Step by step
The question
Question: a sample of 25 has mean 72, and the SD of all the numbers is 10: at the 10% level, is the true mean different from 75?
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 75.
Hₐ is what we want to show.
Here: the mean is different from 75.
Different from means too big or too small: both ends of the bell count.
Two tails.
2 The test number
We act as if H zero is true: the mean is 75.
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.
10/5 is 2.
z is the sample mean minus 75, over 2.
72 minus 75 is −3.
Divided by 2: −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: 75 in the middle, our sample at z = −1.5.
Two tails: we shade both ends, as far out as our sample or more.
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 0.0668.
Careful!
0.0668 is only one end.
Two tails: we double it.
One end is 0.0668.
Two ends: 2 times that is 0.1336.
4 Decide
The p-value, 0.1336, 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 different from 75.
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 split into two ends gives the cut-offs plus and minus 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.