Step by step
The question
Question: a sample of 25 has mean 43.8 and SD 2.4: at the 1% level, is the true mean less than 45?
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 45.
Hₐ is what we want to show.
Here: the mean is less than 45.
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 25 is 5.
2.4/5 is 0.48.
t is the sample mean minus 45, over 0.48.
43.8 minus 45 is −1.2.
Divided by 0.48: −2.5.
3 The cut-off
The t-table needs df: n minus 1, 24.
Careful!
df is 25 minus 1: 24.
One tail: 0.01 in one end.
Row 24, column 0.01.
Row 24, column .01: t* = 2.492.
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, −2.5, is past the cut-off: in the reject part.
So we reject H zero.
5 Say it in words
In words: at the 1% level, there is enough evidence that the mean is less than 45.
6 Check
Let's check with the sample mean itself.
The cut-off for it is 45 minus 2.492 times 0.48: 43.8038.
The sample mean, 43.8, 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.