Step by step
The question
Question: group 1 has mean 7.4, SD 2 and size 64; group 2 has mean 8, SD 2.4 and size 36: at the 5% 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.
2 squared over 64 is 0.0625.
2.4 squared over 36 is 0.16.
Together 0.2225, and the square root is about 0.4717.
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.
7.4 minus 8 is −0.6.
Divided by 0.4717: about −1.27.
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 = −1.27.
We shade the left end, from our z down.
The z-table gives the area to the left of z.
Row −1.2, column .07.
For z = −1.27: the row −1.2 and the column .07 meet at 0.1020.
The left end is the table number itself: 0.1020.
5 Decide
The p-value, 0.1020, is more than alpha, 0.05.
Our sample is not rare enough.
So we do not reject H zero.
6 Say it in words
In words: at the 5% level, there is not enough evidence that the first mean is less than the second.
This does not prove H zero is true.
We just cannot rule it out.
7 Check
Let's check the other way, with the cut-off.
0.05 in the left end gives the cut-off −1.645.
z is −1.27: not 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.