a. A statistics practitioner took a random sample of 50 observations from a population with a standard deviation of 25 and computed the sample mean to be 100. Estimate the population mean with 90% confidence.
b. Repeat part (a) using a 95% confidence level.
c. Repeat part (a) using a 99% confidence level.
d. Describe the effect on the confidence interval estimate of increasing the confidence level.

d) When we increase the confidence level we see that the interval becomes wider and the margin of error given by [tex] ME= t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex] increase.

Step-by-step 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".

[tex]\bar X=100[/tex] represent the sample mean

[tex]\mu[/tex] population mean (variable of interest)

s=25 represent the sample standard deviation

n=50 represent the sample size

Part a

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

[tex]\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex] (1)

In order to calculate the critical value [tex]t_{\alpha/2}[/tex] we need to find first the degrees of freedom, given by:

[tex]df=n-1=50-1=49[/tex]

Since the Confidence is 0.90 or 90%, the value of [tex]\alpha=0.1[/tex] and [tex]\alpha/2 =0.05[/tex], and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.05,49)".And we see that [tex]t_{\alpha/2}=1.68[/tex]

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

[tex]100-1.68\frac{25}{\sqrt{50}}=94.060[/tex]

[tex]100+1.68\frac{25}{\sqrt{50}}=105.940[/tex]

So on this case the 90% confidence interval would be given by (94.060;105.940)

Part b

Since the Confidence is 0.95 or 95%, the value of [tex]\alpha=0.05[/tex] and [tex]\alpha/2 =0.025[/tex], and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.025,49)".And we see that [tex]t_{\alpha/2}=2.01[/tex]

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

[tex]100-2.01\frac{25}{\sqrt{50}}=92.894[/tex]

[tex]100+2.01\frac{25}{\sqrt{50}}=107.106[/tex]

So on this case the 95% confidence interval would be given by (92.894;107.106)

Part b

Since the Confidence is 0.95 or 95%, the value of [tex]\alpha=0.05[/tex] and [tex]\alpha/2 =0.025[/tex], and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.025,49)".And we see that [tex]t_{\alpha/2}=2.01[/tex]

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

[tex]100-2.01\frac{25}{\sqrt{50}}=92.894[/tex]

[tex]100+2.01\frac{25}{\sqrt{50}}=107.106[/tex]

So on this case the 95% confidence interval would be given by (92.894;107.106)

Part c

Since the Confidence is 0.99 or 99%, the value of [tex]\alpha=0.01[/tex] and [tex]\alpha/2 =0.005[/tex], and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.005,49)".And we see that [tex]t_{\alpha/2}=2.68[/tex]

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

[tex]100-2.68\frac{25}{\sqrt{50}}=90.525[/tex]

[tex]100+2.68\frac{25}{\sqrt{50}}=109.475[/tex]

So on this case the 99% confidence interval would be given by (90.525;109.475)

Part d

When we increase the confidence level we see that the interval becomes wider and the margin of error given by [tex] ME= t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex] increase.