Question

Algebra 2: A sample of 500 high school student were asked what their GPAS were at the end of their sophomore year. The mean GPA was 3.2 and the standard deviation was 0.25. You can be 95% confident that the mean GPA of all students at the high school is between what values?

Answer 1

The 95% confidence interval for the mean GPA of all high school students, based on a sample mean of 3.2, standard deviation of 0.25, and sample size of 500, is between 3.178 and 3.222.

Step-by-step explanation:

To determine the 95% confidence interval for the mean GPA of all high school students based on the sample, we use the formula for a confidence interval for a population mean. This is given as:

Mean ± (Z-score * (Standard Deviation / sqrt(Sample Size)))

In this case, for a 95% confidence level with a large sample size (>30), we use a Z-score that corresponds to the standard normal distribution's critical value, which is approximately 1.96 for a two-tailed test.

Using the values provided:

• Sample Mean (X): 3.2
• Standard Deviation (s): 0.25
• Sample Size (n): 500

Confidence Interval = 3.2 ± (1.96 * (0.25 / sqrt(500)))

The margin of error = 1.96 * (0.25 / sqrt(500)) = 0.022

Therefore, the 95% confidence interval is 3.2 ± 0.022, which means we can be confident that the mean GPA is between 3.178 and 3.222.