Question

You are studying the height of a species of bird. You know that the population of heights has a mean of 8 cm, a variance of 9 cm, the population is large, and the shape of the population is slightly skewed. You randomly collect a sample of 45 birds. The likelihood that the mean of this sample of birds is between 7.5 cm and 8 cm is

Answer 1

Solution:

We are given

µ = 8

σ^2 = 9

σ = sqrt(9) = 3

n = 45

We have to find P(7.5 < x̄ < 8)

P(7.5 < x̄ < 8) = P(x̄<8) - P(x̄<7.5)

Find P(x̄<8)

Z = (x̄ - µ)/[σ/sqrt(n)]

Z = (8 - 8)/[3/sqrt(45)]

Z = 0

P(Z<0) = P(x̄<8) = 0.50000

(by using z-table)

Now find P(x̄< 7.5)

Z = (x̄ - µ)/[σ/sqrt(n)]

Z = (7.5 - 8)/(3/sqrt(45))

Z = -1.12

P(Z<-1.12) = P(x̄<7.5) = 0.13136

(by using z-table)

P(7.5 < x̄ < 8) = P(x̄<8) - P(x̄<7.5)

P(7.5 < x̄ < 8) = 0.50000 - 0.13136

P(7.5 < x̄ < 8) = 0.36864

Required probability = 0.36864