Step by step
The question
Question: a sample of 9 has mean 22 and SD 3: at the 5% level, is the true mean more than 20?
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 20.
Hₐ is what we want to show.
Here: the mean is more than 20.
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.
The square root of 9 is 3.
3/3 is 1.
t is the sample mean minus 20, over 1.
22 minus 20 is 2.
Divided by 1: 2.
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, 2, is past the cut-off: in the reject part.
So we reject H zero.
5 Say it in words
In words: at the 5% level, there is enough evidence that the mean is more than 20.
6 Check
Let's check with the sample mean itself.
The cut-off for it is 20 plus 1.860 times 1: 21.86.
The sample mean, 22, is 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.