Step by step
The question
Question: junior: 30 bus, 20 car, 10 walk; senior: 20 bus, 35 car, 15 walk: 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: junior, 60; senior, 70.
Each column's total: bus, 50; car, 55; walk, 25.
And everyone: 130.
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.
Junior, bus: 60 times 50, over 130, is about 23.0769.
Junior, car: 60 times 55, over 130, is about 25.3846.
Junior, walk: 60 times 25, over 130, is about 11.5385.
Senior, bus: 70 times 50, over 130, is about 26.9231.
Senior, car: 70 times 55, over 130, is about 29.6154.
Senior, walk: 70 times 25, over 130, is about 13.4615.
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.
30 minus 23.0769 is 6.9231.
Squared, 47.9293.
Over 23.0769: 2.0769.
20 minus 25.3846 is −5.3846.
Squared, 28.9939.
Over 25.3846: 1.1422.
10 minus 11.5385 is −1.5385.
Squared, 2.367.
Over 11.5385: 0.2051.
20 minus 26.9231 is −6.9231.
Squared, 47.9293.
Over 26.9231: 1.7802.
35 minus 29.6154 is 5.3846.
Squared, 28.9939.
Over 29.6154: 0.979.
15 minus 13.4615 is 1.5385.
Squared, 2.367.
Over 13.4615: 0.1758.
5 Add them
2.0769 plus 1.1422 plus 0.2051 plus 1.7802 plus 0.979 plus 0.1758 is 6.3592.
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 2 is 2.
Careful!
Not the number of cells minus 1.
Rows minus 1, times columns minus 1: 2.
The right way: df = (rows − 1)(columns − 1) = (2 − 1)(3 − 1) = 2.
Chi-square is always a right-tail test: big means far from what we expected.
Row 2, column 0.05.
Row 2, column .05: the cut-off is 5.991.
7 Decide
The chi-square curve for df 2.
Past the cut-off, 5.991, is the reject part.
6.3592 is past the cut-off.
So we reject H zero.
8 Say it in words
In words: at the 5% level, there is enough evidence that the group and the answer are linked.
9 Check
Let's check: the expected counts give the same row totals.
Junior: 23.0769 plus 25.3846 plus 11.5385 is 60.
Senior: 26.9231 plus 29.6154 plus 13.4615 is 70.
So it is right!
10 Pick the choice
Our answer is reject H₀.
That is choice B.
11 Why not the others
A is Do not reject H0.
Wrong: χ² = 6.36 is more than 5.991.
C is Accept H0.
Wrong: a test never shows H0 is true.
D is Accept Ha because more seniors drive.
Wrong: the test decides.
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.