Question

The distribution of room and board expenses per year at a four-year college is normally distributed with a mean of $5850 and standard deviation of $1125. Random samples of size 20 are drawn from this population and the mean of each sample is determined. Which of the following mean expenses would be considered unusual?

Answer 1

Solution:

Given that mean μ = 5850, σ = 1125 and
n = 20
In this question we have
Xbar ~ Normal( μ = 5850 , σ = 1125 / sqrt( 20 ) )
Xbar ~ Normal( μ = 5850 , σ = 251.5576 )

Find P( Xbar > 6180 )
P( ( Xbar - μ ) / σ > ( 6180 - 5850 ) / 251.5576 )
= P( Z > 1.312)
= P( Z < -1.312)
= 0.0984

Find P( Xbar < 5180 )
P( ( Xbar - μ ) / σ < ( 5180 - 5850 ) / 251.5576 )
= P( Z < -2.663)
= 0.0039

Find P( Xbar > 6350 )
P( ( Xbar - μ ) / σ > ( 6350 - 5850 ) / 251.5576 )
= P( Z > 1.987)
= 0.0234

Thus 5180 and 6350 would be considered fairly extreme, but 6180 is relatively close to the mean.