Question

Suppose your statistics professor reports test grades as​ z-scores, and you got a score of 1.57 on an exam. ​a) Write a sentence explaining what that means. ​b) Your friend got a​ z-score of negative 2. If the grades satisfy the Nearly Normal​ Condition, about what percent of the class scored lower than your​ friend?

Answer 1

Answer:

a) For this case we have a z score of 1.57, we need to remember that the z score is defined as:

$z =\frac{x -\mu}{\sigma}$

And this value represent that our score is 1.57 deviations above the mean of all the test grades scores

b) $P(Z<-2)$

And using the normal standard distribution or excel we got:

$P(Z<-2) = 0.02275$

And that represent 2.275% of the data.

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Solution to the problem

Part a

For this case we have a z score of 1.57, we need to remember that the z score is defined as:

$z =\frac{x -\mu}{\sigma}$

And this value represent that our score is 1.57 deviations above the mean of all the test grades scores

Part b

For this case a z score of z=-2 represent that the score for your friend is 2 deviation below all the other test scores.

And if we assume that the distribution is normal we can find the following probability:

$P(Z<-2)$

And using the normal standard distribution or excel we got:

$P(Z<-2) = 0.02275$

And that represent 2.275% of the data.