Question

The sick days of employees every two years in a company are normally distributed with a population standard deviation of 7 days and an unknown population mean. If a random sample of 20 employees is taken and results in a sample mean of 21 days, find a 95% confidence interval for the population mean. Round your answer to TWO decimal places. z0.10 z0.05 z0.025 z0.01 z0.005 1.282 1.645 1.960 2.326 2.576 You may use a calculator or the common z values above.

Answer 1

`21-1.96(7)/(√(20))=17.93`

`21+1.96(7)/(√(20))=24.07`

So on this case the 95% confidence interval would be given by (17.93;24.07)

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=21` represent the sample mean for the sample

`\mu` population mean (variable of interest)

`\sigma=7` represent the population standard deviation

n=20 represent the sample size

Solution to the problem

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 0.95 or 95%, the value of
`\alpha=0.05` and
`\alpha/2 =0.025`, and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-NORM.INV(0.025,0,1)".And we see that
`z_(\alpha/2)=1.96`

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

`21-1.96(7)/(√(20))=17.93`

`21+1.96(7)/(√(20))=24.07`

So on this case the 95% confidence interval would be given by (17.93;24.07)

Answer 2

The 95% confidence interval for the population mean is between 17.93 days and 24.07 days.

Explanation:

We have that to find our
`\alpha` level, that is the subtraction of 1 by the confidence interval divided by 2. So:

`\alpha = (1-0.95)/(2) = 0.025`

Now, we have to find z in the Ztable as such z has a pvalue of
`1-\alpha`.

So it is z with a pvalue of
`1-0.025 = 0.975`, so
`z = 1.96`

Now, find M as such

`M = z*(\sigma)/(√(n))`

In which
`\sigma` is the standard deviation of the population and n is the size of the sample.

`M = 1.96(7)/(√(20)) = 3.07`

The lower end of the interval is the sample mean subtracted by M. So it is 21 - 3.07 = 17.93 days

The upper end of the interval is the sample mean added to M. So it is 21 + 3.07 = 24.07 days

The 95% confidence interval for the population mean is between 17.93 days and 24.07 days.