Question

A soccer ball manufacturer wants to estimate the mean circumference of soccer balls within 0.1 in. Determine the minimum sample size required to construct a 95% confidence interval for the population mean. Assume the population standard deviation is 0.25 in.

Answer 1

`n=((z_(\alpha/2) \sigma)/(ME))^2` (2)

The critical value for 95% of confidence interval now can be founded using the normal distribution. And in excel we can use this formla to find it:"=-NORM.INV(0.025,0,1)", and we got
`z_(\alpha/2)=1.96`, replacing into formula (2) we got:

`n=((1.96(0.25))/(0.1))^2 =24.01`

So the answer for this case would be n=25 rounded up to the nearest integer

Explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

`\bar X` represent the sample mean for the sample

`\mu` population mean (variable of interest)

`\sigma=0.25` represent the population standard deviation

n represent the sample size (variable of interest)

Solution to the problem

The confidence interval for the mean is given by the following formula:

`\bar X \pm z_(\alpha/2)(\sigma)/(√(n))`

The margin of error is given by this formula:

`ME=z_(\alpha/2)(\sigma)/(√(n))` (1)

And on this case we have that ME =0.1 and we are interested in order to find the value of n, if we solve n from equation (1) we got:

`n=((z_(\alpha/2) \sigma)/(ME))^2` (2)

The critical value for 95% of confidence interval now can be founded using the normal distribution. And in excel we can use this formla to find it:"=-NORM.INV(0.025,0,1)", and we got
`z_(\alpha/2)=1.96`, replacing into formula (2) we got:

`n=((1.96(0.25))/(0.1))^2 =24.01`

So the answer for this case would be n=25 rounded up to the nearest integer