Step by step
The question
Question: men: 40 yes, 60 no; women: 50 yes, 50 no: at the 5% level, is the answer linked to the group?
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Write the hypotheses
H₀ is the "nothing new" idea.
Here: the group and the answer are independent.
Hₐ is what we want to show.
Here: the group and the answer are linked.
Independent means no link: the answer does not depend on the group.
2 Totals
The counts in a table.
Each row's total: men, 100; women, 100.
Each column's total: yes, 90; no, 110.
And everyone: 200.
3 Expected counts
If there is no link, every row has the same share of each column as everyone does.
The column total over the grand total is that share.
Times the row total: the row total times the column total, over the grand total.
Men, yes: 100 times 90, over 200, is 45.
Men, no: 100 times 110, over 200, is 55.
Women, yes: 100 times 90, over 200, is 45.
Women, no: 100 times 110, over 200, is 55.
4 Each part
Chi-square adds up, for each cell, the observed minus the expected, squared, over the expected.
Squared, so misses up and down do not cancel.
Over the expected, so a miss counts more when we expected only a few.
A table: each cell, what we saw, O, and what we expected, E.
40 minus 45 is −5.
Squared, 25.
Over 45: 0.5556.
60 minus 55 is 5.
Squared, 25.
Over 55: 0.4545.
50 minus 45 is 5.
Squared, 25.
Over 45: 0.5556.
50 minus 55 is −5.
Squared, 25.
Over 55: 0.4545.
5 Add them
0.5556 plus 0.4545 plus 0.5556 plus 0.4545 is 2.0202.
6 The cut-off
df is rows minus 1, times columns minus 1: with the totals known, only that many cells can be filled freely.
1 times 1 is 1.
Careful!
Not the number of cells minus 1.
Rows minus 1, times columns minus 1: 1.
The right way: df = (rows − 1)(columns − 1) = (2 − 1)(2 − 1) = 1.
Chi-square is always a right-tail test: big means far from what we expected.
Row 1, column 0.05.
Row 1, column .05: the cut-off is 3.841.
7 Decide
The chi-square curve for df 1.
Past the cut-off, 3.841, is the reject part.
2.0202 is not past the cut-off.
So we do not reject H zero.
8 Say it in words
In words: at the 5% level, there is not enough evidence that the group and the answer are linked.
This does not prove H zero is true.
We just cannot rule it out.
9 Check
Let's check: the expected counts give the same row totals.
Men: 45 plus 55 is 100.
Women: 45 plus 55 is 100.
So it is right!
10 Pick the choice
Our answer is do not reject H₀.
That is choice C.
11 Why not the others
A is Reject H0.
Wrong: χ² ≈ 2.02 is less than 3.841.
B is Accept H0.
Wrong: a test never shows H0 is true.
D is Accept Ha because more women said yes.
Wrong: the test decides, not the look of the counts.
Let's look back.
First: write the hypotheses.
Then: totals.
Then: expected counts.
Then: each part.
Then: add them.
Then: the cut-off.
Then: decide.
Then: say it in words.
Then: check.
Then: pick the choice.
Last: why not the others.