We want to compare the grades on an introductory biology exam for freshman versu

Question

We want to compare the grades on an introductory biology exam for freshman versus
sophomores. Suppose that 81% percent of the class are freshman, while the remaining
are sophomores. The exam grades for freshman were normally distributed, with a mean
of 81 and a standard deviation of 12. The exam grades for sophomores were also
normally distributed, and 30% of the entire class got an “A” on the test (scored at least
90).
a. What grade corresponds to the 65th percentile of the distribution of freshman exam
grades?
b. What proportion of sophomores got an “A” on the test?
c. If the standard deviation of exam grades for sophomores is the same as it is for
freshman, what is the mean graph sophomores got on the exam?

Explanation / Answer

ath ) here for 65th percentile ; zscore =0.3853

therefore corresponding score =mean +z*std deviation =81+0.3853*12 =85.62

b)  proportion of freshman got an “A” on the test =P(X>90)=P(Z>(90-81)/12)=P(Z>0.75)=0.2266

therefore proportion of sophomores got an “A” on the test

= (weighted proportion who got A-proportion of getting A and freshman)/proportion of sophomores

(0.3-0.81*0.2266)/0.19=0.6128

c)

for 0.6128 proprotion ; z score =-0.2866

therefore corresponding mean score =X-z*std deviation =90-(-0.2866)*12 =93.44

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