Question

The average annual cost (including tuition, room, board, books and fees) to attend a public college takes nearly a third of the annual income of a typical family with college-age children (Money, April 2012). At private colleges, the average annual cost is equal to about 60% of the typical family's income. The following random samples show the annual cost of attending private and public colleges. Data are in thousands of dollars. Click on the datafile logo to reference the data. Round degrees of freedom to the preceding whole number.

a. Compute the sample mean and sample standard deviation for private and public colleges. Round your answers to two decimal places.

b. What is the point estimate of the difference between the two population means? Round your answer to one decimal place.

Interpret this value in terms of the annual cost of attending private and public colleges.

The mean annual cost to attend private colleges is $________ more than the mean annual cost to attend public colleges.

c. Develop a 95% confidence interval of the difference between the annual cost of attending private and pubic colleges.

95% confidence interval, private colleges have a population mean annual cost $______ to $_______ more

expensive than public colleges.

Answer 1

The sample size is n = 10.

	private college	private college2
	52.8	2787.84
	43.2	1866.24
	45.0	2025
	33.3	1108.89
	44.	1936
	30.6	936.36
	45.8	2097.64
	37.8	1428.84
	50.5	2550.25
	42	1764
Sum =	425	18501.06

The sample mean Xˉ is computed as follows:

Also, the sample variance s^2 is

Therefore, the sample standard deviation s is

The sample size is n = 12.

	X	X2
	20.3	412.09
	22.0	484
	28.2	795.24
	15.6	243.36
	24.1	580.81
	28.5	812.25
	22.8	519.84
	25.8	665.64
	18.5	342.25
	25.6	655.36
	14.4	207.36
	21.8	475.24
Sum =	267.6	6193.44

The sample mean Xˉ is computed as follows:

Also, the sample variance s^2 is

\

Therefore, the sample standard deviation ss is

b)

X ́ 1 − ́X 2 = 42.5−22.3

= 20.2 or $20,200.

The mean annual cost to attend private college is $20,200 more than the mean annual cost to attend public colleges.+

c).

We need to construct the 95% confidence interval for the difference between the population means μ1​−μ2​,

Sample Mean 1 (Xˉ1​) =	42.5
Sample Standard Deviation 1 (s1​) =	6.98
Sample Size 1 (N1​) =	10
Sample Mean 2 (Xˉ2​) =	22.3
Sample Standard Deviation 2 (s2​) =	4.53
Sample Size 2 (N2​) =	12

Based on the information provided, we assume that the population variances are unequal, so then the number of degrees of freedom is computed as follows:

The critical value for α=0.05 and df = 16.182

the corresponding confidence interval is computed as shown below:

Since we assume that the population variances are unequal, the standard error is computed as follows:

Now, we finally compute the confidence interval:

Therefore, based on the data provided, the 95% confidence interval for the difference between the population means μ1​−μ2​ is 14.766<μ1​−μ2​<25.634, which indicates that we are 95% confident that the true difference between population means is contained by the interval (14.766, 25.634)

This means that at 95% confidence interval, private colleges have a population mean annual cost of $14,766 to $25,634 more expensive than public colleges.

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