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.
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 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.
Each sample mean has variance sigma squared over n, and for a difference the variances add.
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 at least 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, from the table, as in the cut-offs lesson.
z is 1.58: not past the cut-off.
The same decision.
So it is right!
8 Pick the choice
Our answer is do not reject H₀.
That is choice D.
9 Why not the others
A is Reject H0.
Wrong: the p-value 0.1142 is more than 0.05.
B is Accept Ha because 16.2 is more than 15.
Wrong: the test decides.
C is Accept H0.
Wrong: a test never shows H0 is true.
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.