Goodness Of Fit Chi Square Test Example

The chi-square goodness-of-fit test is a powerful statistical tool used to determine whether observed sample data matches an expected distribution. It's a versatile test applicable in various fields, from genetics and marketing to social sciences and quality control. This article will delve into the mechanics of the chi-square goodness-of-fit test, providing a step-by-step guide on how to perform it, explaining the underlying concepts, and illustrating its application with a detailed example.

Understanding the Chi-Square Goodness-of-Fit Test

The core purpose of the chi-square goodness-of-fit test is to assess whether the differences between observed frequencies and expected frequencies are statistically significant. In other words, it helps us determine if the data we've collected from a sample aligns with a specific theoretical distribution or a predefined expectation.

Key Concepts:

• Observed Frequencies (O): These are the actual counts or frequencies of each category obtained from your sample data.
• Expected Frequencies (E): These are the frequencies you would expect to see in each category if the null hypothesis is true. The null hypothesis usually states that the sample data follows a specific distribution or matches a certain set of proportions.
• Null Hypothesis (H0): This is the statement being tested. It usually assumes that there is no significant difference between the observed and expected frequencies.
• Alternative Hypothesis (H1): This is the statement that contradicts the null hypothesis. It suggests that there is a significant difference between the observed and expected frequencies.
• Chi-Square Statistic (χ²): This statistic measures the discrepancy between the observed and expected frequencies. A larger chi-square value indicates a greater difference.
• Degrees of Freedom (df): This represents the number of independent pieces of information used to calculate the chi-square statistic. It's typically calculated as the number of categories minus the number of constraints (usually 1, if the expected frequencies are calculated based on the total sample size).
• P-value: This is the probability of obtaining a chi-square statistic as extreme as, or more extreme than, the one calculated from the sample data, assuming the null hypothesis is true. A small p-value (typically less than 0.05) provides evidence against the null hypothesis.
• Significance Level (α): This is a pre-determined threshold used to decide whether to reject the null hypothesis. Commonly used values are 0.05 (5%) or 0.01 (1%).

Steps to Perform a Chi-Square Goodness-of-Fit Test

Here's a step-by-step guide on how to conduct a chi-square goodness-of-fit test:

1. State the Hypotheses:

• Null Hypothesis (H0): The observed frequencies are consistent with the specified distribution or expected proportions.
• Alternative Hypothesis (H1): The observed frequencies are not consistent with the specified distribution or expected proportions.

2. Define the Significance Level (α):

• Choose a significance level (α) that represents the probability of rejecting the null hypothesis when it is actually true. Common values are 0.05 or 0.01.

3. Calculate the Expected Frequencies (E):

• Determine the expected frequency for each category based on the hypothesized distribution or proportions.
  • If you are testing against a uniform distribution (where each category is expected to have the same frequency), divide the total sample size by the number of categories.
  • If you are testing against a specific set of proportions, multiply the total sample size by the proportion for each category.

4. Calculate the Chi-Square Statistic (χ²):

• Use the following formula to calculate the chi-square statistic:

  χ² = Σ [(Oᵢ - Eᵢ)² / Eᵢ]

  Where:

  • χ² is the chi-square statistic
  • Σ is the summation symbol (summing across all categories)
  • Oᵢ is the observed frequency for category i
  • Eᵢ is the expected frequency for category i

5. Determine the Degrees of Freedom (df):

• Calculate the degrees of freedom using the following formula:

  df = k - c

  Where:

  • df is the degrees of freedom
  • k is the number of categories
  • c is the number of constraints (usually 1, if the expected frequencies are based on the total sample size)

6. Determine the P-value:

• Use a chi-square distribution table or a statistical software package to find the p-value associated with the calculated chi-square statistic and the degrees of freedom. The p-value represents the probability of obtaining a chi-square statistic as extreme as, or more extreme than, the one calculated from the sample data, assuming the null hypothesis is true.

7. Make a Decision:

• Compare the p-value to the significance level (α).
  • If the p-value is less than or equal to α (p ≤ α), reject the null hypothesis. This suggests that there is a statistically significant difference between the observed and expected frequencies.
  • If the p-value is greater than α (p > α), fail to reject the null hypothesis. This suggests that there is no statistically significant difference between the observed and expected frequencies.

8. Draw a Conclusion:

• Based on your decision, state your conclusion in the context of the problem. If you reject the null hypothesis, conclude that the observed data does not fit the expected distribution. If you fail to reject the null hypothesis, conclude that the observed data is consistent with the expected distribution.

Chi-Square Goodness-of-Fit Test Example: M&M Colors

Let's illustrate the chi-square goodness-of-fit test with an example involving M&M candies. Suppose the Mars company claims that a standard bag of M&Ms contains the following color distribution: 24% blue, 13% brown, 16% green, 20% orange, 13% red, and 14% yellow.

Scenario:

You buy a bag of M&Ms and want to test if the color distribution in your bag matches the company's claimed distribution. You count the number of each color and obtain the following observed frequencies:

• Blue: 50
• Brown: 25
• Green: 30
• Orange: 40
• Red: 20
• Yellow: 35

The total number of M&Ms in your bag is 200.

Let's perform the chi-square goodness-of-fit test step-by-step:

1. State the Hypotheses:

• Null Hypothesis (H0): The color distribution of M&Ms in the bag matches the company's claimed distribution (24% blue, 13% brown, 16% green, 20% orange, 13% red, and 14% yellow).
• Alternative Hypothesis (H1): The color distribution of M&Ms in the bag does not match the company's claimed distribution.

