Step by step
The question
Question: a sample of 9 has mean 47 and SD 6: at the 5% level, is the true mean different from 50?
We only know s, the SD of the sample.
So we use t, not z.
1 Write the hypotheses
H₀ is the "nothing new" idea: it has the equal sign.
Here: the mean is 50.
Hₐ is what we want to show.
Here: the mean is different from 50.
Different from means too big or too small: both ends of the bell count.
Two tails.
2 The test number
First the SD of the sample means: s over the square root of n.
The square root of 9 is 3.
6/3 is 2.
t is the sample mean minus 50, over 2.
47 minus 50 is −3.
Divided by 2: −1.5.
3 The cut-off
The t-table needs df: n minus 1, 8.
Careful!
df is 9 minus 1: 8.
Two tails: 0.05 is split, 0.025 in each end.
Row 8, column 0.025.
Row 8, column .025: t* = 2.306.
The t curve, with the cut-offs and our t.
Past the cut-offs 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.
5 Say it in words
In words: at the 5% level, there is not enough evidence that the mean is different from 50.
This does not prove H zero is true.
We just cannot rule it out.
6 Check
Let's check with the sample mean itself.
The cut-off for it is 50 plus or minus 2.306 times 2: 45.388 and 54.612.
The sample mean, 47, is not past it.
The same decision.
So it is right!
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.