Step by step
The question
Question: in group 1, 30 of 150 say yes, and in group 2, 45 of 150: at the 5% level, is the first proportion less than the second?
1 Write the hypotheses
H₀ is the "nothing new" idea: it has the equal sign.
Here: the two proportions are the same, so the difference is 0.
Hₐ is what we want to show.
Here: the first proportion is less than the second.
Less than: one tail, the left end.
2 The two parts
Group 1: 30/150 is 0.2.
Group 2: 45/150 is 0.3.
3 Pool the groups
If H zero is true, the two groups are the same.
So we put them together: the pooled proportion.
All the yeses over everyone: 75/300 is 0.25.
Careful!
We pool the counts, not the two proportions.
And q c is 1 minus 0.25: 0.75.
4 Standard error
The standard error, the SD of the difference: the square root of p c times q c, times 1/n 1 plus 1/n 2.
0.25 times 0.75 is 0.1875.
1/150 plus 1/150 is 0.0133.
Multiply, then the square root: about 0.05.
5 The test number
z is the difference over the SE.
0.2 minus 0.3 is −0.1.
Divided by 0.05: about −2.
6 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.
We shade the left end, from our z down.
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 0.0228.
The left end is the table number itself: 0.0228.
7 Decide
The p-value, 0.0228, is less than alpha, 0.05.
Our sample would be rare if H zero were true.
So we reject H zero.
8 Say it in words
In words: at the 5% level, there is enough evidence that the first proportion is less than the second.
9 Check
Let's check the other way, with the cut-off.
0.05 in the left end gives the cut-off −1.645.
z is −2: past the cut-off.
The same decision.
So it is right!
Let's look back.
First: write the hypotheses.
Then: the two parts.
Then: pool the groups.
Then: standard error.
Then: the test number.
Then: the p-value.
Then: decide.
Then: say it in words.
Last: check.