In: Statistics and Probability
At the prestigious university, they found their mean of the 30 accepted students was in the 98thpercentile of all sample means. Individual scores, X, are normally distributed with a mean of 550 and a standard deviation of 100.
a.) What was their mean for the 30 observations?
b.) What is the probability of getting a mean of 30 students that scored higher than 685?
Solution :
mean = = 550
standard deviation = =100
n = 30
= 550
= / n = 100 30 = 37.94733
Using standard normal table,
a ) P(Z < z) = 98%
P(Z < z) = 0.98
P(Z < 2.054 ) = 0.98
z = 2.05
Using z-score formula,
= z * +
= 2.05 * 18.2574 + 550
= 587.42
= 587.42
b ) P ( X > 685 )
= 1 - P ( X < 685 )
= 1 - P ( X - /) < (685 - 550 / 100 )
= 1 - P( z < 135 / 100 )
= 1 - P ( z < 1.35 )
Using z table
= 1 - 0.9115
= 0.0885
Probability = 0.0885