuracy 0.35 0.25 Enter your answer in each of the answer boxes

Solution

First, the z score to use for a 90% confidence interval is 1.645.

How this is obtained: If the confidence level is .90, alpha is .10 (1-.90), and alpha/2 would be .05. Looking up .05 on the standard normal table gives a z of 1.645.

a. The minimum sample size required to construct a 90% confidence interval using a population standard deviation of 0.25 inches is _____

The formula for the sample size required:

n = (z*sigma/E)^2.

z is 1.645.
sigma is 0.25
E is 0.05, the error given at the beginning of the problem.

Substituting these values:

n = (1.645*0.25/0.05)^2 = 68 (always round up)

answer: 68

b. The minimum sample size required to construct a 90% confidence interval using a standard deviation of 0.35 in is ______

The formula for the sample size required:

n = (z*sigma/E)^2.

z is 1.645
sigma is 0.35
E is 0.05

Substituting these values:

n = (1.645*0.35/0.05)^2 = 133 (always round up)

answer: 133

c. A population standard deviation of 0.35 in requires a larger sample size. Due to the increased variability in the population, a larger sample size is needed to ensure the desired accuracy.