Question

You should explain the confidence intervals you create along with explanations of the meaning of your answers and business implications for each problem.

Scenario:

You have been asked once again to study the mean tuition at private universities throughout the United States. You will also again study the proportions of universities throughout the United States that regularly award more than 50% of their students some form of financial aid. The specific questions you will be asked to answer are stated below. In addition, appropriate sample data for the studies you will be accomplishing is given below. Answer the following questions concerning the situations posed.

2.) Once again, as has happened in the past, you are very much in doubt concerning the validity of the known population standard deviations in the tuitions charged by private universities. Therefore, you wish to conduct your study with the knowledge that the population standard deviations are unknown. These unknown population standard deviations will be estimated using sample standard deviations. You collect random samples of the tuitions at private universities east and west of the Mississippi River. The data that has been collected is shown in appendix one. At both the 5% and 10% levels of significance, are there any differences in the mean tuition rates at private universities east and west of the Mississippi River? Based upon the procedure you use to compare the means in this problem, once again, if the software allows for it, find 95% and 90% confidence intervals for the difference in the mean tuition rates for universities both east and west of the Mississippi River. Explain their meanings in the context of the problem. As you did in the previous problem, of the procedure allows for it, use these intervals to confirm the results of the hypothesis tests in the problem. Use the 95% confidence interval to confirm the results at which you arrived at the 5% level of significance and use the 90% confidence interval to confirm the results at which you arrived at the 10% level of significance.

Appendix One (Tuition)

East of the Mississippi River:

$21,165 $20,435 $29,110 $11,890 $19,500 $21,450

$22,250 $18,450 $22,225 $21,550 $19,990 $18,255

$23,350 $24,110 $20,000 $21,340 $22,500 $23,185

$19,080 $24,320 $17,850 $19,990 $16,760 $12,460

$18,150

West of the Mississippi River:

$17,740 $13,000 $19,550 $18,450 $15,050 $18,600

$18,000 $17,500 $16,900 $21,000 $19,990 $14,520

$18,400 $15,000 $19,750 $17,450 $15,575 $13,260

$12,375 $19,750 $15,560 $17,880 $13,450 $16,660

$20,000

Answer 1

In order to solve this question I used R software.

R codes and output:

> east=scan('clipboard');east
Read 25 items
[1] 21165 20435 29110 11890 19500 21450 22250 18450 22225 21550 19990 18255
[13] 23350 24110 20000 21340 22500 23185 19080 24320 17850 19990 16760 12460
[25] 18150
> west=scan('clipboard');west
Read 25 items
[1] 17740 13000 19550 18450 15050 18600 18000 17500 16900 21000 19990 14520
[13] 18400 15000 19750 17450 15575 13260 12375 19750 15560 17880 13450 16660
[25] 20000

> var.test(east,west)

F test to compare two variances

data: east and west
F = 2.095, num df = 24, denom df = 24, p-value = 0.0762
alternative hypothesis: true ratio of variances is not equal to 1
95 percent confidence interval:
0.9232212 4.7542380
sample estimates:
ratio of variances
2.095045

By using var.test we check equality of variances.

Hypothesis:

Since p-value is  0.0762 which is greater than 0.05, hence at 5% LOS we conclude that samples have equal variances.

95% and 90% confidence intervals for the difference in the mean tuition rates for universities both east and west of the Mississippi River:

Hypothesis:

95% confidence interval for difference in mean is (1606.625 , 5109.775). The interval does not contain the value zero, hence we reject null hypothesis at 5% level of significance and conclude that there is significant difference between mean tuition rates for universities both east and west of the Mississippi River.

We are 95% confident that the true mean difference between mean tuition rates for universities both east and west of the Mississippi River lie in the interval (1606.625 , 5109.775).

90% confidence interval for difference in mean is (1897.077 , 4819.323). The interval does not contain the value zero, hence we reject null hypothesis at 10% level of significance and conclude that there is significant difference between mean tuition rates for universities both east and west of the Mississippi River.

We are 90% confident that the true mean difference between mean tuition rates for universities both east and west of the Mississippi River lie in the interval  (1897.077 , 4819.323).