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 $A, a B, a C$, 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 anstudent for a B is (Round up to the nearest integer as needed.)

Step 1 ：The problem is asking for the lowest score that would qualify a student for a B. This corresponds to the top 30% of scores, because the top 10% get A's and the next 20% get B's.

Step 2 ：To find this, we need to find the z-score that corresponds to the top 30% of a standard normal distribution. The z-score is a measure of how many standard deviations an element is from the mean.

Step 3 ：Once we have the z-score, we can use the formula for a z-score to find the corresponding score in the class. The formula is: \(z = \frac{X - \mu}{\sigma}\) where: \(z\) is the z-score, \(X\) is the score in the class, \(\mu\) is the mean score in the class, and \(\sigma\) is the standard deviation of the scores in the class.

Step 4 ：We can rearrange this formula to solve for X: \(X = z\sigma + \mu\)

Step 5 ：We know that \(\mu = 72\) and \(\sigma = 8\). We need to find the z-score that corresponds to the top 30% of a standard normal distribution.

Step 6 ：Using the z-score table, we find that the z-score that corresponds to the top 30% of a standard normal distribution is approximately 0.5244.

Step 7 ：Substituting the values into the formula, we get: \(X = 0.5244 * 8 + 72\)

Step 8 ：Solving for X, we get: \(X = 76.19520410166433\)

Step 9 ：Rounding up to the nearest integer, the lowest score that would qualify a student for a B is 77.

Step 10 ：Final Answer: The lowest score that would qualify a student for a B is \(\boxed{77}\)