The SAT scores have an average of 1200 with a standard devia

The SAT scores have an average of 1200 with a standard deviation of 60. A sample of 36 scores is selected. What is the probability that the sample mean will be larger than 1224? What is the probability that the sample mean will be less than 1230? What is the probability that the sample mean will be between 1200 and 1214? What is the probability that the sample mean will be greater than 1200? What is the probability that the sample mean will be larger than 73.46?

Solution

Standard error = sd/sqrt(n) = 60/sqrt(36) =10

a).

Z value for 1224, z=(1224-1200)/10 =2.4

P( x >1224) = P( z >2.4) = 0.0082

b).

Z value for 1230, z=(1230-1200)/10 =3.0

P( x <1230) = P( z <3.0) = 0.9987

c).

Z value for 1200, z=(1200-1200)/10 =0

Z value for 1214, z=(1214-1200)/10 =1.4

P( 1200<x<1214) = P( 0<z<1.4)

=p( z <1.4) – p( z< 0)

= 0.9192 -0.5

=0.4192

d)

Z value for 1200, z=(1200-1200)/10 =0

P( x >1200) = P( z >0) =0.5

e).

Z value for 73.46, z=(73.46-1200)/10 =-112

P( x >73.46) = P( z >-112) =1

Or

this may be for x= 1206.2,

For which

P( x >1206.2) =0.7346

X value is 1206.2