MATH SOLVE

4 months ago

Q:
# There are two college entrance exams that are often taken by students, Exam A and Exam B. The composite score on Exam A is approximately normally distributed with mean 20.1 and standard deviation 5.1. The composite score on Exam B is approximately normally distributed with mean 1031 and standard deviation 215. Suppose you scored 24 on Exam A and 1167 on Exam B. Which exam did you score better on? Justify your reasoning using the normal model.A.The score on Exam B is better, because the score is higher than the score for Exam A. B.The score on Exam A is better, because the percentile for the Exam A score is higher. C.The score on Exam A is better, because the difference between the score and the mean is lower than it is for Exam B. D.The score on Exam B is better, because the percentile for the Exam B score is higher.

Accepted Solution

A:

Answer:B.The score on Exam A is better, because the percentile for the Exam A score is higher. Step-by-step explanation:Problems of normally distributed samples can be solved using the z-score formula.In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:[tex]Z = \frac{X - \mu}{\sigma}[/tex]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:Two exams. The exam that you did score better is the one in which you had a higher zscore.The composite score on Exam A is approximately normally distributed with mean 20.1 and standard deviation 5.1.This means that [tex]\mu = 20.1, \sigma = 5.1[/tex].You scored 24 on Exam A. So[tex]Z = \frac{X - \mu}{\sigma}[/tex][tex]Z = \frac{24 - 20.1}{5.1}[/tex][tex]Z = 0.76[/tex]The composite score on Exam B is approximately normally distributed with mean 1031 and standard deviation 215.This means that [tex]\mu = 1031, \sigma = 215[/tex].You scored 1167 on Exam B, s:[tex]Z = \frac{X - \mu}{\sigma}[/tex][tex]Z = \frac{1167 - 1031}{215}[/tex][tex]Z = 0.632[/tex]You had a better Z-score on exam A, so you did better on that exam.The correct answer is:B.The score on Exam A is better, because the percentile for the Exam A score is higher.