Step by step
The question
Question: a sample of 25 has mean 43 and SD 5: at the 5% level, is the true mean different from 40?
We only know s, the SD of the sample.
So we use t, not z.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Write the hypotheses
H₀ is the "nothing new" idea: it has the equal sign.
Here: the mean is 40.
Hₐ is what we want to show.
Here: the mean is different from 40.
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.
5/5 is 1.
t is the sample mean minus 40, over 1.
43 minus 40 is 3.
Divided by 1: 3.
3 The cut-off
The t-table needs df: n minus 1, 24.
Careful!
df is 25 minus 1: 24.
The right way: df = n − 1 = 25 − 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 curve is the same on both sides, so the cut-offs are plus and minus 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, 3, is past the cut-off: in the reject part.
So we reject H zero.
5 Say it in words
In words: at the 5% level, there is enough evidence that the mean is different from 40.
6 Check
Let's check with the sample mean itself.
The cut-off for it is 40 plus or minus 2.064 times 1: 37.936 and 42.064.
The sample mean, 43, is past it.
The same decision.
So it is right!
7 Pick the choice
Our answer is reject H₀.
That is choice C.
8 Why not the others
A is Accept H0.
Wrong: a test never shows H0 is true.
B is Do not reject H0.
Wrong: |t| = 3 is more than 2.064.
D is Accept Ha because 43 is not 40.
Wrong: the test decides.
Let's look back.
First: write the hypotheses.
Then: the test number.
Then: the cut-off.
Then: decide.
Then: say it in words.
Then: check.
Then: pick the choice.
Last: why not the others.