Question

A soccer ball manufacturer wants to estimate the mean circumference of soccer balls within 0.05 in.

(a) Determine the minimum sample size required to construct a 99% confidence interval for the population mean. Assume the population standard deviation is 0.20 in.
(b) Repeat part (a) using a standard deviation of 0.40 in. Which standard deviation requires a larger sample size? Explain.
(a) The minimum sample size required to construct a 99% confidence interval using a population standard deviation of 0.20 in. is balls. (Round up to the nearest integer.)
(b) The minimum sample size required to construct a 99% confidence interval using a standard deviation of 0.40 in. is balls. (Round up to the nearest integer.)
A population standard deviation of in. requires a larger sample size. Due to the increased variability in the population, a sample size is needed to ensure the desired accuracy.

Answer 1

107,426, bigger

Explanation:

Given that a soccer ball manufacturer wants to estimate the mean circumference of soccer balls within 0.05 in.

Margin of error = 0.05 inches

Since population std deviation is known we can use z critical value.

(a) i.e. for 99% confidence interval

Z critical = 2.58

`2.58((0.20)/(√(n) ) )<0.05\\n>106.50\\n>107`

A minimum sample size of 107 needed.

b)
`2.58((0.40)/(√(n) ) )<0.05\\\\\\n>426`

Here minimum sample size = 426

Due to the increased variability in the population, a bigger sample size is needed to ensure the desired accuracy.