Chi-square independence: boys 20 yes 30 no, girls 30 yes 20 no… Easy

The question

Question: boys: 20 yes, 30 no; girls: 30 yes, 20 no: at the 5% level, is the answer linked to the group?

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 not independent.

Independent means no link: the answer does not depend on the group.

2 Totals

The counts in a table.

Each row's total: boys, 50; girls, 50.

Each column's total: yes, 50; no, 50.

And everyone: 100.

3 Expected counts

If there is no link, each cell expects its row total times its column total, over the grand total.

Boys, yes: 50 times 50, over 100, is 25.

Boys, no: 50 times 50, over 100, is 25.

Girls, yes: 50 times 50, over 100, is 25.

Girls, no: 50 times 50, over 100, is 25.

4 Each part

Chi-square adds up, for each cell, the observed minus the expected, squared, over the expected.

A table: each cell, what we saw, O, and what we expected, E.

20 minus 25 is −5.

Squared, 25.

Over 25: 1.

30 minus 25 is 5.

Squared, 25.

Over 25: 1.

30 minus 25 is 5.

Squared, 25.

Over 25: 1.

20 minus 25 is −5.

Squared, 25.

Over 25: 1.

5 Add them

1 plus 1 plus 1 plus 1 is 4.

6 The cut-off

df is rows minus 1, times columns minus 1: 1 times 1 is 1.

Careful!

Not the number of cells minus 1.

Rows minus 1, times columns minus 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.

4 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 not independent.

9 Check

Let's check: the expected counts give the same row totals.

Boys: 25 plus 25 is 50.

Girls: 25 plus 25 is 50.

So it is right!

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.

Last: check.

▶ Watch it solved

Chi-square independence: boys 22 sport 18 music 20 art, girls 18… Medium

The question

Question: boys: 22 sport, 18 music, 20 art; girls: 18 sport, 22 music, 20 art: at the 5% level, is the answer linked to the group?

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 not independent.

Independent means no link: the answer does not depend on the group.

2 Totals

The counts in a table.

Each row's total: boys, 60; girls, 60.

Each column's total: sport, 40; music, 40; art, 40.

And everyone: 120.

3 Expected counts

If there is no link, each cell expects its row total times its column total, over the grand total.

Boys, sport: 60 times 40, over 120, is 20.

Boys, music: 60 times 40, over 120, is 20.

Boys, art: 60 times 40, over 120, is 20.

Girls, sport: 60 times 40, over 120, is 20.

Girls, music: 60 times 40, over 120, is 20.

Girls, art: 60 times 40, over 120, is 20.

4 Each part

Chi-square adds up, for each cell, the observed minus the expected, squared, over the expected.

A table: each cell, what we saw, O, and what we expected, E.

22 minus 20 is 2.

Squared, 4.

Over 20: 0.2.

18 minus 20 is −2.

Squared, 4.

Over 20: 0.2.

20 minus 20 is 0.

Squared, 0.

Over 20: 0.

18 minus 20 is −2.

Squared, 4.

Over 20: 0.2.

22 minus 20 is 2.

Squared, 4.

Over 20: 0.2.

20 minus 20 is 0.

Squared, 0.

Over 20: 0.

5 Add them

0.2 plus 0.2 plus 0 plus 0.2 plus 0.2 plus 0 is 0.8.

6 The cut-off

df is rows minus 1, times columns minus 1: 1 times 2 is 2.

Careful!

Not the number of cells minus 1.

Rows minus 1, times columns minus 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.

0.8 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 not independent.

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.

Boys: 20 plus 20 plus 20 is 60.

Girls: 20 plus 20 plus 20 is 60.

So it is right!

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.

Last: check.

▶ Watch it solved

Chi-square independence: junior 30 bus 20 car 10 walk, senior 20 bus… Harder

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?

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 not independent.

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, each cell expects its row total times its 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.

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: 1 times 2 is 2.

Careful!

Not the number of cells minus 1.

Rows minus 1, times columns minus 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 not independent.

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!

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.

Last: check.

▶ Watch it solved