The scores of students on the ACT college entrance examination in a recent year

Question

The scores of students on the ACT college entrance examination in a recent year had the normal distribution with a national average of 18.4 and population SD of 5.0.

a) What is the chance that a single student, randomly chosen from all those taking the test, scores 22 or higher?

b) The average score of the 30 students at Northside High who took the test was 22. School officials want to know what the chance is that the average sore for 30 students randomly selected from all who took the test nationally is 22 or higher. What is the chance?

Explanation / Answer

(a) Let X denote the score obtained by the randomly selected student. Then, we wish to calculate P(X >= 22). From the information given, X~ N(18.4, 5.0^2). So P(X >= 22) = P[(X-18.4)/5.0 >= (22 - 18.4)/5.0] = P(Z >= 0.72), where Z = (X - 18.4)/5.0 has a standard normal distribution N(0,1). P(Z >= 0.72) = 1 - Pr( Z < 0.72) - 1 - Phi( 0.72) = 0.24 (2 dp) [ from tables of Phi] (b) We now are interested in the distribution of the sample mean score M from the 30 randomly selected students. Given that the score for each of the 30 students is N(18.4, 5.0^2), it follows using a standard result that M ~ N(18.4 25.0/30) = N(18.4, 0.8333). Therefore, P(M > = 22) = Pr[ (M - 18.4)/sqrt(0.8333) >= (22- 18.4)/sqrt(0.8333) = 3.9437) = 1 - Phi(3.9437) = 0.000040.

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