Chi-square independence: spring 20 rain 30 dry, summer 10 rain 40… Step by Step

Step by step

The question

Question: spring: 20 rain, 30 dry; summer: 10 rain, 40 dry; autumn: 25 rain, 25 dry: at the 5% level, is the answer linked to the group?

spring: 20 rain, 30 dry; summer: 10 rain, 40 dry; autumn: 25 rain, 25 dry; α = 0.05

1 Write the hypotheses

H₀ is the "nothing new" idea.

Here: the group and the answer are independent.

H₀ : the group and the answer are independent

Hₐ is what we want to show.

Here: the group and the answer are not independent.

H_a : the group and the answer are not independent

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

independent: no link, the answer does not depend on the group

2 Totals

The counts in a table.

Each row's total: spring, 50; summer, 50; autumn, 50.

Each column's total: rain, 55; dry, 95.

And everyone: 150.

3 Expected counts

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

E = row total × column totaltotal

Spring, rain: 50 times 55, over 150, is about 18.3333.

= spring, rain: 50 × 55 ÷ 150 ≈ 18.3333

Spring, dry: 50 times 95, over 150, is about 31.6667.

= spring, dry: 50 × 95 ÷ 150 ≈ 31.6667

Summer, rain: 50 times 55, over 150, is about 18.3333.

= summer, rain: 50 × 55 ÷ 150 ≈ 18.3333

Summer, dry: 50 times 95, over 150, is about 31.6667.

= summer, dry: 50 × 95 ÷ 150 ≈ 31.6667

Autumn, rain: 50 times 55, over 150, is about 18.3333.

= autumn, rain: 50 × 55 ÷ 150 ≈ 18.3333

Autumn, dry: 50 times 95, over 150, is about 31.6667.

= autumn, dry: 50 × 95 ÷ 150 ≈ 31.6667

4 Each part

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

χ² = add up all (O − E)²E

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

20 minus 18.3333 is 1.6667.

Squared, 2.7779.

Over 18.3333: 0.1515.

30 minus 31.6667 is −1.6667.

Squared, 2.7779.

Over 31.6667: 0.0877.

10 minus 18.3333 is −8.3333.

Squared, 69.4439.

Over 18.3333: 3.7879.

40 minus 31.6667 is 8.3333.

Squared, 69.4439.

Over 31.6667: 2.193.

25 minus 18.3333 is 6.6667.

Squared, 44.4449.

Over 18.3333: 2.4243.

25 minus 31.6667 is −6.6667.

Squared, 44.4449.

Over 31.6667: 1.4035.

5 Add them

0.1515 plus 0.0877 plus 3.7879 plus 2.193 plus 2.4243 plus 1.4035 is 10.0479.

χ² = 0.1515 + 0.0877 + 3.7879 + 2.193 + 2.4243 + 1.4035 = 10.0479

6 The cut-off

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

df = (rows − 1)(columns − 1) = (3 − 1)(2 − 1) = 2

Careful!

Not the number of cells minus 1.

Rows minus 1, times columns minus 1: 2.

Careful: not df = 5 ✗

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.

10.0479 is past the cut-off.

So we reject H zero.

= χ² = 10.0479 > 5.991 → reject H₀

8 Say it in words

At the 5% level, there is enough evidence that

In words: at the 5% level, there is enough evidence that the group and the answer are not independent.

the group and the answer are not independent.

9 Check

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

Spring: 18.3333 plus 31.6667 is 50.

spring: E 18.3333 + 31.6667 = 50 = 50

Summer: 18.3333 plus 31.6667 is 50.

summer: E 18.3333 + 31.6667 = 50 = 50

Autumn: 18.3333 plus 31.6667 is 50.

So it is right!

autumn: E 18.3333 + 31.6667 = 50 = 50

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.

Answer