Test the claim mean < 60: sample mean 57, sample sd 8, n 16, alpha… Step by Step

Step by step

The question

Question: a sample of 16 has mean 57 and SD 8: at the 5% level, is the true mean less than 60?

μ < 60? x̄ = 57, s = 8, n = 16, α = 0.05

We only know s, the SD of the sample.

So we use t, not z.

σ not known: we use s and t

1 Write the hypotheses

H₀ is the "nothing new" idea: it has the equal sign.

Here: the mean is 60.

H₀ : μ = 60

Hₐ is what we want to show.

Here: the mean is less than 60.

H_a : μ < 60

Less than means only the left end counts.

One tail.

one tail: the left end

2 The test number

First the SD of the sample means: s over the square root of n.

The square root of 16 is 4.

8/4 is 2.

s√n = 8√16 = 84 = 2

t is the sample mean minus 60, over 2.

57 minus 60 is −3.

Divided by 2: −1.5.

t = x̄ − μ₀SD = 57 − 602 = −32 = −1.5

3 The cut-off

The t-table needs df: n minus 1, 15.

df = n − 1 = 16 − 1 = 15

Careful!

df is 16 minus 1: 15.

Careful: not df = 16 ✗

One tail: 0.05 in one end.

Row 15, column 0.05.

Row 15, column .05: t* = 1.753.

The t curve, with the cut-off and our t.

Past the cut-off is the reject part: the rare samples.

4 Decide

Our t, −1.5, is not past the cut-off.

So we do not reject H zero.

= t = −1.5 > −1.753 → do not reject H₀

5 Say it in words

At the 5% level, there is not enough evidence that

In words: at the 5% level, there is not enough evidence that the mean is less than 60.

the mean is less than 60.

This does not prove H zero is true.

We just cannot rule it out.

not rejecting H₀ does not prove H₀

6 Check

Let's check with the sample mean itself.

The cut-off for it is 60 minus 1.753 times 2: 56.494.

cut-off for the sample mean: 60 − 1.753 × 2 = 56.494

The sample mean, 57, is not past it.

The same decision.

So it is right!

57: not past it, do not reject

Let's look back.

First: write the hypotheses.

Then: the test number.

Then: the cut-off.

Then: decide.

Then: say it in words.

Last: check.

Answer