Step by step
The question
Question: a sample of 12 has mean 67 and SD 4: at the 5% level, is the true mean more than 65?
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 65.
Hₐ is what we want to show.
Here: the mean is more than 65.
More than means only the right end counts.
One tail.
2 The test number
First the SD of the sample means: s over the square root of n.
4 over the square root of 12 is about 1.1547.
t is the sample mean minus 65, over 1.1547.
67 minus 65 is 2.
Divided by 1.1547: about 1.732.
3 The cut-off
The t-table needs df: n minus 1, 11.
Careful!
df is 12 minus 1: 11.
One tail: 0.05 in one end.
Row 11, column 0.05.
Row 11, column .05: t* = 1.796.
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, 1.732, 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 more than 65.
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 65 plus 1.796 times 1.1547: 67.0738.
The sample mean, 67, 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.