Step by step
The question
Question: a sample of 15 has mean 19.5 and SD 3: at the 1% level, is the true mean different from 18?
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 18.
Hₐ is what we want to show.
Here: the mean is different from 18.
Different from means too big or too small: both ends of the bell count.
Two tails.
2 The test number
First the SD of the sample means: s over the square root of n.
3 over the square root of 15 is about 0.7746.
t is the sample mean minus 18, over 0.7746.
19.5 minus 18 is 1.5.
Divided by 0.7746: about 1.936.
3 The cut-off
The t-table needs df: n minus 1, 14.
Careful!
df is 15 minus 1: 14.
Two tails: 0.01 is split, 0.005 in each end.
Row 14, column 0.005.
Row 14, column .005: t* = 2.977.
The t curve, with the cut-offs and our t.
Past the cut-offs is the reject part: the rare samples.
4 Decide
Our t, 1.936, 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 different from 18.
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 18 plus or minus 2.977 times 0.7746: 15.694 and 20.306.
The sample mean, 19.5, 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.