A researcher was interested in comparing the GPAs of students at two different colleges. Independent simple random samples of 8 students from college A and 13 students from college B yielded the following GPAs.

College A

College B

3.7

3.8

2.8

3.2

3.2

4.0

3.0

3.0

3.6

2.5

3.9

2.6

2.7

3.8

4.0

3.6

2.5

3.6

2.8

3.9

3.4

Construct a 95% confidence interval for the difference between the mean GPA of college A students and the mean GPA of college B students.

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To calculate a confidence interval for the population mean (average), under these conditions, do the following: Determine the confidence
level and degrees of freedom and then find the appropriate t*-value. Find the sample mean. Multiply t* times s
and divide that by the square root of n

You estimate the population mean,

by using a sample mean,

plus or minus a margin of error. The result is called a confidence interval for the population mean,

In many situations, you don’t know

so you estimate it with the sample standard deviation, s; and/or
the sample size is small (less than 30), and you can’t be sure your
data came from a normal distribution. (In the latter case, the Central
Limit Theorem can’t be used.) In either situation, you can’t use a z*-value from the standard normal (Z-) distribution as your critical value anymore; you have to use a larger critical value than that, because of not knowing what

is and/or having less data.

The formula for a confidence interval for one population mean in this case

is the critical t*-value from the t-distribution with n – 1 degrees of freedom (where n is the sample size).

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