A professor at a local college noted that the grades of his students were normally distributed with an average percentage of 74 and a population standard deviation of 10. He found that 6.3% of his students received A's while 2.5% of his students failed the course and received F's. Based upon the professor's grading scale, what was the minimun percentage needed to get an A in the course?

Question

A professor at a local college noted that the grades of his students were normally distributed with an average percentage of 74 and a population standard deviation of 10. He found that 6.3% of his students received A's while 2.5% of his students failed the course and received F's. Based upon the professor's grading scale, what was the minimun percentage needed to get an A in the course?

Answer 1

Z = (score-mean)/SD

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the Z score related to your proportion. Put that value along with the values of mean and SD to find the score.

Answer 2

To determine the minimum percentage needed to get an A in the course, we need to use the concept of z-scores.

A z-score measures how many standard deviations a data point is above or below the mean. In this case, we want to find the z-score that corresponds to the A grade, which is the cutoff point.

To do this, we need to convert the given percentages (6.3% and 2.5%) into their respective z-scores.

First, let's find the z-score for the upper cutoff point of 6.3%. We need to find the percentage of students scoring below this cutoff point, which is 100% - 6.3% = 93.7%.

The formula to calculate the z-score is:
z = (x - μ) / σ

Where:
x = the score we want to convert to a z-score
μ = the mean of the distribution
σ = the standard deviation of the distribution

In this case, x = 93.7, μ = 74, and σ = 10.

Let's plug in the values:
z = (93.7 - 74) / 10
z ≈ 1.97

Now, we need to find the z-score for the lower cutoff point of 2.5%. We need to find the percentage of students scoring below this cutoff point, which is 100% - 2.5% = 97.5%.

Using the same formula:
z = (x - μ) / σ

In this case, x = 97.5, μ = 74, and σ = 10.

Let's plug in the values:
z = (97.5 - 74) / 10
z ≈ 2.45

Now that we have the z-scores for the upper and lower cutoff points, we need to find the raw score corresponding to the z-score for the upper cutoff point. We can do this by rearranging the formula for the z-score:

x = z * σ + μ

Plugging in the values:
x = 1.97 * 10 + 74
x ≈ 95.7

So, a minimum percentage of approximately 95.7% is needed to get an A in the course.