Step by step
The question
Question: group 1 has mean 102, SD 15 and size 25; group 2 has mean 95, SD 12 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.
15 squared over 25 is 9.
12 squared over 36 is 4.
Together 13, and the square root is about 3.6056.
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.
102 minus 95 is 7.
Divided by 3.6056: about 1.94.
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 = 1.94.
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 1.9, column .04.
For z = 1.94: the row 1.9 and the column .04 meet at 0.9738.
One end is 1 minus 0.9738: 0.0262.
Two ends: 2 times that is 0.0524.
5 Decide
The p-value, 0.0524, 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 different from 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 split into two ends gives the cut-offs plus and minus 1.96.
z is 1.94: 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.