Chi-Square Goodness-of-Fit Tests
Time Allocation: 15 minutes total (5 min reading, 10 min individual work)
Part 1: Chi-Square Calculations (12 minutes)
Perform chi-square goodness-of-fit test calculations for the following 4 scenarios:
Example 1: Fair Die Test
Context: Testing if a six-sided die is fair
- Observed counts: 1:12, 2:7, 3:14, 4:15, 5:4, 6:8 (total = 60 rolls)
- Expected under \(H_0\): Each side = 10 (equal probability)
Calculate:
- \(\chi^2\) =
- \(df\) =
- Decision at \(\alpha = 0.05\):
Example 2: Gender Representation in STEM
Context: Testing if university STEM program has equal gender representation
- Observed: Female = 200, Male = 300 (total = 500 students)
- Expected under \(H_0\): Female = 250, Male = 250 (equal representation)
Calculate:
- \(\chi^2\) =
- \(df\) =
- Decision at \(\alpha = 0.05\):
Example 3: Genetic Inheritance Test
Context: Testing if genetic traits follow Mendelian 3:1 ratio
- Observed: Dominant = 75, Recessive = 25 (total = 100 offspring)
- Expected under \(H_0\): Dominant = 75, Recessive = 25 (3:1 ratio)
Calculate:
- \(\chi^2\) =
- \(df\) =
- Decision at \(\alpha = 0.05\):
Example 4: Survey Response Distribution
Context: Testing if survey responses follow uniform distribution
- Observed: A=12, B=15, C=18, D=10 (total = 55 responses)
- Expected under \(H_0\): Each category = 13.75 (equal probability)
Calculate:
- \(\chi^2\) =
- \(df\) =
- Decision at \(\alpha = 0.05\):
Show your work for one calculation:
# R code for chi-square goodness-of-fit calculations (for reference)
# Example 1: Fair Die Test
observed_die <- c(12, 7, 14, 15, 4, 8)
expected_probs_die <- rep(1/6, 6)
die_test <- chisq.test(x = observed_die, p = expected_probs_die)
die_test
Chi-squared test for given probabilities
data: observed_die
X-squared = 9.4, df = 5, p-value = 0.09413
# Example 2: Gender Representation
observed_gender <- c(200, 300)
expected_probs_gender <- c(0.5, 0.5)
gender_test <- chisq.test(x = observed_gender, p = expected_probs_gender)
gender_test
Chi-squared test for given probabilities
data: observed_gender
X-squared = 20, df = 1, p-value = 7.744e-06
# Example 3: Genetic Inheritance
observed_genetic <- c(75, 25)
expected_probs_genetic <- c(0.75, 0.25)
genetic_test <- chisq.test(x = observed_genetic, p = expected_probs_genetic)
genetic_test
Chi-squared test for given probabilities
data: observed_genetic
X-squared = 0, df = 1, p-value = 1
# Example 4: Survey Response Distribution
observed_survey <- c(12, 15, 18, 10)
expected_probs_survey <- rep(1/4, 4)
survey_test <- chisq.test(x = observed_survey, p = expected_probs_survey)
survey_test
Chi-squared test for given probabilities
data: observed_survey
X-squared = 2.6727, df = 3, p-value = 0.4449
Part 2: Decision Making and Interpretation (3 minutes)
Make decisions and interpret results:
- For the fair die test (Example 1), what does your conclusion tell us about the die’s fairness?
- For the gender representation study (Example 2), what practical implications does your finding have for the STEM program?
Critical Thinking: Why is it important to check the expected counts assumption before interpreting chi-square test results?