The scores on an Economics examination are normally distributed with a mean of 77 and a standard deviation of 18. If the instructor assigns a grade of A to 16% of the class, what is the lowest score a student may have and still obtain an A? (Give your answer to two decimal places.)

Question

The scores on an Economics examination are normally distributed with a mean of 77 and a standard deviation of 18. If the instructor assigns a grade of A to 16% of the class, what is the lowest score a student may have and still obtain an A? (Give your answer to two decimal places.)

Answer 1

This type of question can be answered using a table or website of the normal-distribution function.

Using (Broken Link Removed)
I get 95 to be the cutoff grade for an A

Answer 2

To find the lowest score a student may have and still obtain an A, we need to identify the z-score corresponding to the top 16% of the distribution.

Step 1: Standardize the score
The formula for calculating the z-score is:
z = (x - μ) / σ

Where:
- x is the raw score
- μ is the mean of the distribution
- σ is the standard deviation of the distribution

Step 2: Find the z-score corresponding to the top 16%
Since we want to find the z-score for the top 16% of the distribution (which is the same as the z-score for the lowest score to obtain an A), we need to find the z-score that corresponds to the cumulative area of 0.84 (100% - 16%).

Step 3: Use the z-score to find the raw score
After finding the z-score, we can use it to calculate the raw score (x) using the formula:
x = z * σ + μ

Now let's calculate the z-score and the lowest score a student may have to obtain an A:

Step 1: Standardize the score
We're looking for the lowest score to obtain an A, so we need to find the z-score for the top 16% of the distribution. This is equivalent to finding the z-score for the cumulative area of 0.84.

Step 2: Find the z-score corresponding to the top 16%
Using the standard normal distribution table or a calculator, we find that the z-score for a cumulative area of 0.84 is approximately 0.97.

Step 3: Use the z-score to find the raw score
Now we can calculate the lowest score a student may have to obtain an A using the formula:
x = z * σ + μ

x = 0.97 * 18 + 77
x ≈ 17.46 + 77
x ≈ 94.46

Therefore, the lowest score a student may have and still obtain an A is approximately 94.46 (rounded to two decimal places).