Step by step
The question
Question: a sample of 16 has mean 31.2, and the SD of all the numbers is 4: at the 10% level, is the true mean more than 30?
Alpha, 0.1, is the level: a sample rarer than 10% counts as too rare.
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 more than 30.
More than means only the right end counts.
One tail.
2 The test number
We act as if H zero is true: the mean is 30.
How far is our sample mean from it?
First the SD of the sample means: σ over the square root of n.
The square root of 16 is 4.
4/4 is 1.
z is the sample mean minus 30, over 1.
31.2 minus 30 is 1.2.
Divided by 1: 1.2.
3 The p-value
The p-value is the chance of getting a sample this far out, if H zero is true.
On the bell of H zero: 30 in the middle, our sample at z = 1.2.
We shade the right end, from our z up.
The z-table gives the area to the left of z.
Row 1.2, column .00.
For z = 1.2: the row 1.2 and the column .00 meet at 0.8849.
Careful!
0.8849 is the area to the left.
The p-value is the end past our z.
The right end is 1 minus 0.8849: 0.1151.
4 Decide
The p-value, 0.1151, is more than alpha, 0.1.
Our sample is not rare enough.
So we do not reject H zero.
5 Say it in words
In words: at the 10% level, there is not enough evidence that the mean is more than 30.
This does not prove H zero is true.
We just cannot rule it out.
6 Check
Let's check the other way, with the cut-off.
0.1 in the right end gives the cut-off 1.282.
z is 1.2: not past the cut-off.
The same decision.
So it is right!
Let's look back.
First: write the hypotheses.
Then: the test number.
Then: the p-value.
Then: decide.
Then: say it in words.
Last: check.