Step by step
The question
Question: a sample of 100 has mean 1012, and the SD of all the numbers is 40: at the 1% level, is the true mean different from 1000?
Alpha, 0.01, is the level: a sample rarer than 1% counts as too rare.
1 Write the hypotheses
H₀ is the "nothing new" idea: it has the equal sign.
Here: the mean is 1000.
Hₐ is what we want to show.
Here: the mean is different from 1000.
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 1000.
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 100 is 10.
40/10 is 4.
z is the sample mean minus 1000, over 4.
1012 minus 1000 is 12.
Divided by 4: 3.
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: 1000 in the middle, our sample at z = 3.
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 3.0, column .00.
For z = 3: the row 3.0 and the column .00 meet at 0.9987.
Careful!
0.0013 is only one end.
Two tails: we double it.
One end is 1 minus 0.9987: 0.0013.
Two ends: 2 times that is 0.0026.
4 Decide
The p-value, 0.0026, is less than alpha, 0.01.
Our sample would be rare if H zero were true.
So we reject H zero.
5 Say it in words
In words: at the 1% level, there is enough evidence that the mean is different from 1000.
6 Check
Let's check the other way, with the cut-off.
0.01 split into two ends gives the cut-offs plus and minus 2.576.
z is 3: 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.