Last week, Mrs. Thomas, the AP Statistics teacher at Green Valley High School, gave her students an assessment on material they had recently studied. The students' scores on the assessment followed an approximately normal distribution with mean 83.54 and standard deviation 11.21. Andrew, a student in Mrs. Thomas' class, had a standardized score of z = 0.88. What was Andrew's score on the test, and in which quarter of the distribution would Andrew's score fall?

We need to take in count that the first quarter accumulates 0.25 of the area on the left, the second quarter 0.5, the third 0.75 and the fourth 1.0 of the area on the left. Based on the value obtained for the probability of 0.811 we see that we are on the last quarter of the distribution or the fourth quarter.

Step-by-step explanation:

1) 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".

2) Solution to the problem

Let X the random variable that represent the scores of a population, and for this case we know the distribution for X is given by:

[tex]X \sim N(83.54,11.21)[/tex]

Where [tex]\mu=83.54[/tex] and [tex]\sigma=11.21[/tex]

And the best way to solve this problem is using the normal standard distribution and the z score given by:

[tex]z=\frac{x-\mu}{\sigma}[/tex]

If we use the formula we got:

[tex]0.88=\frac{X-83.54}{11.21}[/tex]

And if we solve for th value of X we got:

[tex]X=83.54 + 0.88*11.21=93.405[/tex]

So then the score for Andrew is 93.405.

In order to find on which quarter is the score we just need to find this probability:

[tex]P(Z<0.88)=0.811[/tex]

We need to take in count that the first quarter accumulates 0.25 of the area on the left, the second quarter 0.5, the third 0.75 and the fourth 1.0 of the area on the left. Based on the value obtained for the probability of 0.811 we see that we are on the last quarter of the distribution or the fourth quarter.