Step by step
The question
Question: a sample of 16 has mean 28 and SD 4: at the 5% level, is the true mean less than 30?
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 30.
Hₐ is what we want to show.
Here: the mean is less than 30.
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 16 is 4.
4/4 is 1.
t is the sample mean minus 30, over 1.
28 minus 30 is −2.
Divided by 1: −2.
3 The cut-off
The t-table needs df: n minus 1, 15.
Careful!
df is 16 minus 1: 15.
The right way: df = n − 1 = 16 − 1 = 15.
One tail: 0.05 in one end.
Row 15, column 0.05.
Row 15, column .05: t* = 1.753.
The table gives 1.753 for the right end.
The t curve is the same on both sides, so the left end's cut-off is −1.753.
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 less than 30.
6 Check
Let's check with the sample mean itself.
The cut-off for it is 30 minus 1.753 times 1: 28.247.
The sample mean, 28, is past it.
The same decision.
So it is right!
7 Pick the choice
Our answer is reject H₀.
That is choice B.
8 Why not the others
A is Do not reject H0.
Wrong: t = −2 is below −1.753.
C is Accept H0.
Wrong: a test never shows H0 is true.
D is Accept Ha because 28 is less than 30.
Wrong: a sample mean below 30 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.