Unit 8: Inference for Chi Square

Chi-Square distribution

A density curve that takes only positive values and is skewed right

• The larger the df get → closer the distribution becomes normal-like

• The mean of the chi-square is the the degrees of freedom

• The X² statistic is always positive

Chi-square distribution

(observed-expected)² / expected

• Whichever observed sample produces the largest contribute, affects the X² the most.

Calculating Expected Values

Goodness of Fit Test: (expected proportion)(total sample size)

Chi Square Test: (row total)(column) / (grand total)

• Create Matrices

  • Press 2nd, x¹

  • Edit row x column and input numbers

  • Press X²-Test calculator and check matrice B

Chi-Square Test for Goodness of Fit

Compares observed values to expected values. Its used to determine if the difference is statistically significant

1) STATE

• Null Hypothesis H₀: The claimed distribution of …(context).. is true

  • Proportions (expected)→ p₁=p₂=p₃

• Alternative Hypothesis H_a: The claimed distribution of …(context) is NOT true

  • Proportions (observed)→ p₁≠p₂≠p₃

• Significance level

2) PLAN

• Random: They took a random sample sample → establish generalization

• Independence: sample size<10% of population → assume independence

• Large counts: All expected values are greater than 5 → can use X² distribution

  • State the expected values

3) DO

• X² statistic: Σ(observed - expected)² / expected

  • Calculator : Create a data table, L₁ is observed, L₂ is expected→ Use STAT: X²-GOF TEST

• p-value: (X², 999, df) → df = (#of categories)-1

4) Conclude

If p < 0.05 →Reject H₀

If p > 0.05 → Fail to reject H₀

Chi-Square Test for Homogeneity

Tests if different populations have the same proportions for a categorical value (EX: Hair color→ black, blonde, brown)

1) STATE

• Null Hypothesis H₀: There is no difference in (variable) between (population 1) and (population 2)

• Alternative Hypothesis H_a: There is a difference in (variable) between (population 1) and (population 2)

• Significance level

2) PLAN

• Random: They took a random sample from each population→ establish generalization

• Independence: each sample size<10% of population → assume independence

• Large counts: All expected values are greater than 5 → can use X² distribution

  • Expected values: (row total)(column total) / (grand total)

3) DO

• X² statistic: Σ(observed - expected)² / expected

  • Calculator : Create Matrices → Press 2nd, x^{1 →} Edit row x column and input numbers → Press X²-Test calculator

• p-value: (X², 999, df) → df = (row - 1)(column-1)

4) CONCLUDE

If p < 0.05 →Reject H₀

If p > 0.05 → Fail to reject H₀

Chi-Square Test for Independence

To determine whether two categorical variables are associated within a single population (EX→Is gender related to favorite food)

1) STATE

• Null Hypothesis H₀: There is no association in between (variable 1) and (variable 2)

• Alternative Hypothesis H_a: There is a association in between (variable 1) and (variable 2)

• Significance level

2) PLAN

• Random: They took a random sample from population→ establish generalization

• Independence: sample size<10% of population → assume independence

• Large counts: All expected values are greater than 5 → can use X² distribution

  • Expected values: (row total)(column total) / (grand total)

3) DO

• X² statistic: Σ(observed - expected)² / expected

  • Calculator : Create Matrices → Press 2nd, x^{1 →} Edit row x column and input numbers → Press X²-Test calculator

• p-value: (X², 999, df) → df = (row - 1)(column-1)

4) CONCLUDE

If p < 0.05 →Reject H₀

If p > 0.05 → Fail to reject H₀