MATH SOLVE

2 months ago

Q:
# The grades received by 200 students follow a normal distribution. The mean of the grades is 70%, and the standard deviation is 7%. The number of students who received a grade greater than 70% is about ___, and the number of students who got a grade higher than 84% is about ___.

Accepted Solution

A:

It may help you to consider the standard normal curve. If the mean here is 70%, then 0.5 of the area under this curve. In other words, half the students received a grade > 70%. That would be 100 students.

Note that 84 is 2 std dev above the mean.

Recall the Empirical Rule: 95% of scores lie within 2 std. dev of the mean.

Measuring from the mean, that would be 47.5% above the mean and 47.5% below the mean. The area to the right of the mean is 0.5. Subtract 0.475 from that to obtain the fraction of students who rec'd a grade greater than 2 std dev from the mean: It's 0.025.

Then the # of students who rec'd a grade greater than 84% would be

0.025(200 students) = 5 students.

Note that 84 is 2 std dev above the mean.

Recall the Empirical Rule: 95% of scores lie within 2 std. dev of the mean.

Measuring from the mean, that would be 47.5% above the mean and 47.5% below the mean. The area to the right of the mean is 0.5. Subtract 0.475 from that to obtain the fraction of students who rec'd a grade greater than 2 std dev from the mean: It's 0.025.

Then the # of students who rec'd a grade greater than 84% would be

0.025(200 students) = 5 students.