Step by step
The question
Question: a sample of 40 has mean 52, and the SD of all the numbers is 8: at the 5% level, is the true mean more than 50?
Alpha, 0.05, is the level: a sample rarer than 5% counts as too rare.
1 Write the hypotheses
H₀ is the "nothing new" idea: it has the equal sign.
Here: the mean is 50.
Hₐ is what we want to show.
Here: the mean is more than 50.
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 50.
How far is our sample mean from it?
First the SD of the sample means: σ over the square root of n.
8 over the square root of 40 is about 1.2649.
z is the sample mean minus 50, over 1.2649.
52 minus 50 is 2.
Divided by 1.2649: about 1.58.
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: 50 in the middle, our sample at z = 1.58.
We shade the right end, from our z up.
The z-table gives the area to the left of z.
Row 1.5, column .08.
For z = 1.58: the row 1.5 and the column .08 meet at 0.9429.
Careful!
0.9429 is the area to the left.
The p-value is the end past our z.
The right end is 1 minus 0.9429: 0.0571.
4 Decide
The p-value, 0.0571, is more than alpha, 0.05.
Our sample is not rare enough.
So we do not reject H zero.
5 Say it in words
In words: at the 5% level, there is not enough evidence that the mean is more than 50.
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.
0.05 in the right end gives the cut-off 1.645.
z is 1.58: 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.