Question

A soccer ball manufacturer wants to estimate the mean circumference of​ mini-soccer balls within 0.05 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 0.25 inch.

​(b) Repeat part​ (a) using a population standard deviation of 0.35 inch.

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

Answer 1

Given that, the circumference of mini-soccer balls is normally distributed.

A soccer ball manufacturer wants to estimate the mean circumference of mini-soccer balls within 0.05 inch.

a)

We know that the minimum sample size required to construct a (1- ) 100% confidence interval for the population mean is,

where,

Here the mean value within E

For this problem, E = 0.05, = 0.25, and confidence coefficient = (1 - ) = 0.95

Therefore, = 0.05 and

Therefore the minimum sample size required to construct a 95 % confidence interval for the population mean is

Hence the minimum sample size required to construct a 95 % confidence interval for the population mean is 96.

b)

Now the population standard deviation is

Therefore the minimum sample size required to construct a 95 % confidence interval for the population mean is

Hence the minimum sample size required to construct a 95 % confidence interval for the population mean is 188.

c)

From the above two results, we can say that the standard deviation of 0.35 inch requires a larger sample size.

Actually, the formula for sample size is

From the above formula, the sample size is directly proportional to the population standard deviation, therefore as population standard deviation increases the sample size also increases.

Hence the population standard deviation of 0.35 inch requires larger sample size as compared to the population standard deviation of 0.25 inch.