Question

The SAT scores of entering freshmen at University X have a N(1200, 90) distribution and the SAT scores of entering freshmen at University Y have a N(1215, 110) distribution. A random sample of 100 freshmen is sampled from each University, with ?̅the sample mean of the 100 scores from University X and ?̅the sample mean of the 100 scores from University Y. The probability that ?̅ is less than 1190 is

Answer 1

If X follows N() then by central limit theorem that sampling distribution of sample mean with sample size : n   follows  N()

X: The SAT scores of entering freshmen at University X

X follows N(1200, 90)

A random sample of 100 freshmen is sampled from each University, with the sample mean of the 100 scores from University X

Sample size : n = 100

Therefore ,  sampling distribution of sample mean follows N(1200, 90/) i.e follows N(1200, 9)

The probability that is less than 1190 = P( <1190)

Z-score for 1190 = (1190 - mean)/Standard deviation = (1190-1200)/9 = -10/9=-1.11

P( <1190) = P(Z< -1.11)

From Standard normal tables, P(Z<-1.11) = 0.1335

P( <1190) = P(Z< -1.11) = 0.1335

The probability that is less than 1190 = P( <1190) = 0.1335

The probability that is less than 1190 = 0.1335