Question

The scores of individual students on the American College Testing (ACT) composite college entrance examination have a normal distribution with mean 17.3 and standard deviation 5.4.

(a) What is the probability that a single student randomly chosen from all those taking the test scores 25 or higher?


(b) Now take an SRS of 67 students who took the test. What are the mean and standard deviation of the average (sample mean) score for the 67 students?

μ =
σ = Do your results depend on the fact that individual scores have a normal distribution? yesno      (c) What is the probability that the mean score x of these students is 19.47 or higher?

Answer 1

Solution :

Given that ,

mean = = 17.3

standard deviation = = 5.4

a) P(x 25 ) = 1 - P(x   25 )

= 1 - P[(x - ) / (25-17.3) /5.4 ]

= 1 -  P(z 1.43 )  

= 1 - 0.9236 = 0.0764

Probability = 0.0764

b)

n = 67

= = 17.3

= / n = 5.4/ 67 = 0.6597

Yes .

c) P( ≥ 19.47) = 1 - P( ≤ 19.47)

= 1 - P[( - ) / ≤ (19.47 - 17.3) / 0.6597 ]

= 1 - P(z ≤ 3.29 )

= 1-0.9995 = 0.0005

Probability = 0.0005