Example Of Chi Square Goodness Of Fit

Chi-Square Goodness-of-Fit Test: Examples and Applications

The Chi-Square Goodness-of-Fit test is a powerful statistical tool used to determine if a sample data set matches a population. It assesses whether observed frequencies differ significantly from expected frequencies, allowing us to test hypotheses about the distribution of a categorical variable. This article will delve into the intricacies of this test, providing clear explanations, worked examples, and practical applications. We'll move beyond the mere formulas and explore the real-world implications and interpretations of this crucial statistical method.

Understanding the Chi-Square Goodness-of-Fit Test

At its core, the Chi-Square Goodness-of-Fit test compares observed frequencies (the counts you actually observe in your data) with expected frequencies (the counts you'd expect if your null hypothesis were true). The null hypothesis typically posits that the observed data follows a specific distribution (e.g., uniform, normal, binomial). The test statistic, denoted as χ², measures the discrepancy between these observed and expected frequencies. A larger χ² value indicates a greater difference between observed and expected frequencies, suggesting a potential rejection of the null hypothesis.

The test relies on several assumptions:

• Independence: Observations should be independent of each other. This means that one observation's outcome doesn't influence another.
• Sample Size: Expected frequencies in each category should generally be at least 5. This ensures the chi-square distribution is a reasonable approximation. Smaller expected frequencies can lead to inaccurate results.
• Categorical Data: The data must be categorical; it deals with frequencies or counts, not continuous variables.

Calculating the Chi-Square Statistic

The formula for calculating the chi-square statistic is:

χ² = Σ [(Oᵢ - Eᵢ)² / Eᵢ]

Where:

• Oᵢ = Observed frequency for category i
• Eᵢ = Expected frequency for category i
• Σ = Summation across all categories

The degrees of freedom (df) are calculated as:

df = k - p - 1

Where:

• k = number of categories
• p = number of parameters estimated from the data (often 0 for simple goodness-of-fit tests)

Once the χ² statistic and degrees of freedom are calculated, a p-value is obtained using a chi-square distribution table or statistical software. If the p-value is less than a predetermined significance level (typically α = 0.05), the null hypothesis is rejected, indicating a significant difference between observed and expected frequencies.

Example 1: Dice Rolling Experiment

Let's consider a classic example: testing the fairness of a six-sided die. We roll the die 60 times and record the following observed frequencies:

Face	Observed Frequency (Oᵢ)
1	8
2	12
3	9
4	10
5	11
6	10

Null Hypothesis (H₀): The die is fair; each face has an equal probability of appearing (1/6).

Expected Frequency (Eᵢ): If the die is fair, we expect each face to appear 60/6 = 10 times.

Now, we calculate the chi-square statistic:

χ² = [(8-10)²/10] + [(12-10)²/10] + [(9-10)²/10] + [(10-10)²/10] + [(11-10)²/10] + [(10-10)²/10] = 0.4

Degrees of freedom (df) = 6 - 1 = 5

Using a chi-square distribution table or statistical software, we find the p-value associated with χ² = 0.4 and df = 5. The p-value will be much greater than 0.05. Therefore, we fail to reject the null hypothesis. There's not enough evidence to conclude that the die is unfair.

Example 2: Distribution of Blood Types

Suppose a researcher wants to investigate whether the distribution of blood types in a particular population conforms to the expected proportions known from larger population studies. The expected proportions are:

• Type A: 42%
• Type B: 10%
• Type AB: 4%
• Type O: 44%

A sample of 200 individuals yields the following observed frequencies:

Blood Type	Observed Frequency (Oᵢ)	Expected Frequency (Eᵢ)
A	75	84
B	15	20
AB	10	8
O	100	88

We calculate the chi-square statistic:

χ² = [(75-84)²/84] + [(15-20)²/20] + [(10-8)²/8] + [(100-88)²/88] ≈ 6.2

Degrees of freedom (df) = 4 - 1 = 3

With df = 3 and a calculated χ² ≈ 6.2, we consult a chi-square table or statistical software. If the resulting p-value is less than 0.05, we reject the null hypothesis, suggesting the observed distribution of blood types significantly differs from the expected proportions. Otherwise, we fail to reject the null hypothesis.

Example 3: Testing for a Uniform Distribution

A company produces candies in five different colors. They claim that the colors are equally distributed. A random sample of 100 candies reveals the following distribution:

Color	Observed Frequency (Oᵢ)
Red	25
Blue	18
Green	20
Yellow	22
Purple	15

Null Hypothesis (H₀): The colors are uniformly distributed (each color has a probability of 1/5).

Expected Frequency (Eᵢ): 100/5 = 20 for each color.

Calculating the chi-square statistic:

χ² = [(25-20)²/20] + [(18-20)²/20] + [(20-20)²/20] + [(22-20)²/20] + [(15-20)²/20] ≈ 3.5

Degrees of freedom (df) = 5 - 1 = 4

We then consult a chi-square table or software to determine the p-value associated with χ² ≈ 3.5 and df = 4. If the p-value exceeds 0.05, we fail to reject the null hypothesis, meaning there isn't enough evidence to contradict the company's claim of uniform color distribution.

Interpreting the Results

The interpretation of the Chi-Square Goodness-of-Fit test hinges on the p-value.

• P-value ≤ α (Significance Level): Reject the null hypothesis. There is sufficient evidence to conclude that the observed frequencies differ significantly from the expected frequencies. The data does not support the proposed distribution.

• P-value > α: Fail to reject the null hypothesis. There is not enough evidence to conclude that the observed frequencies differ significantly from the expected frequencies. The data is consistent with the proposed distribution.

Important Considerations:

• Effect Size: While the p-value indicates statistical significance, it doesn't necessarily reflect the practical significance of the difference. A large sample size can lead to statistically significant results even when the difference between observed and expected frequencies is small. Considering effect size measures (like Cramer's V) can provide a more complete picture.

• Limitations: The Chi-Square test assumes relatively large expected frequencies. If expected frequencies are too small (generally less than 5), the test might yield inaccurate results. In such cases, alternative tests, like Fisher's exact test, might be more appropriate.

Conclusion

The Chi-Square Goodness-of-Fit test is a valuable tool for analyzing categorical data and assessing the fit of observed frequencies to a hypothesized distribution. By carefully considering the assumptions, calculating the test statistic, and interpreting the p-value, researchers can draw meaningful conclusions about the distribution of their data and test specific hypotheses regarding population parameters. Understanding its strengths and limitations is essential for proper application and interpretation in various fields, from genetics and medicine to market research and quality control. Remember to always consider effect sizes and potential limitations to obtain a thorough understanding of your results.