Question

The article "First Year Academic Success: A Prediction Combining Cognitive and Psychosocial Variables for Caucasian and African American Students" (Journal of College Student Development \([1999]: 599-610\) ) reported that the sample mean and standard deviation for high school grade point average (GPA) for students enrolled at a large research university were \(3.73\) and \(0.45\), respectively. Suppose that the mean and standard deviation were based on a random sample of 900 students at the university. a. Construct a \(95 \%\) confidence interval for the mean high school GPA for students at this university. b. Suppose that you wanted to make a statement about the range of GPAs for students at this university. Is it reasonable to say that \(95 \%\) of the students at the university have GPAs in the interval you computed in Part (a)? Explain.

Short answer

a. The 95% confidence interval for the mean high school GPA for students at this university is estimated to be between 3.70 and 3.76. b. It is not reasonable to say that 95% of the students at the university have GPAs in the interval computed in Part (a). This is because the calculated confidence interval reflects where we believe the true mean GPA lies, not the individual students' GPA.

Step-by-step solution

Calculate the Confidence Interval

First, calculate the standard error. The standard error (\( SE \)) is given by: \[ SE = \frac{{\text{{Standard deviation}}}}{{\sqrt{{\text{{Sample size}}}} }} \]where the standard deviation (SD) is \( 0.45 \) and the sample size (N) is 900. So the standard error is \[ SE = \frac{{0.45}}{{\sqrt{{900}} }} = 0.015 \]For a 95% confidence level, the Z-value is approximately 1.96 (You can find this value in Z-table or online).Now, calculate the confidence interval using the given formula:\[\text{{Confidence Interval}} = \text{{Sample mean}} \pm (z \times SE ) = 3.73 \pm (1.96 \times 0.015)\]This gives the 95% Confidence Interval as \( (3.73 - 0.0294 , 3.73 + 0.0294) = (3.70, 3.76) \).

Interpretation and Discussion of Part (b)

To answer part (b), it's crucial to understand that even though the Confidence Interval has been calculated to be 95%, the interpretation of this does not mean that 95% of all students have a GPA falling within this range. The Confidence Interval is a reflection of where we expect the true mean GPA of the entire student population to fall, given our sample data, not the individual GPAs. Hence it is not reasonable to state that 95% of the students at the university have GPAs in the interval computed in Part (a).