Question

The mean height of men in the US (ages 20-29) is 69.5 inches and the standard deviation is 3.0 inches. A random sample of 49 men between ages 20-29 is drawn from this population.

Find the probability that the sample height x is more than 70.5 inches.

Answer 1

Solution:

Let X be a random variable which represents the height of men in the US (ages 20 -29)

Given that, μ = 69.5 inches and σ = 3.0 inches

We have to obtain P(x̄ > 70.5 inches). (where, x̄ is sample mean)

According to central limit theorem, if we have a population with mean μ and standard deviation σ and if we draw samples of sufficiently large size from this population then sampling distribution of sample mean follows approximately normal distribution with mean μ and standard deviation σ/√n (variance σ^{​​​​​​2}/n).

i.e. x̄ ~ N(μ, σ^{​​​​​​2}/n)

(Where, x̄ is sample mean and n is sample size.)

We have to obtain P(x̄ > 70.5 inches).

We know that if x̄ ~ N(μ, σ^{​​​​​​2}/n) then

We have, μ = 69.5 inches, σ = 3.0 inches and n = 49

Using "pnorm" function of R we get, P(Z > 2.3333) = 0.0098

The probability that the sample height x̄ is more than 70.5 inches is 0.0098.