Step by step
The question
Question: group 1 has mean 81, SD 12 and size 36; group 2 has mean 76, SD 9 and size 36: at the 5% level, is the first mean different from 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 different from the second.
Different from: two tails.
2 Standard error
The standard error of the difference: add the two variances, sigma squared over n, then the square root.
12 squared over 36 is 4.
9 squared over 36 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.
81 minus 76 is 5.
Divided by 2.5: about 2.
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.
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 2.0, column .00.
For z = 2: the row 2.0 and the column .00 meet at 0.9772.
One end is 1 minus 0.9772: 0.0228.
Two ends: 2 times that is 0.0456.
5 Decide
The p-value, 0.0456, is less than alpha, 0.05.
Our sample would be rare if H zero were true.
So we reject H zero.
6 Say it in words
In words: at the 5% level, there is enough evidence that the first mean is different from the second.
7 Check
Let's check the other way, with the cut-off.
0.05 split into two ends gives the cut-offs plus and minus 1.96.
z is 2: 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.