Question

A random sample is selected from a population with mean μ = 99 and standard deviation σ = 10. Determine the mean and standard deviation of the x sampling distribution for each of the following sample sizes. (Round the answers to three decimal places.)

(a) n=9 μ = σ =
(b) n=14 μ = σ =
(c) n=35 μ = σ =
(d) n=60 μ = σ =
(e) n=120 μ = σ =
(f) n=480 μ = σ =

Answer 1

a) μ = 99 σ = 3.333

b) μ = 99 σ = 2.673

c) μ = 99 σ = 1.69

d) μ = 99 σ = 1.291

e) μ = 99 σ = 0.913

f) μ = 99 σ = 0.456

Explanation:

The Central Limit Theorem estabilishes that, for a random variable X, with mean
`\mu` and standard deviation
`\sigma`, the sample means with size n of at least 30 can be approximated to a normal distribution with mean
`\mu` and standard deviation
`s = (\sigma)/(√(n))`

In this problem, we have that:

`\mu = 99, \sigma = 10`

(a) n=9 μ = σ =

`\mu = 99, s = (10)/(√(9)) = 3.333`

(b) n=14 μ = σ =

`\mu = 99, s = (10)/(√(14)) = 2.673`

(c) n=35 μ = σ =

`\mu = 99, s = (10)/(√(35)) = 1.69`

(d) n=60 μ = σ =

`\mu = 99, s = (10)/(√(60)) = 1.291`

(e) n=120 μ = σ =

`\mu = 99, s = (10)/(√(120)) = 0.913`

(f) n=480 μ = σ =

`\mu = 99, s = (10)/(√(480)) = 0.456`