7. ACT scores are found to be normally distributed with a mean of 21 and a stand

Question

7. ACT scores are found to be normally distributed with a mean of 21 and a standard deviation of 4.5. Answer the following: (a) Determine the z-score for a person from this population that has a ACT score of 23. Then find the z-score for someone whose ACT score is 15 (b) If x represents a possible ACT score from this population, find P(x > 23). (c) Find P(20 < x < 25) and write a sentence for the interpretation of this value. (d) The top 10% of all people in this group have ACT scores high enough to earn a scholarship. Determine the ACT score which is high enough to earn the scholarship. 7. ACT scores are found to be normally distributed with a mean of 21 and a standard deviation of 4.5. Answer the following: (a) Determine the z-score for a person from this population that has a ACT score of 23. Then find the z-score for someone whose ACT score is 15 (b) If x represents a possible ACT score from this population, find P(x > 23). (c) Find P(20 < x < 25) and write a sentence for the interpretation of this value. (d) The top 10% of all people in this group have ACT scores high enough to earn a scholarship. Determine the ACT score which is high enough to earn the scholarship.

Explanation / Answer

Mean = 21

Standard deviation = 4.5

a) Z score = (X - mean)/standard deviation)

Z score for 23 = (23 - 21)/4.5 = 0.44

Z score for 15 = (15 - 21)/4.5 = -1.33

b) P(X > 23) = 1 - P(X < 23)

= 1 - P(Z < 0.44)

= 1 - 0.6700

= 0.3300

c) P(20 < X < 25) = P(X < 25) - P(X < 20)

= P(Z < (25-21)/4.5) - P(Z < (20-21)/4.5)

= P(Z < 0.89) - P(Z < -0.22)

= 0.8133 - 0.4129

= 0.4004

There is a probability of 0.4004 that a randomly selected ACT score is between 20 and 25

d) Let the ACT score above which the top 10% lies be A

P(X > A) = 0.1

So, P(X < A) = 1 - 0.1 = 0.9

P(Z < (A - 21)/4.5) = 0.9

From standard normal distribution table,

(A - 21)/4.5 = 1.28

A = 26.76

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