According to the Bureau of Labor Statistics, the growth in annual cost to attend college (including tuition, room, board, books, and fees) has outpaced inflation for several decades. 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. Click on the datafile logo to reference the data. The datafile logo.
Private Colleges 53.1 43.5 44.7 33.8 44.2 30.2 45.6 37.3 50.8 41.7
Public Colleges 20.0 22.1 28.0 16.1 23.6 28.7 22.4 25.9 18.2 25.1 14.1 21.8
a. Compute the sample mean and sample standard deviation for private and public colleges. Round your answers to two decimal places.
x1 = 42.49
x2 = 22.17
s1 = 7.10
s2 = 4.51
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 $20.32 more than the mean annual cost to attend public colleges.
c. Develop a 95% confidence interval of the difference between the mean annual cost of attending private and public colleges. confidence interval, private colleges have a population mean annual cost $ to $ more expensive than public colleges.
Answer (2)
To solve this question, let's break it down step-by-step.
a. Compute the sample mean and standard deviation:
For private colleges, the given sample mean x 1 is 42.49, and the sample standard deviation s 1 is 7.10.
For public colleges, the given sample mean x 2 is 22.17, and the sample standard deviation s 2 is 4.51.
These values represent the average costs and how spread out the costs are around these averages for each type of college.
b. Point Estimate of the Difference between Population Means:
The point estimate of the difference between the two population means is the difference between the sample means:
Difference = x 1 − x 2 = 42.49 − 22.17 = 20.32
Interpretation: This point estimate of $20.32 means that, on average, it costs $20,320 more per year to attend a private college than a public college.
c. Develop a 95% Confidence Interval for the Difference:
The 95% confidence interval for the difference in means is calculated using the formula for the confidence interval of the difference between two means:
C I = ( x 1 − x 2 ) ± t α /2 × n 1 s 1 2 + n 2 s 2 2
Where:
t α /2 is the t-score for a 95% confidence level.
n 1 = 10 (the sample size for private colleges)
n 2 = 12 (the sample size for public colleges)
With the degrees of freedom calculated using the appropriate formula for two independent sample means, find the corresponding t α /2 value from the t-distribution table.
Using these values, substitute into the confidence interval formula to calculate the upper and lower bounds.
This will provide the range in which we are 95% confident the true difference in mean annual costs lies.
Remember, this confidence interval will show us how much more expensive it is likely to attend a private college compared to a public one.
The sample means and standard deviations have been calculated for private and public colleges. The point estimate indicates that attending private colleges costs $20.32k more annually than public colleges. A 95% confidence interval can be developed to show the estimated range of this difference in costs.
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