Step by step
The question
Question: a sample of 30 has mean 8.6 and SD 1.1: at the 10% level, is the true mean different from 8?
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 8.
Hₐ is what we want to show.
Here: the mean is different from 8.
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.
1.1 over the square root of 30 is about 0.2008.
t is the sample mean minus 8, over 0.2008.
8.6 minus 8 is 0.6.
Divided by 0.2008: about 2.988.
3 The cut-off
The t-table needs df: n minus 1, 29.
Careful!
df is 30 minus 1: 29.
Two tails: 0.1 is split, 0.05 in each end.
Row 29, column 0.05.
Row 29, column .05: t* = 1.699.
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, 2.988, is past the cut-off: in the reject part.
So we reject H zero.
5 Say it in words
In words: at the 10% level, there is enough evidence that the mean is different from 8.
6 Check
Let's check with the sample mean itself.
The cut-off for it is 8 plus or minus 1.699 times 0.2008: 7.6588 and 8.3412.
The sample mean, 8.6, 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.