Chi Square Test

χ²GOF Test

• 1 Sample

• 1 Variable

χ²Test for Homogeneity

• 2+ Samples

• 1 Variable

χ²Test for Independence

• 1 Sample

• 2+ Variables

Key Words: “Association”, “Relation”

χ²Test for Independence

Where can you find expected counts?

Matrix B

Solve Ratio 1:3:3:9

(Ratio) 1:3:3:9 = 16

(Samples) 25, 92, 68, 295 = 480

480/16 = 30

30:90:90:270

If χ² sample is very close to what is expected it is…

It would be small

If χ² sample is very far to what is expected it is…

It would be big

A small chi-square test statistic was calculated. What does this finding suggest?

The null hypothesis may not be rejected, since the difference between the observed and expected values are relatively small.

How to solve a die rolled 24 times?

24/6=4

-Stats→List

4 2

4 8

4 2

Symetric χ² Distribution

Big DF

Right Skew χ² Distribution

Esp Low Df

The bigger the χ² is the more it….

Shifts right increasing the median and mean

For all χ² distribution, P(χ² >0)=1

True, χ² values cannot be negative the probabilirt that χ² is at least 0 is always 0

Big P-Value gives you…

A small value of χ²

Small P-Value gives you…

A big χ^{2 essentially 0}

Scenario: A cereal company claims that 25% of its cereal boxes contain a toy car, 25% a toy plane, 25% a toy boat, and 25% a toy robot. [H0]

• H₀ (Null Hypothesis):
  The proportion of toy types in the cereal boxes is 25% for each type (car, plane, boat, robot). The observed distribution matches the expected distribution.

Scenario: A cereal company claims that 25% of its cereal boxes contain a toy car, 25% a toy plane, 25% a toy boat, and 25% a toy robot. [HA]

• Hₐ (Alternative Hypothesis):
  The proportions of toys in the cereal boxes are not all 25%. The observed distribution does not match the expected distribution.

You survey students from three different high schools about their preferred study method (solo, group, or tutoring) to see if the preferences differ by school. [H0]

• H₀ (Null Hypothesis):
  The distribution of preferred study methods is the same across all three schools.

You survey students from three different high schools about their preferred study method (solo, group, or tutoring) to see if the preferences differ by school. [HA]

• Hₐ (Alternative Hypothesis):
  The distribution of preferred study methods is not the same across the schools. At least one school’s distribution differs.

Scenario: You collect data on people’s preferred social media platform (Instagram, TikTok, or YouTube) and their age group (teen, young adult, adult) to see if there is a relationship. [H0]

• H₀ (Null Hypothesis):
  Social media preference is independent of age group. There is no association between age and preferred platform.

You collect data on people’s preferred social media platform (Instagram, TikTok, or YouTube) and their age group (teen, young adult, adult) to see if there is a relationship. [HA]

• Hₐ (Alternative Hypothesis):
  Social media preference is dependent on age group. There is an association between age and preferred platform.

: A candy company claims their candies are evenly distributed across five colors.

a.

Reject H₀

Based on our results, we reject the null hypothesis. There is sufficient evidence to conclude that the distribution of candy colors differs from the expected uniform distribution.

b.

Fail to Reject H₀

We fail to reject the null hypothesis. There is not enough evidence to say the color distribution of the candies is different from what was expected. The data support the claim of a uniform distribution.

You survey students from three schools about their favorite school subject to see if the preferences are the same across schools.

a.

Reject H₀

We reject the null hypothesis. There is sufficient evidence to suggest that the distribution of favorite school subjects differs between the schools.

b.

Fail to Reject H₀

We fail to reject the null hypothesis. There is not enough evidence to suggest that subject preferences differ across the schools. The data are consistent with the distributions being the same.

You want to know if there’s a relationship between gender and preferred music genre.

Reject H₀

We reject the null hypothesis. There is evidence of an association between gender and preferred music genre. The two variables are not independent.

b.

Fail to Reject H₀

We fail to reject the null hypothesis. There is no evidence of a relationship between gender and music preference. The variables appear to be independent.