Chi Square Goodness of Fit Test: Understanding Statistical Significance in Intro Stats / AP Statistics

What is the Chi-Square Goodness-of-Fit Test in Mathematics?

The Chi-Square Goodness-of-Fit Test is a statistical hypothesis test used to determine whether a sample data set comes from a population with a specific distribution. It compares the observed frequencies of outcomes within a categorical variable to the expected frequencies derived from a theoretical distribution.

When is the Chi-Square Goodness-of-Fit Test Used?

This test is employed when you have categorical data and want to see if the distribution of your observed data matches an expected distribution. For example, you might use the test to determine if a die is fair by comparing the observed frequency of each outcome to the expected frequency (assuming the die is fair, each outcome should occur with equal probability).

How is the Chi-Square Goodness-of-Fit Test Performed?

To perform the Chi-Square Goodness-of-Fit Test, follow these steps:

1. Formulate the Hypotheses:
- Null Hypothesis (H0): The observed frequencies follow the specified distribution.
- Alternative Hypothesis (H1): The observed frequencies do not follow the specified distribution.

2. Determine the Expected Frequencies:
- Calculate the expected frequency for each category based on the assumed distribution. For example, if you assume that a die is fair and roll it 60 times, the expected frequency for each face (1 through 6) would be 60 / 6 = 10.

3. Calculate the Test Statistic:
- Use the formula:
Chi-Square Statistic = ?((Observed Frequency - Expected Frequency)² / Expected Frequency)
- Sum this calculation across all categories.

4. Determine the Degrees of Freedom (df):
- Degrees of freedom = Number of categories - 1

5. Compare to the Critical Value:
- Find the critical value from the Chi-Square distribution table based on the degrees of freedom and the desired significance level (commonly 0.05).
- Compare the calculated Chi-Square statistic to the critical value from the table.

6. Make a Decision:
- If the Chi-Square statistic exceeds the critical value, reject the null hypothesis.
- If the Chi-Square statistic is less than or equal to the critical value, do not reject the null hypothesis.

Example Problem:

Question:
Suppose you roll a die 60 times, and the results are as follows:
- 1: 8 times
- 2: 8 times
- 3: 10 times
- 4: 14 times
- 5: 12 times
- 6: 8 times

Conduct a Chi-Square Goodness-of-Fit Test at the 0.05 significance level to determine if the die is fair.

Answer:

1. Formulate the Hypotheses:
- H0: The die is fair (each number appears with equal probability, 1/6).
- H1: The die is not fair.

2. Determine the Expected Frequencies:
- For a fair die, the expected frequency for each outcome is:
- Expected frequency = Total rolls / Number of outcomes = 60 / 6 = 10

3. Calculate the Test Statistic:
- ?² = ?((O - E)² / E)
- Where O = observed frequency, E = expected frequency.
- ?² = ((8 - 10)² / 10) + ((8 - 10)² / 10) + ((10 - 10)² / 10) + ((14 - 10)² / 10) + ((12 - 10)² / 10) + ((8 - 10)² / 10)
- ?² = (4 / 10) + (4 / 10) + (0 / 10) + (16 / 10) + (4 / 10) + (4 / 10)
- ?² = 0.4 + 0.4 + 0 + 1.6 + 0.4 + 0.4
- ?² = 3.2

4. Determine the Degrees of Freedom:
- df = Number of categories - 1 = 6 - 1 = 5

5. Compare to the Critical Value:
- At the 0.05 significance level and 5 degrees of freedom, the critical value from the Chi-Square distribution table is approximately 11.07.

6. Make a Decision:
- Since 3.2 < 11.07, we do not reject the null hypothesis.

Conclusion:
- There is not enough evidence to conclude that the die is not fair based on the given data set.

Intro Stats / AP Statistics: Chi Square Goodness of Fit Test: Understanding Statistical Significance

Related

Recommended Videos