A humanities professor assigns letter grades on a test according to the following scheme.

A: Top 12%
 of scores

B: Scores below the top 12%
 and above the bottom 64%

C: Scores below the top 36%
 and above the bottom 23%

D: Scores below the top 77%
 and above the bottom 8%

F: Bottom 8%
 of scores

Scores on the test are normally distributed with a mean of 65.7
 and a standard deviation of 8.9
. Find the minimum score required for an A grade. Round your answer to the nearest whole number, if necessary.

To find the minimum score required for an A grade, we need to determine the score that corresponds to the top 12% of the distribution.

Since the scores are normally distributed, we can use the Z-score formula:

\[
Z = \frac{X - \mu}{\sigma}
\]

where:
- \( X \) is the score we need to find,
- \( \mu = 65.7 \) is the mean of the distribution,
- \( \sigma = 8.9 \) is the standard deviation.

### Step 1: Determine the Z-score for the Top 12%

The top 12% corresponds to the 88th percentile of the normal distribution. We look up the Z-score that corresponds to the 88th percentile, which is approximately:

\[
Z \approx 1.175
\]

### Step 2: Solve for \( X \) Using the Z-score Formula

Rearrange the Z-score formula to solve for \( X \):

\[
X = Z \cdot \sigma + \mu
\]

Substitute the values:

\[
X = 1.175 \cdot 8.9 + 65.7
\]

### Step 3: Calculate \( X \)

\[
X \approx 1.175 \times 8.9 + 65.7 \approx 10.4575 + 65.7 \approx 76.2
\]

Rounding to the nearest whole number, the minimum score required for an A grade is:

\[
\boxed{76}
\]

Would you like further explanation or have questions? 

---

Here are some related questions you might consider:
1. What score would be required to secure a B grade?
2. How do you find the score that corresponds to the bottom 8%?
3. How would you find the score threshold for a C grade?
4. What percentile corresponds to the score of 65.7?
5. How would changes in mean or standard deviation affect grade cutoffs?

**Tip:** Remember, percentiles indicate the percentage of scores below a specific value in a normal distribution.

A humanities professor assigns letter grades on a test according to the following scheme. A: Top 12% of scores, B: Scores below the top 12% and above the bottom 64%, C: Scores below the top 36% and above the bottom 23%, D: Scores below the top 77% and above the bottom 8%, F: Bottom 8% of scores. Scores on the test are normally distributed with a mean of 65.7 and a standard deviation of 8.9. Find the minimum score required for an A grade. Round your answer to the nearest whole number, if necessary.

Learn how to calculate the minimum score required for an A grade based on a normal distribution of test scores with a mean of 65.7 and a standard deviation of 8.9. This step-by-step solution covers Z-scores, percentiles, and normal distribution properties, ideal for college-level students.

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