Step by step
The question
Question: a sample of 9 has mean 53 and SD 6: at the 5% level, is the true mean more than 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 more than 50.
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.
6/3 is 2.
t is the sample mean minus 50, over 2.
53 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.
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, 1.5, 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 more than 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 1.860 times 2: 53.72.
The sample mean, 53, 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.