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

Answer:

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

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.

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

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

$Z = \frac{320 - 290}{37}$

$Z = 0.81$

$Z = 0.81$ 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

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

$Z = \frac{300 - 290}{37}$

$Z = 0.27$

$Z = 0.27$ has a pvalue of 0.6064

X = 250

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

$Z = \frac{250 - 290}{37}$

$Z = -1.08$

$Z = -1.08$ 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

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

$Z = \frac{200 - 290}{37}$

$Z = -2.43$

$Z = -2.43$ 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.

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

$1.645 = \frac{X - 290}{37}$

$X - 290 = 37*1.645$

$X = 350.865$

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

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

$-0.675 = \frac{X - 290}{37}$

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

$X = 265.025$