Step by step
The question
Question: in group 1, 45 of 100 say yes, and in group 2, 30 of 100: at the 5% level, is the first proportion more than the second?
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 proportions are the same, so the difference is 0.
Hₐ is what we want to show.
Here: the first proportion is more than the second.
More than: one tail, the right end.
2 The two parts
Group 1: 45/100 is 0.45.
Group 2: 30/100 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, p c, with c for combined.
All the yeses over everyone: 75/200 is 0.375.
Careful!
We pool the counts, not the two proportions.
And q c is 1 minus 0.375: 0.625.
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.
Each group has its own spread, and for a difference the two spreads add.
0.375 times 0.625 is 0.2344.
1/100 plus 1/100 is 0.02.
Multiplied: 0.004687.
Then the square root: about 0.0685.
5 The test number
z is the difference over the SE.
0.45 minus 0.3 is 0.15.
Divided by 0.0685: about 2.19.
6 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 = 2.19.
We shade the right end, from our z up.
The z-table gives the area to the left of z.
Row 2.1, column .09.
For z = 2.19: the row 2.1 and the column .09 meet at 0.9857.
The right end is 1 minus 0.9857: 0.0143.
7 Decide
The p-value, 0.0143, 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 more than the second.
9 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 2.19: past the cut-off.
The same decision.
So it is right!
10 Pick the choice
Our answer is reject H₀.
That is choice C.
11 Why not the others
A is Do not reject H0.
Wrong: the p-value 0.0143 is less than 0.05.
B is Accept H0.
Wrong: a test never shows H0 is true.
D is Accept Ha because 45 is more than 30.
Wrong: the difference must be checked by the test.
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.
Then: check.
Then: pick the choice.
Last: why not the others.