Step by step
The question
Question: group 1 has mean 16.2, SD 3 and size 50; group 2 has mean 15, SD 4 and size 40: 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.
3 squared over 50 is 0.18.
4 squared over 40 is 0.4.
Together 0.58, and the square root is about 0.7616.
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.
16.2 minus 15 is 1.2.
Divided by 0.7616: about 1.58.
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 = 1.58.
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 1.5, column .08.
For z = 1.58: the row 1.5 and the column .08 meet at 0.9429.
One end is 1 minus 0.9429: 0.0571.
Two ends: 2 times that is 0.1142.
5 Decide
The p-value, 0.1142, 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 1.58: 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.