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

Answer:

107,426, bigger

Step-by-step 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

$2.58(\frac{0.20}{\sqrt{n} } )<0.05\\n>106.50\\n>107$

A minimum sample size of 107 needed.

b) $2.58(\frac{0.40}{\sqrt{n} } )<0.05\\\\\\n>426$

Here minimum sample size = 426

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