Step 1 :The problem is asking for the probability that the average of a random sample of 38 test scores is greater than 75. The scores are normally distributed with a mean of 76 and a standard deviation of 16.
Step 2 :We can use the Central Limit Theorem to solve this problem. The Central Limit Theorem states that the distribution of sample means approximates a normal distribution as the sample size gets larger, regardless of the shape of the population distribution.
Step 3 :We calculate the z-score for the sample mean of 75. The z-score is the number of standard deviations the sample mean is from the population mean. The formula for the z-score is: \(z = \frac{X - \mu}{\sigma / \sqrt{n}}\), where \(X\) is the sample mean, \(\mu\) is the population mean, \(\sigma\) is the standard deviation, and \(n\) is the sample size.
Step 4 :Substituting the given values into the formula, we get: \(z = \frac{75 - 76}{16 / \sqrt{38}} = -0.385\)
Step 5 :We then use the standard normal distribution (z-distribution) to find the probability that a z-score is greater than the calculated value. The probability is approximately 0.65.
Step 6 :Thus, the probability that a random sample of 38 test scores has an average greater than 75 is approximately 0.65.
Step 7 :Final Answer: \(\boxed{0.65}\)