Question

So I have a rather long question for Stats and I'm not really even sure where to begin let alone finish it. I am supposed to answer this using PhStat in Excel so if anyone can help me solve this and also explain how they did that would be awesome! Any help is appreciated :)

Scenarios of the Problems:

1. A researcher wishes to estimate the mean yearly tuition of the population of private universities throughout the United States. A random and representative sample of universities of this type is to be selected to provide the required estimate. To provide estimates of the population mean tuition, what minimum number of universities will be sampled under the following conditions to provide the needed estimate?

1. The estimate desired will need to be computed with 95% confidence to within ±$4,000 when it is felt that the population standard deviation in the tuition amounts is $9,000.  

2. The estimate desired will now need to be computed with 90% confidence to within ±$4,000 when it is felt that the population standard deviation in the tuition amounts is $9,000.

3. The estimate desired will now need to be computed with 95% confidence to within ±$4,000 when it is felt that the population standard deviation in the tuition amounts is $10,000.

4. The estimate desired will now need to be computed with 95% confidence to within ±$5,000 when it is felt that the population standard deviation in the tuition amounts is $9,000.

* note left Be sure to provide explanations of your answers. In addition, be sure to comment on any differences you observe in your answers. Give reasons for these differences. Do not use only mathematical justifications as you explain any differences in your answers. Make all comparisons relative to the answer found in the first part of the problem.

2. You now need to construct confidence intervals for the mean tuition charged by this population of universities. You select a random sample of 40 universities that are members of the population you are studying. You find that the mean tuition for this sample of universities is $19,457. You believe that the population standard deviation of the tuition amounts is approximately $9,000. Construct both 95% and 90% confidence intervals for the mean tuition for this population of universities. Be sure to explain the meaning of these intervals. Comment upon the effect of the change in confidence level on the width of your interval.

3. You now no longer believe that the population standard deviation in the tuition amounts charged by this population of universities is a known quantity. You therefore will use the sample standard deviation of the tuition amounts as an estimate of this unknown population standard deviation as you attempt to estimate the mean tuition of this population of universities. You collect tuition information from a random sample of universities. This data is shown in appendix one below. Find both 95% and 90% confidence intervals for the mean yearly tuition of this population of universities. Explain the meaning of these intervals. At each confidence level, comment upon the change in the results of this problem from the results of the previous problem.

            Appendix One (Yearly Tuition)

            $15,678         $21,409         $32,311         $20,945         $22,375         $10,456         $25,674         $35,151         $18,904            $30,500

            $23,571         $28,612         $21,786         $17,776         $18,905

            $23,678         $25,892         $21,090         $16,736         $19,999

            $21,689         $36,792         $23,893         $15,430         $22,229

            $26,903         $21,378         $26,500         $30,000         $22,287

            $38,017         $23,765         $21,001         $28,739         $25,000

            $23,001`        $24,590         $21,017         $17,450         $19,540

4. In addition to studying the mean tuition of this population of universities, the researcher wishes to study the proportion of universities in this population that regularly awards more than 50% of their student body some form of financial aid. What minimum numbers of universities are necessary to estimate the desired proportions under the following conditions? Explain the meanings of the minimum sample sizes you compute.

1. The estimate is desired to within ±15% with 98% confidence when the population proportion of universities that regularly awards financial aid to more than 50% of their student body is thought to equal 40%.

2. The estimate is desired to within ±15% with 98% confidence when the population proportion of universities that regularly awards financial aid to more than 50% of their student body is thought to be unknown.

3. The estimate is desired to within ±15% with 98% confidence when the population proportion of universities that regularly awards financial aid to more than 50% of their student body is thought to equal 70%.

*note Comment upon the changes in the minimum sample sizes you have computed based upon the changes in the information in the parts of this problem. Again, compare your answers to the answer of the first part of the problem. Also, do not use only mathematical justifications to explain why there are differences in the results at which you arrived for the different parts of the problem.

5. You now need to estimate with 98% confidence the population proportion of universities that regularly award more than 50% of their student body financial aid. You randomly sample the population of universities. You then record whether the university sampled regularly awards more than 50% of their student body some form of financial aid. The results of that sampling process are shown in appendix two below. Using this information, construct the required confidence interval and explain its meaning.

Appendix Two:

Regularly awards more than 50% of their student body financial aid?  

(Y = yes, N = no)

Y         N         Y         Y         Y         N         N         N         Y         Y         Y         

Y         N         N         N         Y         N         N         Y         Y         N         Y

N         Y         N         Y         Y         Y         Y         Y         N         N         Y

Y         N         Y         N         Y         N         N         Y         Y         N         Y

N         Y         Y         Y         Y         Y         N         N         N         Y         N

Y         N         N         N         Y         N         N         Y         Y         Y         N

Answer 1

I.The minimum number of universities that will be sampled for different confidence intervals (CI-95% and 90%) and margin of error (ME) when population standard deviation is known are as follows :-

II 95% and 90% confidence intervals for 40 sample universities with mean tuition fees of $19457 and population standard deviation of $9000 are as follows

We are 95% confident that mean tuition fees from the population of universities will vary between $16668 and $22246

We are 90% confident that mean tuition fees from the population of universities will vary between $17116 and $21798

From the above data,one can see that increase the confidence interval will increase the margin of error this resulting in more wider interval as shown above

III From the given sample data of 40 universities yearly tuition fees,we find that sample mean and sample standard deviation of the data from the excel are as follows.One can use AVERAGE() and STDEV.S() function from excel for this purpose. 95% and 90% confidence interval for the sample tuition fees for the population of universities are given as under :-

From the sample data,one can find that average tuition fees is $23517 which is higher from population mean of $19457 and however,sample standard deviation from 40 universities tuition fees data,it is lesser and only $5788 against the population standard deviation of $9000.Based on this,we can say with 95% confidence that population tuition fees will vary between [$21666,$25368] but with 90% confidence level,it will vary between [$21975,$25059]