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Lecture – Chi Square Goodness of Fit Test

PSYC 3510

Matthew Babb

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Review and Reminders

Reminders

Workshop due this Friday (5/26)
Reading Quiz due next Monday (5/29)

Review

Null Hypothesis
Test of Significance
p-value and what it means

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Chi Square Test

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Goals for Today

Chi-Square Test

What it is, when to use it, and how to use it

How to calculate Chi-Square Value
Solve a Chi-Square Example

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Chi Square Test

Test of Association between Categorical Variables
Compares Actual Data to Expected Data
Only works with Categorical Data

(Observed Data – Expected Data)

Expected

=

2

Σ

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Types of Chi Square Tests

𝝌²Goodness of Fit Test

Examines one variable
Compares actual frequencies to the expected frequencies
Compares sample of data to population of data

2. 𝝌²Test of Independence

Examines two variables
Determines if membership in one category is independent of membership in another category
Compares frequencies of variable 1 to two categories of variable 2

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Types of Chi Square Tests

𝝌²Goodness of Fit Test

EX: “Does the sex distribution in Congress reflect the sex distribution of the entire United States?“
EX: “Do the five flavors of skittles occur evenly throughout every bag?”

2. 𝝌²Test of Independence

EX: “Is the sex distribution in Congress the same between Democrats and Republicans?”
EX: “Does peanut butter preference (creamy or crunchy) determine if drive slow in the fast lane?”

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Chi Square Example

Question: Does the gender distribution of those affected with COVID-2.0 reflect the gender distribution of the entire United States?
Variable: Gender

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Null Hypothesis and Significance Testing

State the hypotheses
Set the criteria for a decision
Collect the data & compute test statistic

- In this case it’s a Chi Square Value

Decide to reject or accept the null

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1. State Hypotheses

Question: Does the gender distribution of those affected with COVID-2.0 reflect the gender distribution of the entire United States?

Null Hypothesis

There is no difference between the gender distribution of those infected by COVID-2.0 and the US Population

Alternative Hypothesis

There is a difference between the gender distribution of those infected by COVID-2.0 and the US Population

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2. Set Criteria for Decision

P-value

To set criteria, we need to use the Chi Square Distribution

Which line?
Depends upon number of categories in your variable of interest
Degrees of Freedom (df)
df = k – 1 (where k is the number of categories)

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Chi Square Distribution

P-value

1 2 3

Male

Female

*

df = k – 1

(k is the number of categories)

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How many degrees of freedom?

PollEv.com/mattbabb632

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Chi Square Distribution

P-value

1 2 3

Male

Female

*

df = k – 1

(k is the number of categories)

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2. Set Criteria for Decision

df = 2

P-value

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2. Set Criteria for Decision

P-value

α = 0.05
When α = 0.05, the χ² value is 5.99
That is our criteria.
If we get a χ²value above 5.99, then we can reject our null hypothesis

α = 0.05

5.99

I will always give you this number! But I do want you to know how to find it

Now that we are only focused on one like we can determine our criteria. Since its standard in psychology to have an alpha level of 0.05, like we talked about a couple classes ago, we are going to set our alpha level at 0.05 – That way if we get a p value that is less than this alpha, the results we found are very very unlikely if our null hypothesis were true. So we go to 0.05 and we trace it across the graph until we intersect with the line. Then we go down and we choose our criteria. Here, in this case it is 5.99.

[CLICK] Now I want you all to know that I will always give you this criteria number when solving a long chi-square problem, but I do want you all to be able to look at the chi sq distribution, determine which line to use, and find out roughly what the criteria would be based on degrees of freedom.

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3. Compute our Statistic

3 Groups in our Category
Formula:

(Obs. Freq. – Exp. Freq.)²

Exp. Freq.

=

Male

Female

*

(Obs. Freq. – Exp. Freq.)²

Exp. Freq.

+

(Obs. Freq. – Exp. Freq.)²

Exp. Freq.

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3. Compute our Statistic

	Male	Female	*	Total
Observed Frequency	407	127	0	534

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3. Compute our Statistic

3 Groups in our Category
Formula:

=

Male

Female

*

(407 – Exp. Freq.)²

Exp. Freq.

(127 – Exp. Freq.)²

Exp. Freq.

+

(0 – Exp. Freq.)²

Exp. Freq.

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3. Compute our Statistic

	Male	Female	*	Total
US Population	0.44	0.46	0.10	1.00
Expected Frequency	0.44 x 534 = 234.96	0.46 x 534 = 245.64	0.10 x 534 = 53.40	534

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3. Compute our Statistic

3 Groups in our Category
Formula:

(407 – 234.96)²

234.96

=

Male

Female

*

(127 – 245.64)²

245.64

+

(0 – 53.40)²

53.40

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3. Compute our Statistic

3 Groups in our Category
Formula:

(407 – 234.96)²

234.96

=

Male

Female

*

(127 – 245.64)²

245.64

+

(0 – 53.40)²

53.40

57.30

53.40

125.97

=

236.67

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3. Compute our Statistic

=

236.67

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4. Decide to accept or reject the null

P-value

α = 0.05

=

236.67

Criteria = 5.99

5.99

If > criteria, we reject the null

If < criteria, we fail to reject the null

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Do we reject or fail to reject the null?

PollEv.com/mattbabb632

=

236.67

Criteria = 5.99

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Results

“The gender distribution of those affected by COVID 2.0 is significantly different from the gender distribution in the US population, (2, N = 534) = 236.67, p < 0.05”

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Key Concepts

What a goodness of fit is used for
Formula for
Chi-Square Distribution
How to calculate Expected frequencies
Compare Chi Square value to criteria
Decide to reject or accept null hypothesis

(Obs– Exp)

Exp

=

2

Σ

P-value

	Cat.1	Cat.2
Expected	234.96	245.64
Observed	407	127

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If we have 2 categories, what is the criteria if α = 0.2?

PollEv.com/mattbabb632