Question

It is reported that the ACT scores of freshman students at NMSU is approximately Normally distributed, with mean of 20.5 and standard deviation of 5.1. (1) (3pts) If the university decides to offer a scholarship to the top 2% of students, how high does a student need to score on ACT test to qualify for that scholarship? (No points will be given without proper steps) (2) (3pts) A math class has 20 freshmen enrolled. If we assume the 20 freshmen in the math class is a random sample, what is the chance that the AVERAGE ACT scores of the math class is above 22? (No points will be given without proper steps)

Answer 1

Answer:

Given,

X ~ N(20.5 , 5.1^2)

1)

P(X > x) = 0.03

P(z > (x - 20.5)/5.1) = 0.03

P(z > 1.88) = 0.03

since from standard normal table

z = 1.88

(x - 20.5)/5.1 = 1.88

x = 30.088

2)

Xbar ~ N(20.5 , 5.1^2/20) ~ N(20.5 , 1.3005)

P(Xbar > 22) = P(z > (22 - 20.5)/1.3005)

= P(z > 1.15)

= 0.1250719 [since from z table]

= 0.1251