Topic 8.2 Notes – Setting Up a Chi-Square Goodness of Fit Test

What a Chi-Square Goodness-of-Fit Test Is

A chi-square goodness-of-fit (GOF) test compares your observed counts from a sample to the expected counts you would get if a claimed distribution were true.

You use it when:

• There is one categorical variable
• There are two or more categories
• You’re testing a claimed distribution of proportions

Example situations:

• A cereal company claims flavors are distributed 20%, 30%, 25%, 25%.
• A die is claimed to be fair (each face 1/6).
• A school claims grade levels are evenly distributed.

If there are only two categories and you're testing a single proportion, that’s a 1-proportion z test, not chi-square.
If you're comparing two categorical variables, that’s a chi-square test of independence, which is a different setup.

Parameters

The parameters are the true population proportions:

$$p_1, p_2, \dots, p_k$$

These must add to 1. Each $p_i$ represents the true proportion in a specific category.

Expected Counts and the Chi-Square Statistic

The whole test is built around comparing what you saw to what you would expect under $H_0$.

Expected Counts

For each category:

$$\text{Expected} = n \times (\text{null proportion})$$

If a sample of 200 customers is tested against a claim that 40% prefer vanilla, the expected number preferring vanilla is:

$$200(0.40) = 80$$

Expected counts are based entirely on:

• Sample size $n$
• Claimed proportions from $H_0$

They are not based on what you actually observed.

The Chi-Square Statistic

The test statistic measures how far observed counts are from expected counts:

$$\chi^2 = \sum \frac{(O - E)^2}{E}$$

• Big differences between observed and expected → large $\chi^2$
• Small differences → small $\chi^2$
• Always 0 or positive (because of squaring)

Large $\chi^2$ values fall in the right tail of the distribution and give small p-values.

Chi-Square Distributions

The chi-square statistic follows a chi-square distribution, which has some unique features:

• Continuous
• Only positive values
• Skewed right
• Shape depends on degrees of freedom

For a GOF test:

$$df = \text{number of categories} - 1$$

Here’s how the distribution changes as degrees of freedom increase:

Chi-square distributions for df = 1, 4, and 10

With $df = 1$, the distribution is very right-skewed and concentrated near 0. As $df$ increases to 4 and then 10, the curve becomes less skewed and more spread out, though it is always right-skewed and stays above 0.

Hypotheses for a GOF Test

Null Hypothesis $H_0$

Lists the claimed proportion for each category.

Example (favorite season survey):

$$H_0: p_{\text{Fall}} = 0.25,\; p_{\text{Winter}} = 0.25,\; p_{\text{Spring}} = 0.25,\; p_{\text{Summer}} = 0.25$$

Each $p$ must be clearly defined in context. On a free-response question, you would say what each proportion represents.

Alternative Hypothesis $H_a$

At least one of the population proportions differs from the stated value.

Important:

• Do not say “all are different.”
• Do not compare categories to each other.
• You are testing the overall distribution, not pairwise differences.

Conditions for Inference

All three must be satisfied.

1. Random

Data come from:

• A random sample, or
• A randomized experiment

Quote the wording if possible.

2. Independence

If sampling without replacement:

$$\frac{n}{N} \le 0.10$$

The population must be at least 10 times the sample size.

3. Large Counts

All expected counts must be:

$$\ge 5$$

You check this using expected counts, not observed counts. This ensures the chi-square distribution is a good model.

Identifying the Correct Test

When you read a problem, ask:

• Is there one categorical variable?
• Are we testing whether the distribution matches a claim?

If yes → chi-square GOF.

Students often misidentify the test when there are only two categories. If it says “Is the proportion different from 0.60?” and there are only two outcomes, that’s a 1-proportion z test.