Question

On a recent analytics survey, the scores were normally distributed, with a mean of 76 and a standard deviation of 10. Letter grades for the survey were based on a classic scale, with 90-100 = A; 80-89 = B; 70-79 = C; 60-69 = D; below 69 = F. What percentage of the class would you expect to have earned an A?(a)

What percentage of the class would you expect to have earned a B?(b)

If you formed a sample from the four people on the front row, what is the probability that the mean of their scores was greater than 80?(c)

Answer 1

On a recent analytics survey, the scores were normally distributed with a mean of 76 and a standard deviation of 10.

Letter grades for the survey were based on a classic scale, with 90-100 giving grade A.

Let, X be the random variable denoting the score; it is given that X follows normal with mean 76, and a standard deviation of 10.

Now, we can say that Z=(X-76)/10 follows standard normal with mean 0 and standard deviation of 1.

Question (a)

We have to find the percentage of students who got an A.

So, basically we have to find

Now, this actually is

Where, phi is the distribution function of the standard normal variate.

From the standard normal table, this becomes

The corresponding percentage is 0.0726*100, ie. 7.26%.

The answer is

7.26% of the class are expected to get grade A.

Question (b)

Scores in the range 80-89 is assigned grade B.

Now, we have to find the percentage of the class that gets a B.

So, basically we have to find

This actually becomes

Where, phi is the distribution function of the standard normal variate.

From the standard normal table, this becomes

The corresponding percentage is 0.2478*100, ie. 24.78%.

The answer is

24.78% of the class are expected to get grade B.

Question (c)

A sample of 4 people were taken from the front row.

We have to find the probability that the mean of these 4 scores is greater than 80.

Now, let the mean of these 4 scores is denoted by

.

Then,

So, we can say that

Now, we have to find

Where, phi is the distribution function of the standard normal variate.

From the standard normal table, this becomes

The answer is

The probability that their mean scores is greater than 80 is 0.2119.