Question

1. A random sample of n=100 observations is selected from a population with m=30 and s=16. Approximate the following probabilities.

a. P( x-bar >= 28)    

b. P(22.1<=x-bar<= 26.8)

c. P( x-bar<= 28.2)

d. P(x-bar>= 27.0)

2. A random sample of n=100 observations is selected from a population with m =100 and s=10.

a. What are the largest and smallest values that you would expect to see in the 100 data points?

b. What are the largest and smallest values you would expect x-bar (the sample mean) to be? …Assuming that we drew subgroups of 100 multiple times.

Answer 1

1)

Given

m = , 30 , s = 16

The centrl limit theorem states that

P( < x) = P( Z < x - m / S / sqrt(n) )

a)

P( >= 28) = P( Z > 28 - 30 / 16 / sqrt(100) )

= P( Z > -1.25)

= P( Z < 1.25)

= 0.8944

b)

P(22.1 <= <= 6.8) = P( < 26.8) - P( < 22.1)

= P( Z < 26.8 -30 / 16 / sqrt(100) ) - P( Z < 22.1 - 30 / 16 / sqrt(100) )

= P( Z < -2) - p( z < -4.9375)

= ( 1 - P( Z < 2) ) - ( 1 - P(Z < 4.9375) )

= ( 1 - 0.9772) - ( 1 - 0.9999)

= 0.0227

c)

P( <= 28.2) = P( Z < 28.2 - 30 / 16 / sqrt(100) )

= P( Z < -1.125)

= ( 1 - P( Z < 1.125) )

= 1 - 0.8697

= 0.1303

d)

P( >= 27) = P( Z > 27 - 30 / 16 / sqrt(100) )

= P( Z > -1.875)

= P( Z < 1.875)

= 0.9696