Question

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 than15?

b) Find the probability that the student’s ACT score is between 18 and25?

c) Find the probability that the student’s ACT score is more than34?

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

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

f) What score represents the 80th percentile?

g) What score represents the third quartile?

Answer 1

P(X < A) = P(Z < (A - mean)standard deviation)

Mean = 20

Standard deviation = 6

(a) P(the student’s ACT score is less than 15) = P(X < 15)

= P(Z < (15 - 20)/6)

= P(Z < -0.83)

= 0.2033

(b) P( the student’s ACT score is between 18 and 25) = P(18 < X < 25)

= P(X < 25) - P(X < 18)

= P(Z < (25 - 20)/6) - P(Z < (18 - 20)/6)

= P(Z < 0.83) - P(Z < -0.33)

= 0.7967 - 0.3707

= 0.4260

(c) P(the student’s ACT score is more than 34) = P(X > 34)

= 1 - P(X < 34)

= 1 - P(Z < (34 - 20)/6)

= 1 - P(Z < 2.33)

= 1 - 0.9901

= 0.0099

(d) Event in part (c) is unusual because the probability of the event is less than 0.05