Question

Suppose your statistics professor reports test grades as​ z-scores, and you got a score of 2.49 on an exam. ​(a) Write a sentence explaining what that means. ​(b) Your friend got a​ z-score of negative 2−2. If the grades satisfy the Nearly Normal​ Condition, about what percent of the class scored lower than your​ friend? ​

Answer 1

Answer:

a) A z-score of 2.49 means that your grade was 2.49 standard deviations above the mean of all grades.

b) 1.39% of the class scored lower than your​ friend.

Step-by-step explanation:

Problems of normally distributed samples are 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.

You got a score of 2.49 on an exam. ​

(a) Write a sentence explaining what that means

A z-score of 2.49 means that your grade was 2.49 standard deviations above the mean of all grades.

(b) Your friend got a​ z-score of negative 2−2. If the grades satisfy the Nearly Normal​ Condition, about what percent of the class scored lower than your​ friend? ​

This percentage is the pvalue of $Z = -2.2$

So 0.0139 = 1.39% of the class scored lower than your​ friend.