Question

Researchers studied the role that the age of workers has in determining the hours per month spent on personal tasks. A sample of 1,686 adults were observed for one month. The data are: Construct an 88 percent confidence interval for the mean hours spent on personal tasks for 18-24 year olds.

Answer 1

`df=n-1=241-1=240`

Since the Confidence is 0.88 or 88%, the value of
`\alpha=0.12` and
`\alpha/2 =0.06`, and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.06,240)".And we see that
`t_(\alpha/2)=1.56`

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

`4.17-1.56(0.75)/(√(241))=4.095`

`4.17+1.56(0.75)/(√(241))=4.245`

So on this case the 88% confidence interval would be given by (4.095;4.245)

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".

The summary statistics are given on the picture attached

`\bar X=4.17` represent the sample mean for the sample

`\mu` population mean (variable of interest)

s=0.75 represent the sample standard deviation

n=241 represent the sample size

Solution to the problem

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

`\bar X \pm t_(\alpha/2)(s)/(√(n))` (1)

The mean calculated for this case is
`\bar X=4.17`

The sample deviation calculated
`s=0.75`

In order to calculate the critical value
`t_(\alpha/2)` we need to find first the degrees of freedom, given by:

`df=n-1=241-1=240`

Since the Confidence is 0.88 or 88%, the value of
`\alpha=0.12` and
`\alpha/2 =0.06`, and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.06,240)".And we see that
`t_(\alpha/2)=1.56`

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

`4.17-1.56(0.75)/(√(241))=4.095`

`4.17+1.56(0.75)/(√(241))=4.245`

So on this case the 88% confidence interval would be given by (4.095;4.245)