Question

In a recent year, students taking a mathematics assessment test had a mean of 290 and a standard deviation of 37. Possible test scores could range from 0 to 500. (hint: use 0 as the lower limit and 500 as the upper limit) [4 pts. Each] a) Find the probability that a student had a score less than 320. b) Find the probability that a student had a score between 250 and 300. c) What percent of the students had a test score greater than 200? d) What is the lowest score that would still place a student in the top 5% of the scores? e) What is the highest score that would still place a student in the bottom 25% of the scores

Answer 1

a) 79.10% probability that a student had a score less than 320.

b) 46.63% probability that a student had a score between 250 and 300.

c) 99.25% of the students had a test score greater than 200

d) 350.865

e) 265.025

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 = (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 = 290, \sigma = 37`

a) Find the probability that a student had a score less than 320.

This is the pvalue of Z when X = 320. So

`Z = (X - \mu)/(\sigma)`

`Z = (320 - 290)/(37)`

`Z = 0.81`

`Z = 0.81` has a pvalue of 0.7910

79.10% probability that a student had a score less than 320.

b) Find the probability that a student had a score between 250 and 300.

This is the pvalue of Z when X = 300 subtracted by the pvalue of Z when X = 250.

X = 300

`Z = (X - \mu)/(\sigma)`

`Z = (300 - 290)/(37)`

`Z = 0.27`

`Z = 0.27` has a pvalue of 0.6064

X = 250

`Z = (X - \mu)/(\sigma)`

`Z = (250 - 290)/(37)`

`Z = -1.08`

`Z = -1.08` has a pvalue of 0.1401

0.6064 - 0.1401 = 0.4663

46.63% probability that a student had a score between 250 and 300.

c) What percent of the students had a test score greater than 200?

This is 1 subtracted by the pvalue of Z when X = 200. So

`Z = (X - \mu)/(\sigma)`

`Z = (200 - 290)/(37)`

`Z = -2.43`

`Z = -2.43` has a pvalue of 0.0075

1 - 0.0075 = 0.9925

99.25% of the students had a test score greater than 200

d) What is the lowest score that would still place a student in the top 5% of the scores?

X when Z has a pvalue of 1-0.05 = 0.95. So X when Z = 1.645.

`Z = (X - \mu)/(\sigma)`

`1.645 = (X - 290)/(37)`

`X - 290 = 37*1.645`

`X = 350.865`

e) What is the highest score that would still place a student in the bottom 25% of the scores

X when Z has a pvalue of 0.25. So X when Z = -0.675

`Z = (X - \mu)/(\sigma)`

`-0.675 = (X - 290)/(37)`

`X - 290 = 37*(-0.675)`

`X = 265.025`