(16 points) In a recent year, the ACT scores for the English portion of the test

Question

(16 points) In a recent year, the ACT scores for the English portion of the test were normally
distributed, with a mean of 20 and a standard deviation of six. A high school student who took
the English portion of the ACT is randomly selected

a) Find the probability that the student’s ACT score is less than 15?

b) Find the probability that the student’s ACT score is between 18 and 25?

c) Find the probability that the student’s ACT score is more than 34?

d) Can any of these events be considered unusual? Explain your reasoning?

e) Find the minimum of the top 5% scores?

f) What score represents the 80th percentile?

g) What score represents the third quartile?

Explanation / Answer

Ans:

mean=20 and standard deviation=6

a)

z=(15-20)/6=-0.833

P(z<-0.833)=0.2023

b)

z(18)=(18-20)/6=-0.333

z(25)=(25-20)/6=0.833

P(-0.333<z<0.833)=P(z<0.833)-P(z<-0.333)

=0.7977-0.3694=0.4282

c)

z=(34-20)/6=2.33

P(z>2.33)=0.0098

d)Yes,event that student’s ACT score is more than 34 is unusual,as Probability for this event is less than 0.05

e)

P(Z>=z)=0.05

z=1.645

minimum score=20+1.645*6=29.87

f)

P(Z<=z)=0.8

z=0.8416

P80=20+0.8416*6=25.05

g)

P(Z<=z)=0.75

z=0.6745

P75=20+0.6745*6=24.05

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