Question

A random sample of n measurements was selected from a population with unknown mean μ and known standard deviation σ. Calculate a 95% confidence interval for μ for the given situation. Round to the nearest hundredth when necessary. n = 125, x-bar = 91, σ = 20 Group of answer choices

Answer 1

`91-1.96(20)/(√(125))=87.494`

`91+1.96(20)/(√(125))=94.506`

So on this case the 95% confidence interval would be given by (87.494; 94.506). We have 95% of confidence that the true mean is between the limits founded.

Explanation:

Notation

`\bar X=91` represent the sample mean

`\mu` population mean (variable of interest)

`\sigma=20` represent the population standard deviation

n=125 represent the sample size

Confidence interval

The confidence interval for the mean if we know the deviation 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 significance is
`\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):

`91-1.96(20)/(√(125))=87.494`

`91+1.96(20)/(√(125))=94.506`

So on this case the 95% confidence interval would be given by (87.494; 94.506). We have 95% of confidence that the true mean is between the limits founded.

Answer 2

The 95% confidence interval for μ for the given situation is between 87.49 and 94.51.

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 the margin of error 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(20)/(√(125)) = 3.51`

The lower end of the interval is the sample mean subtracted by M. So it is 91 - 3.51 = 87.49

The upper end of the interval is the sample mean added to M. So it is 91 + 3.51 = 94.51

The 95% confidence interval for μ for the given situation is between 87.49 and 94.51.