Question

3. The scores for all students taking the ACT test in a recent year are normally distributed with a mean of 21.0 and a standard deviation of 4.7.

1. If a student who takes the ACT is randomly selected, find the probability that their score is above 23.
2. If five ACT scores are randomly selected from the population of students who took the test, find the probability that all five of the scores are above 23. (Remember the Multiplication Rule from Chap 4)
3. If five students who took the ACT are randomly selected, find the probability that their mean score is above 23.
4. Find P₉₀, the score separating the bottom 90% from the top 10%.

Answer 1

For normal distribution, P(X < A) = P(Z < (A - )/)

= 21

= 4.7

a) P(score is above 23) = P(X > 23)

= 1 - P(X < 23)

= 1 - P(Z < (23 - 21)/4.7)

= 1 - P(Z < 0.43)

= 1 - 0.6664

= 0.3336

b) P(all 5 scores are above 23) = 0.3336⁵

= 0.0041

c) For sampling distribution of mean, P( < A) = P(Z < (A - )/)

= = 21

=

=

= 2.10

P(scor90 - e is above 23) = P( > 23)

= 1 - P( < 23)

= 1 - P(Z < (23 - 21)/2.1)

= 1 - P(Z < 0.95)

= 1 - 0.8289

= 0.1711

d) P(X < P90) = 0.90

P(Z < (P90 - 21)/4.7) = 0.90

Take z scores corresponding to probability of 0.90 from standard normal distribution table

(P90 - 21)/4.7 = 1.28

P90 = 27.02