Step by step
The question
Question: a sample of 25 has mean 11.2 and SD 2: at the 5% level, is the true mean different from 12?
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 12.
Hₐ is what we want to show.
Here: the mean is different from 12.
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.
The square root of 25 is 5.
2/5 is 0.4.
t is the sample mean minus 12, over 0.4.
11.2 minus 12 is −0.8.
Divided by 0.4: −2.
3 The cut-off
The t-table needs df: n minus 1, 24.
Careful!
df is 25 minus 1: 24.
Two tails: 0.05 is split, 0.025 in each end.
Row 24, column 0.025.
Row 24, column .025: t* = 2.064.
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, −2, 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 different from 12.
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 12 plus or minus 2.064 times 0.4: 11.1744 and 12.8256.
The sample mean, 11.2, 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.