Step by step
The question
Question: a sample of 25 has mean 26.5, and the SD of all the numbers is 5: at the 5% level, is the true mean more than 25?
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 25.
Hₐ is what we want to show.
Here: the mean is more than 25.
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 25.
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 25 is 5.
5/5 is 1.
z is the sample mean minus 25, over 1.
26.5 minus 25 is 1.5.
Divided by 1: 1.5.
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: 25 in the middle, our sample at z = 1.5.
We shade the right end, from our z up.
The z-table gives the area to the left of z.
Row 1.5, column .00.
For z = 1.5: the row 1.5 and the column .00 meet at 0.9332.
Careful!
0.9332 is the area to the left.
The p-value is the end past our z.
The right end is 1 minus 0.9332: 0.0668.
4 Decide
The p-value, 0.0668, 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 25.
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 1.5: 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.