Weekly spending in a population has mean $120 and standard deviation… Step by Step

Step by step

The question

Question: samples of 36 are taken from numbers with mean 120 and SD 18: find the mean and the SD of the sample means.

μ = 120, σ = 18, n = 36: mean and SD of x̄?

The central limit theorem: the means of many samples make a bell.

many samples of 36: their means make a bell

There are four choices.

First we solve it, the normal way.

Then we pick.

1 The center

Its middle is the same as the numbers' mean: 120.

μ_x̄ = μ = 120

2 The spread

Means of samples are closer together than single numbers: in a mean, high and low numbers cancel out part of the way.

Their SD is σ divided by the square root of n, where n is the sample size.

means are less spread out: SD = σ ÷ √n

The square root of 36 is 6.

18 divided by 6 is 3.

σ_x̄ = σ√n = 18√36 = 186 = 3

Careful!

We divide by the square root of 36, not by 36.

Careful: divide by √n, not by n

The bell of the sample means: middle 120, one SD is 3.

3 Check

Let's check: times the square root of 36 takes us back.

3 times 6 is 18, the SD σ.

So it is right!

3 × √36 = 3 × 6 = 18

4 Pick the choice

Our answer is μx̄ = 120, σx̄ = 3.

That is choice B.

B) mean 120, SD 3

5 Why not the others

A is mean 120, SD 18.

Wrong: the sample mean varies less: divide the SD by √36.

A) mean 120, SD 18 ✗ the sample mean varies less: divide the SD by √36

C is mean 120, SD 0.5.

Wrong: divided by 36: divide by √36.

C) mean 120, SD 0.5 ✗ divided by 36: divide by √36

D is mean 20, SD 3.

Wrong: divided the mean too: it stays 120.

D) mean 20, SD 3 ✗ divided the mean too: it stays 120

Let's look back.

First: the center.

Then: the spread.

Then: check.

Then: pick the choice.

Last: why not the others.

Answer