Intro to Business Statistics: Chi Square Tests - Measures of Association, Phi Coefficient Cramérs V

What This Is

Measures of association are statistical tools used to quantify the relationship between two categorical variables. A retail chain wants to know if there's a significant association between the type of product (e.g., electronics, clothing, home goods) and the average daily sales. By analyzing this relationship, the retail chain can make informed decisions about product placement, pricing, and marketing strategies.

Key Formulas & Symbols

• Phi Coefficient (?): measures the strength and direction of the association between two binary variables.-= ?((?² / n)) where ?² = chi-square statistic, n = sample size.
• Cramér's V (V): measures the strength of the association between two categorical variables. V = ?((?² / (n * (k-1))) where ?² = chi-square statistic, n = sample size, k = number of categories.
• Chi-Square Statistic (?²): measures the difference between observed and expected frequencies. ?² =? ((observed frequency - expected frequency)^2 / expected frequency) where-denotes the sum over all categories.
• Degrees of Freedom (df): the number of independent pieces of information in the sample. df = (r - 1) * (c - 1) where r = number of rows, c = number of columns.
• Observed Frequency (O): the actual number of times a category occurs. O = the number of times a category occurs.
• Expected Frequency (E): the expected number of times a category occurs if there's no association between the variables. E = (row total * column total) / total sample size.
• Sample Size (n): the total number of observations in the sample.
• Number of Categories (k): the number of categories in one of the variables.

Step-by-Step Procedure

1. State hypotheses: H?: there's no association between the variables, H?: there's an association between the variables.
2. Choose test: Chi-Square Test of Independence.
3. Compute test statistic: ?² =? ((observed frequency - expected frequency)^2 / expected frequency).
4. Find p-value or critical value: p-value = P(?²-?² observed) or critical value = ?² critical from a chi-square distribution table with df = (r - 1) * (c - 1).
5. Compare to ?: if p-value <-or ?² observed > ?² critical, reject H?.
6. Conclude: if H? is rejected, conclude that there's a significant association between the variables.

Common Mistakes

• Mistake: Misinterpreting the p-value as the probability that H? is true.
• Correction: The p-value is the probability of observing the data (or more extreme) if H? is true. It's not a probability statement about H? itself.
• Mistake: Failing to check the assumptions of the Chi-Square Test of Independence (e.g., sample size, categorical variables).
• Correction: Make sure the variables are categorical and the sample size is sufficiently large (usually n-20).
• Mistake: Using the wrong degrees of freedom (e.g., df = (r - 1) * (c - 1) instead of df = (r - 1) * (c - 1) - 1).
• Correction: Use the correct formula for degrees of freedom.

Quick Practice Problems

1. A marketing firm wants to know if there's a significant association between the type of product (e.g., electronics, clothing, home goods) and the average daily sales. The observed frequencies are: electronics = 50, clothing = 30, home goods = 20. The expected frequencies are: electronics = 40, clothing = 30, home goods = 30. What is the p-value?

p-value = 0.01 (the observed frequencies are significantly different from the expected frequencies).

1. A quality control team wants to know if there's a significant association between the type of defect (e.g., material, manufacturing, design) and the number of defects per unit. The observed frequencies are: material = 10, manufacturing = 20, design = 30. The expected frequencies are: material = 15, manufacturing = 20, design = 25. What is the p-value?

p-value = 0.05 (the observed frequencies are significantly different from the expected frequencies).

1. A retail chain wants to know if there's a significant association between the type of customer (e.g., male, female, other) and the average daily sales. The observed frequencies are: male = 50, female = 30, other = 20. The expected frequencies are: male = 40, female = 30, other = 30. What is the p-value?

p-value = 0.01 (the observed frequencies are significantly different from the expected frequencies).

Last-Minute Cram Sheet

1. Chi-Square Test of Independence: used to test the association between two categorical variables.
2. Phi Coefficient (?): measures the strength and direction of the association between two binary variables.
3. Cramér's V (V): measures the strength of the association between two categorical variables.
4. Degrees of Freedom (df): df = (r - 1) * (c - 1) where r = number of rows, c = number of columns.
5. Observed Frequency (O): the actual number of times a category occurs.
6. Expected Frequency (E): the expected number of times a category occurs if there's no association between the variables.
7. Sample Size (n): the total number of observations in the sample.
8. Number of Categories (k): the number of categories in one of the variables.
9. Chi-Square Statistic (?²): measures the difference between observed and expected frequencies.
10. p-value: the probability of observing the data (or more extreme) if H? is true.
11. p-value is NOT the probability that H? is true – it’s the probability of observing the data (or more extreme) if H? is true.
12. Assumptions of the Chi-Square Test of Independence: categorical variables, sample size-20.