submit test A teacher gives a test to a large group of students. The results are closely approximated by a normal curve. The mean is 77 with a standard deviation of 6 . The teacher wishes to give A's to the top $8 \%$ of the students and F's to the bottom $8 \%$. A grade of B is given to the next $15 \%$, with D's given similarly. All other students "get Cs. Find the bottom cutoff (rounded to the nearest whole number) for a B. (Hint Use a table of areas under the standard normal curve to find $z$-scores trom known A-values.) The bottom cutoff for a B is $\square$. (Simplify your answer. Round to the nearest whole number as needed.)

Step 1 ：Given that the results of the test are closely approximated by a normal curve, with a mean of 77 and a standard deviation of 6.

Step 2 ：The teacher wishes to give A's to the top $8 \%$ of the students and F's to the bottom $8 \%$. A grade of B is given to the next $15 \%$, with D's given similarly. All other students get Cs.

Step 3 ：We are asked to find the bottom cutoff for a B grade.

Step 4 ：Using a table of areas under the standard normal curve, we find that the $z$-score for the bottom $8 \%$ plus $15 \%$ is $-0.74$.

Step 5 ：We can use the formula for the $z$-score, which is $Z = \frac{X - \mu}{\sigma}$, where $X$ is the score, $\mu$ is the mean, and $\sigma$ is the standard deviation.

Step 6 ：Rearranging the formula, we get $X = Z \cdot \sigma + \mu$.

Step 7 ：Substituting the given values, we get $X = -0.74 \cdot 6 + 77$, which simplifies to $X = 72.56$.

Step 8 ：However, we are asked to round the answer to the nearest whole number. Therefore, the bottom cutoff for a B grade is 73.

Step 9 ：Final Answer: The bottom cutoff for a B is \(\boxed{73}\).