In a large section of a statistics class, the points for the final exam are normally distributed, with a mean of 72 and a standard deviation of 8 . Grades are assigned such that the top $10 \%$ receive A's, the next $20 \%$ received B's, the middle $40 \%$ receive C's, the next $20 \%$ receive D's, and the bottom $10 \%$ receive F's. Find the lowest score on the final exam that would qualify a student for an $A, a B, a$, and a $D$. Click here to view Page 1 of the Standard Normal Table. Click here to view Page 2 of the Standard Normal Table. The lowest score that would qualify a student for an A is 83 (Round up to the nearest integer as needed.) The lowest score that would qualify a student for a B is 77 (Round up to the nearest integer as needed.) The lowest score that would qualify a student for a C is (Round up to the nearest integer as needed.)

Step 1 ：The problem is asking for the lowest scores that would qualify a student for an A, B, C, and D. This is a problem of finding percentiles in a normal distribution. The percentiles for A, B, C, and D correspond to the top 10%, the next 20%, the middle 40%, and the next 20% respectively.

Step 2 ：We can use the z-score formula to find these percentiles. The z-score formula is: \(z = \frac{X - \mu}{\sigma}\) where X is the score, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. We can rearrange this formula to solve for X: \(X = z\sigma + \mu\)

Step 3 ：The z-scores for the top 10%, the next 20%, the middle 40%, and the next 20% are approximately 1.28, 0.84, 0, and -0.84 respectively. We can substitute these z-scores, the given mean of 72, and the given standard deviation of 8 into the formula to find the lowest scores that would qualify a student for an A, B, C, and D.

Step 4 ：The calculated lowest scores that would qualify a student for an A, B, C, and D are approximately 82, 76, 72, and 68 respectively. However, the problem asks for the scores to be rounded up to the nearest integer as needed. Therefore, we need to adjust the scores to meet this requirement.

Step 5 ：Final Answer: The lowest scores that would qualify a student for an A, B, C, and D are \(\boxed{82}\), \(\boxed{76}\), \(\boxed{72}\), and \(\boxed{68}\) respectively.