Step by step
The question
Question: a sample of 25 has mean 22 and SD 5: at the 5% level, is the true mean more than 20?
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 20.
Hₐ is what we want to show.
Here: the mean is more than 20.
More than means only the right 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.
5/5 is 1.
t is the sample mean minus 20, over 1.
22 minus 20 is 2.
Divided by 1: 2.
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.
One tail: 0.05 in one end.
Row 24, column 0.05.
Row 24, column .05: t* = 1.711.
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, 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 more than 20.
6 Check
Let's check with the sample mean itself.
The cut-off for it is 20 plus 1.711 times 1: 21.711.
The sample mean, 22, is past it.
The same decision.
So it is right!
7 Pick the choice
Our answer is reject H₀.
That is choice A.
8 Why not the others
B is Do not reject H0.
Wrong: t = 2 is more than 1.711.
C is Accept H0.
Wrong: a test never shows H0 is true.
D is Accept Ha because 22 is more than 20.
Wrong: a sample mean above 20 is not enough: 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.