Question

A soccer ball manufacturer wants to estimate the mean circumference of​ mini-soccer balls within

0.10

inch. Assume the population of circumferences is normally distributed.

​(a) Determine the minimum sample size required to construct a

95

​%

confidence interval for the population mean. Assume the population standard deviation is

0.20

inch.

​(b) Repeat part​ (a) using a population standard deviation of

0.10

inch.

​(c) Which standard deviation requires a larger sample​ size? Explain.

​(a) The minimum sample size with a population standard deviation of

0.20

inch is

nothing

balls.

​(Round up to the nearest​ integer.)

​(b) The minimum sample size with a population standard deviation of

0.10

inch is

nothing

balls.

​(Round up to the nearest​ integer.)

​(c) A population standard deviation of

▼

0.20

0.10

inch requires a larger sample size because greater variability in the population requires a

▼

smaller

larger

sample size to ensure the desired accuracy.

Answer 1

a) At 95% confidence level, the critical value is z_0.025 = 1.96

Margin of error = 0.1

or, z_0.025 * = 0.1

or, 1.96 * 0.2/ = 0.1

or, n = ((1.96 * 0.2)/0.1)^2

or, n = 16

b) Margin of error = 0.1

or, z_0.025 * = 0.1

or, 1.96 * 0.1/ = 0.1

or, n = ((1.96 * 0.1)/0.1)^2

or, n = 4

c) A population standard deviation of 0.20 inch requires a larger sample size because greater variability in the population requires a larger sample size to ensure the desired accuracy.