The GPA at a particular school has an average of 2.89 with a standard deviation of 0.63. A random sample of 38 students from that school is collected. Find the probability that the average GPA for this sample is greater than 3.0.

Answer 1

To solve this problem, we will use the Central Limit Theorem, which states that the sampling distribution of the sample mean approaches a normal distribution with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size.

First, we calculate the standard error of the sample mean:
standard error = standard deviation / sqrt(sample size)
standard error = 0.63 / sqrt(38) = 0.102

Next, we calculate the z-score for the sample mean:
z = (sample mean - population mean) / standard error
z = (3.0 - 2.89) / 0.102 = 1.078

Using a standard normal distribution table or calculator, we can find the probability associated with a z-score of 1.078. This probability represents the area under the curve to the right of the z-score.

The probability that the average GPA for this sample is greater than 3.0 is approximately 0.1401, or 14.01%.