Test the claim mean < 2.5: sample mean 2.3, sample sd 0.4, n 20… Step by Step

Step by step

The question

Question: a sample of 20 has mean 2.3 and SD 0.4: at the 5% level, is the true mean less than 2.5?

μ < 2.5? x̄ = 2.3, s = 0.4, n = 20, 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 2.5.

H₀ : μ = 2.5

Hₐ is what we want to show.

Here: the mean is less than 2.5.

H_a : μ < 2.5

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.

0.4 over the square root of 20 is about 0.0894.

s√n = 0.4√20 = ≈ 0.0894

t is the sample mean minus 2.5, over 0.0894.

2.3 minus 2.5 is −0.2.

Divided by 0.0894: about −2.237.

t = x̄ − μ₀SD = 2.3 − 2.50.0894 = −0.20.0894 ≈ −2.237

3 The cut-off

The t-table needs df: n minus 1, 19.

df = n − 1 = 20 − 1 = 19

Careful!

df is 20 minus 1: 19.

Careful: not df = 20 ✗

One tail: 0.05 in one end.

Row 19, column 0.05.

Row 19, column .05: t* = 1.729.

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, −2.237, is past the cut-off: in the reject part.

So we reject H zero.

= t = −2.237 < −1.729 → reject H₀

5 Say it in words

At the 5% level, there is enough evidence that

In words: at the 5% level, there is enough evidence that the mean is less than 2.5.

the mean is less than 2.5.

6 Check

Let's check with the sample mean itself.

The cut-off for it is 2.5 minus 1.729 times 0.0894: 2.3454.

cut-off for the sample mean: 2.5 − 1.729 × 0.0894 = 2.3454

The sample mean, 2.3, is past it.

The same decision.

So it is right!

2.3: past it, 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