Step by step
The question
Question: a sample of 9 has mean 4.4 and SD 1: at the 5% level, is the true mean less than 5?
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 5.
Hₐ is what we want to show.
Here: the mean is less than 5.
Less than means only the left end counts.
One tail.
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.
1/3 is 0.3333.
t is the sample mean minus 5, over 0.3333.
4.4 minus 5 is −0.6.
Divided by 0.3333: about −1.8.
3 The cut-off
The t-table needs df: n minus 1, 8.
Careful!
df is 9 minus 1: 8.
One tail: 0.05 in one end.
Row 8, column 0.05.
Row 8, column .05: t* = 1.860.
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, −1.8, 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 less than 5.
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 5 minus 1.860 times 0.3333: 4.3801.
The sample mean, 4.4, 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.