A humarities professor assigns letter grades on a test according to the following scheme.
A Top $12 \%$ of scores
B. Scares below the top $12 \%$ and above the bortom $61 \%$
c scores below the top $39 \%$ and above the bottom $21 \%$
D. Scores below the top $79 \%$ and above the bottom $6 \%$
F. Bottom $6 \%$ of scores
Scores on the test are normally distributed with a mean of 67.7 and a standard deviation of 7.8. Find the minimum score required for an A grade. Round your answer to the nearest whole number, if necessary.

Step 1 ：The problem is asking for the minimum score required for an A grade. According to the grading scheme, an A grade is given to the top 12% of scores. Since the scores are normally distributed, we can use the properties of the normal distribution to find the score corresponding to the top 12%.

Step 2 ：The z-score corresponding to the top 12% can be found using a z-table or a statistical function. The z-score tells us how many standard deviations away from the mean a particular score is. Once we have the z-score, we can use the formula for a z-score to find the corresponding score.

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

Step 4 ：We can rearrange this formula to solve for X: \(X = z\sigma + \mu\). We can use this formula to find the minimum score required for an A grade.

Step 5 ：Given that the mean is 67.7, the standard deviation is 7.8, and the z-score is 1.1749867920660904, we can substitute these values into the formula to find the score.

Step 6 ：Doing the calculation gives us a score of 77.

Step 7 ：Final Answer: The minimum score required for an A grade is \(\boxed{77}\).