Step by step
The question
Question: group 1 has mean 120, SD 20 and size 100; group 2 has mean 126, SD 15 and size 100: at the 1% level, is the first mean less than the second?
The SDs of all the numbers in each group are known.
1 Write the hypotheses
H₀ is the "nothing new" idea: it has the equal sign.
Here: the two means are the same, so the difference is 0.
Hₐ is what we want to show.
Here: the first mean is less than the second.
Less than: one tail, the left end.
2 Standard error
The standard error of the difference: add the two variances, sigma squared over n, then the square root.
20 squared over 100 is 4.
15 squared over 100 is 2.25.
Together 6.25, and the square root is 2.5.
Careful!
Add the variances, then take the square root.
Do not add the two SDs.
3 The test number
z is the difference of the means over the SE.
120 minus 126 is −6.
Divided by 2.5: about −2.4.
4 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: 0 in the middle, our sample at z = −2.4.
We shade the left end, from our z down.
The z-table gives the area to the left of z.
Row −2.4, column .00.
For z = −2.4: the row −2.4 and the column .00 meet at 0.0082.
The left end is the table number itself: 0.0082.
5 Decide
The p-value, 0.0082, is less than alpha, 0.01.
Our sample would be rare if H zero were true.
So we reject H zero.
6 Say it in words
In words: at the 1% level, there is enough evidence that the first mean is less than the second.
7 Check
Let's check the other way, with the cut-off.
0.01 in the left end gives the cut-off −2.326.
z is −2.4: past the cut-off.
The same decision.
So it is right!
Let's look back.
First: write the hypotheses.
Then: standard error.
Then: the test number.
Then: the p-value.
Then: decide.
Then: say it in words.
Last: check.