Question

A professor has found that the grades on the Statistics Final are normally distributed with a mean of 68 and a standard deviation of 15. If only the best 14 % of the students in the class will receive an A, what grade must a student obtain in order to get an A?

Answer 1

Answer:

A student must obtain a grade of at least 84.2 in order to get an A.

Step-by-step explanation:

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

$\mu = 68, \sigma = 15$

If only the best 14 % of the students in the class will receive an A, what grade must a student obtain in order to get an A?

This is the value of X when Z is in the (100-14) = 86th percentile.

So it is the value of X when $Z = 1.08$, and higher values of X. So

$Z = \frac{X - \mu}{\sigma}$

$1.08 = \frac{X - 68}{15}$

$X - 68 = 15*1.08$

$X = 84.2$

A student must obtain a grade of at least 84.2 in order to get an A.