Question

2 multiple answer 1 point a high school randomly selected 75 of the 200 seniors at the school to take a sample college entrance exam. the mean grade point average (gpa) of the seniors selected was 2.85, and the standard deviation was 0.4. select all of the statements that are true. (note: margin of error ≈ 2*(s/√n), where s is the standard deviation and n is the sample size.) the margin of error for the mean grade point average is about 0.092. the margin of error would increase if the sample size were changed to 125 seniors. the margin of error for the mean grade point average is about 0.057. the margin of error would decrease if the sample size were changed to 125 seniors.

Explanation:

Step1: Calculate Margin of Error for n=75

Given \( s = 0.4 \), \( n = 75 \). Use formula \( \text{Margin of error} \approx 2\frac{s}{\sqrt{n}} \).
First, compute \( \sqrt{75} \approx 8.660 \). Then, \( \frac{0.4}{8.660} \approx 0.0462 \). Multiply by 2: \( 2 \times 0.0462 \approx 0.092 \). So first statement is true, third is false.

Step2: Analyze Sample Size Effect

Margin of error formula: \( 2\frac{s}{\sqrt{n}} \). As \( n \) increases, \( \sqrt{n} \) increases, so \( \frac{s}{\sqrt{n}} \) decreases, hence margin of error decreases. If \( n \) changes from 75 to 125 (increase), margin of error decreases. So second statement (increase) is false, fourth (decrease) is true.

Answer:

The margin of error for the mean grade point average is about 0.092, The margin of error would decrease if the sample size were changed to 125 seniors.