37. The scores on a statistics test are normally distributed with a mean of 77 and a standard deviation of 3. What percent of students would you expect to score between 77 and 83 ?

Step 1 ：The problem is asking for the percentage of students who scored between 77 and 83 on a statistics test. The scores are normally distributed with a mean of 77 and a standard deviation of 3.

Step 2 ：To solve this problem, we need to calculate the z-scores for 77 and 83, and then find the area under the normal distribution curve between these two z-scores. The z-score is a measure of how many standard deviations an element is from the mean.

Step 3 ：The formula for calculating the z-score is: \(z = \frac{X - \mu}{\sigma}\) where: \(X\) is the value we are interested in, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.

Step 4 ：After calculating the z-scores, we can use a z-table or a function to find the area under the curve between these two z-scores, which will give us the percentage of students who scored between 77 and 83.

Step 5 ：Given that the mean is 77, the standard deviation is 3, the lower limit is 77 and the upper limit is 83, the z-scores for these limits are 0.0 and 2.0 respectively.

Step 6 ：The percentage of students who scored between 77 and 83 on the test is approximately 47.72%. This is the area under the normal distribution curve between the z-scores of 0 (corresponding to a score of 77) and 2 (corresponding to a score of 83).

Step 7 ：Final Answer: The percentage of students expected to score between 77 and 83 is approximately \(\boxed{47.72\%}\).