Step by step
The question
Question: group 1 has mean 79, SD 10 and size 40; group 2 has mean 74, SD 12 and size 45: at the 5% level, is the first mean more than the second?
The SDs of all the numbers in each group are known.
There are four choices.
First we solve it, the normal way.
Then we pick.
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 more than the second.
More than: one tail, the right end.
2 Standard error
The standard error of the difference: add the two variances, sigma squared over n, then the square root.
Each sample mean has variance sigma squared over n, and for a difference the variances add.
10 squared over 40 is 2.5.
12 squared over 45 is 3.2.
Together 5.7, and the square root is about 2.3875.
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.
79 minus 74 is 5.
Divided by 2.3875: about 2.09.
4 The p-value
The p-value is the chance of getting a sample at least this far out, if H zero is true.
On the bell of H zero: 0 in the middle, our sample at z = 2.09.
We shade the right end, from our z up.
The z-table gives the area to the left of z.
Row 2.0, column .09.
For z = 2.09: the row 2.0 and the column .09 meet at 0.9817.
The right end is 1 minus 0.9817: 0.0183.
5 Decide
The p-value, 0.0183, 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 more than the second.
7 Check
Let's check the other way, with the cut-off.
0.05 in the right end gives the cut-off 1.645, from the table, as in the cut-offs lesson.
z is 2.09: past the cut-off.
The same decision.
So it is right!
8 Pick the choice
Our answer is reject H₀.
That is choice B.
9 Why not the others
A is Do not reject H0.
Wrong: the p-value 0.0183 is less than 0.05.
C is Accept H0.
Wrong: a test never shows H0 is true.
D is Accept Ha because 79 is more than 74.
Wrong: the difference must be checked by the test.
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.
Then: check.
Then: pick the choice.
Last: why not the others.