Question

An educational psychologist wishes to know the mean number of words a third grader can read per minute. She wants to make an estimate at the 99% level of confidence. For a sample of 4657 third graders, the mean words per minute read was 24.9. Assume a population standard deviation of 5.7. Construct the confidence interval for the mean number of words a third grader can read per minute. Round your answers to one decimal place.

Answer 1

The 99% confidence interval is given by (24.7;25.2)

Explanation:

1) 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=24.9` represent the sample mean for the sample

`\mu` population mean (variable of interest)

`\sigma=5.7` represent the population standard deviation

n=4657 represent the sample size

Confidence =0.99 or 99%

2) Confidence interval

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

`\bar X \pm z_(\alpha/2)(\sigma)/(√(n))` (1)

Since the Confidence is 0.99 or 99%, the value of
`\alpha=0.01` and
`\alpha/2 =0.005`, and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-NORM.INV(0.005,0,1)".And we see that
`z_(\alpha/2)=2.58`

Now we have everything in order to replace into formula (1):

`24.9-2.58(5.7)/(√(4657))=24.685`

`24.9+2.58(5.7)/(√(4657))=25.115`

So on this case the 99% confidence interval would be given by (24.7;25.2)