Chi-Square Goodness-of-Fit Test

• The Chi-Square goodness-of-fit test assesses if the observed frequency distribution for a single categorical variable significantly differs from an expected distribution.
• Null hypothesis (H₀) in a Chi-Square goodness-of-fit test states that the observed frequencies follow the specified distribution.
• Alternative hypothesis (H₁) indicates that the observed frequencies do not follow the specified distribution.

Testing a Claim with Sample Data

• Collect sample data on the number of products of each type.
• Calculate the expected frequencies using the proportions claimed.
• Compute the test statistic using the Chi-Square formula.
• Compare the test statistic to the critical value or use the p-value to assess significance.

Conditions for Chi-Square Goodness-of-Fit

• Data must be frequencies (counts).
• Categories must be mutually exclusive.
• The expected frequency in each category should be at least 5.

Expected Frequencies in Chi-Square Test

• Expected frequencies represent what one would expect if the null hypothesis is true.
• They help assess if the observed data significantly deviates from this expectation.

Chi-Square Test Statistic Interpretation

• A large Chi-Square statistic suggests a significant difference between observed and expected frequencies, indicating evidence against the null hypothesis.

Degrees of Freedom

• Degrees of freedom (df) for a goodness-of-fit test = (number of categories) - 1

P-Value Indication

• A small p-value (e.g., p < 0.05) suggests strong evidence against the null hypothesis.
• A small p-value indicates the observed distribution that does not match the expected distribution.

Chi-Square Statistic

• No, it cannot be negative.
• It is always non-negative due to squaring the differences between observed and expected frequencies.

Proportions and Expected Frequencies

• Proportions allow calculating expected frequencies in terms of counts, which are needed for the Chi-Square formula.

Sample Size Impact

• With a large sample size, even minor differences between observed and expected frequencies can become statistically significant.
• Large sample sizes can lead to rejection of the null hypothesis even if the practical difference is minor.

P-Value of 0.87

• A p-value of 0.87 is very high.
• A p-value of 0.87 suggests no evidence to reject the null hypothesis.
• A p-value of 0.87 indicates the observed distribution likely matches the expected distribution.

Nature of Chi-Square Test

• Considered non-parametric because it does not require assumptions about the population distribution.
• Based on categorical data.

Real-Life Applications

• Determining if a die is fair.
• Assessing customer preferences across product categories.
• Analyzing election vote proportions.

Addressing Low Frequencies

• Combine categories if appropriate, or use an exact test or simulation-based alternative.

Chi-Square Test of Independence

• Determines if there is a significant association between two categorical variables.

Hypotheses

• H₀: The two variables are independent.
• H₁: The two variables are not independent (they are associated).

Calculating Expected Frequencies

• (row total * column total) / grand total.

Assumptions

• Observations are independent.
• Categories are mutually exclusive.
• Expected cell counts are at least 5 in most cells.

Real-World Examples

• Testing if smoking status relates to gender.
• Assessing if educational level associates with employment status.

Degrees of Freedom

• df = (number of rows - 1) * (number of columns - 1)

Significant Test of Independence

• Implies a statistically significant association between the two variables.
• Two variables imply they are not independent.

Non-Significant Test of Independence

• Indicates there is no evidence of association between the variables.
• Two variables imply they are likely independent.

Data Type Usage

• Chi-Square test of independence is not used for continuous data.
• Requires categorical data.
• Continuous data must be categorized before applying the test.

Causation

• Does not determine causation.
• Only detects association, not causality.

Data Table

• A contingency table (cross-tabulation) of observed frequencies is used for Chi-Square test of independence.

Observed vs. Expected Frequencies

• If all observed frequencies are close to expected frequencies, it suggests the two variables are likely independent.
• A result of observed frequencies close to expected frequencies could result in failure to reject the null hypothesis.

Distribution

• The sampling distribution of the test statistic under H₀ approximates the Chi-Square distribution.
• The Chi-Square distribution is used to evaluate the test statistic.

Result of x² = 0

• Observed frequencies equal expected frequencies exactly.
• Concludes perfect independence between variables.

Limitations

• Requires large sample sizes.
• Does not measure strength or direction of association.
• Can be misleading with small expected frequencies.

Test Purpose Difference

• Goodness-of-Fit tests whether the distribution of a single categorical variable matches a specific expected distribution.
• Test of Independence tests whether there is an association between two categorical variables.

Hypotheses Difference

• Goodness-of-Fit: H₀ states the observed distribution fits the expected distribution, while H₁ states it does not.
• Test of Independence: H₀ states the two variables are independent, while H₁ states they are associated (dependent).

Frequency Calculation Difference

• Goodness-of-Fit: Expected frequencies are calculated from a known or hypothesized distribution.
• Test of Independence: Expected frequencies are calculated from the product of row and column totals divided by the grand total

Degrees of Freedom Difference

• Goodness-of-Fit: df = (number of categories – 1)
• Test of Independence: df = (number of rows – 1) × (number of columns – 1)

Data Type Difference

• Goodness-of-Fit: One categorical variable with multiple levels.
• Test of Independence: Two categorical variables cross-tabulated into a contingency table.

Situational Appropriate Examples

• Goodness-of-Fit: Testing whether a die is fair (each number 1–6 has equal probability).
• Test of Independence: Testing whether gender is associated with voting preference.

Distribution Table Usage

• Both tests use the Chi-Square distribution table to find critical values.
• The specific value depends on the degrees of freedom, which are calculated differently for each test.

Test Choice for Blood Types

• The Goodness-of-Fit test should be used to assess whether the distribution of blood types in a city matches national proportions.
• Test choice because you are comparing observed frequencies in a single categorical variable (blood type) to known national proportions.

Test Choice for Survey Data

• The Chi-Square Test of Independence should be used to analyze survey data about people's favorite fruit and their region.
• Test choice because you are examining the relationship between two categorical variables.

Conclusion with P-value > 0.05

• In both cases, a p-value greater than 0.05 means we fail to reject the null hypothesis.
• For Goodness-of-Fit: The observed distribution matches the expected.
• For Test of Independence: There is no evidence of association between the variables.