2. Define the Significance Level (α):

• Let's choose a significance level of α = 0.05.

3. Calculate the Expected Frequencies (E):

• Multiply the total number of M&Ms (200) by the company's claimed proportion for each color:

  • Blue: 200 * 0.24 = 48
  • Brown: 200 * 0.13 = 26
  • Green: 200 * 0.16 = 32
  • Orange: 200 * 0.20 = 40
  • Red: 200 * 0.13 = 26
  • Yellow: 200 * 0.14 = 28

4. Calculate the Chi-Square Statistic (χ²):

• Using the formula χ² = Σ [(Oᵢ - Eᵢ)² / Eᵢ], we calculate the chi-square statistic:

  χ² = [(50 - 48)² / 48] + [(25 - 26)² / 26] + [(30 - 32)² / 32] + [(40 - 40)² / 40] + [(20 - 26)² / 26] + [(35 - 28)² / 28]

  χ² = [4 / 48] + [1 / 26] + [4 / 32] + [0 / 40] + [36 / 26] + [49 / 28]

  χ² = 0.083 + 0.038 + 0.125 + 0 + 1.385 + 1.75

  χ² = 3.381

5. Determine the Degrees of Freedom (df):

• df = k - c = 6 (number of colors) - 1 (constraint: total sample size) = 5

6. Determine the P-value:

• Using a chi-square distribution table or a statistical software package, we find the p-value associated with χ² = 3.381 and df = 5. The p-value is approximately 0.641.

7. Make a Decision:

• Since the p-value (0.641) is greater than the significance level (α = 0.05), we fail to reject the null hypothesis.

8. Draw a Conclusion:

• Based on our analysis, we conclude that there is no statistically significant evidence to suggest that the color distribution of M&Ms in the bag is different from the company's claimed distribution. In other words, the observed color distribution in your bag of M&Ms is consistent with the expected distribution claimed by the Mars company.

Assumptions of the Chi-Square Goodness-of-Fit Test

Like all statistical tests, the chi-square goodness-of-fit test relies on certain assumptions:

• Random Sampling: The data should be obtained through random sampling to ensure that the sample is representative of the population.
• Independence: The observations should be independent of each other. One observation should not influence another.
• Expected Frequencies: All expected frequencies should be at least 5. This is a general rule of thumb. If some expected frequencies are less than 5, consider combining categories or using a different test. This is crucial for the validity of the chi-square approximation.
• Categorical Data: The data should be categorical, meaning that the variables are divided into distinct categories.

Applications of the Chi-Square Goodness-of-Fit Test

The chi-square goodness-of-fit test is a versatile tool with applications in numerous fields:

• Genetics: Testing if observed genetic ratios match expected Mendelian ratios.
• Marketing: Assessing if customer preferences for different products align with predicted market shares.
• Social Sciences: Analyzing if survey responses are distributed according to a theoretical model.
• Quality Control: Determining if the number of defects in a manufacturing process follows a Poisson distribution.
• Ecology: Examining if the distribution of plant species in a habitat matches a predicted pattern.
• Education: Analyzing if grade distributions in a class follow a normal distribution or a pre-determined grading scheme.

Advantages and Disadvantages of the Chi-Square Goodness-of-Fit Test

Advantages:

• Simple to Understand and Apply: The test is relatively straightforward to understand and perform, even without advanced statistical knowledge.
• Versatile: Applicable to a wide range of categorical data analysis problems.
• Non-Parametric: It does not require assumptions about the underlying distribution of the data (beyond the categorical nature of the data itself).

Disadvantages:

• Sensitivity to Sample Size: With very large sample sizes, even small differences between observed and expected frequencies can lead to statistically significant results, even if the differences are practically insignificant. Conversely, with small sample sizes, the test may lack the power to detect real differences.
• Requirement for Expected Frequencies: The rule of thumb that all expected frequencies should be at least 5 can be a limitation in some cases, requiring researchers to combine categories, which can reduce the granularity of the analysis.
• Only Applicable to Categorical Data: The test cannot be used with continuous data.

Alternatives to the Chi-Square Goodness-of-Fit Test

While the chi-square goodness-of-fit test is a powerful tool, there are alternative tests that may be more appropriate in certain situations:

• Kolmogorov-Smirnov Test: This test can be used to compare an observed distribution to a continuous theoretical distribution. It is often used as an alternative to the chi-square test when dealing with continuous data that has been grouped into categories.
• Anderson-Darling Test: Similar to the Kolmogorov-Smirnov test, the Anderson-Darling test is used to assess the goodness of fit of a sample to a specified distribution. It gives more weight to the tails of the distribution, making it more sensitive to differences in the tails.
• Shapiro-Wilk Test: This test is specifically designed to test whether a sample comes from a normally distributed population.
• Cramér-von Mises Test: Another alternative to the Kolmogorov-Smirnov test, offering different weighting schemes for deviations between the observed and expected distributions.

The choice of which test to use depends on the nature of the data, the specific hypothesis being tested, and the assumptions of each test.

Conclusion

The chi-square goodness-of-fit test is a fundamental statistical tool for assessing the agreement between observed data and expected frequencies. By understanding the steps involved, the underlying assumptions, and the limitations of the test, researchers can effectively use it to analyze categorical data and draw meaningful conclusions. The M&M example provides a practical illustration of how to apply the test and interpret the results. Remember to consider the context of your research question and the characteristics of your data when choosing the appropriate statistical test. While the chi-square goodness-of-fit test is a valuable tool, it's important to be aware of its limitations and to consider alternative tests when necessary.