Step by step
The question
Question: a sample of 16 has mean 77 and SD 6: at the 1% level, is the true mean less than 80?
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 80.
Hₐ is what we want to show.
Here: the mean is less than 80.
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 16 is 4.
6/4 is 1.5.
t is the sample mean minus 80, over 1.5.
77 minus 80 is −3.
Divided by 1.5: −2.
3 The cut-off
The t-table needs df: n minus 1, 15.
Careful!
df is 16 minus 1: 15.
One tail: 0.01 in one end.
Row 15, column 0.01.
Row 15, column .01: t* = 2.602.
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, is not past the cut-off.
So we do not reject H zero.
5 Say it in words
In words: at the 1% level, there is not enough evidence that the mean is less than 80.
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 80 minus 2.602 times 1.5: 76.097.
The sample mean, 77, 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.