Chi-Square Tests and Contingency Tables MCQs 2026

Pearson chi-square statistic for goodness of fit is

1. A χ² = Σ[(Oi − Ei)²/Ei] ✓
2. B Σ(Oi·Ei) ✓
3. C Σ(Oi−Ei) ✓
4. D Oi/Ei only ✓

For testing Poisson goodness of fit, cells with small λ may need

1. A combining categories to ensure adequate Ei ✓
2. B ignoring df ✓
3. C using z-test on means only ✓
4. D dropping χ² ✓

χ²_{0.05, 1} is approximately

1. A 1.96 ✓
2. B 6.635 ✓
3. C 3.841 ✓
4. D 0 ✓

χ² test p-value is area to the right of observed χ² under

1. A normal with n−1 df ✓
2. B t distribution always ✓
3. C χ² distribution with appropriate df ✓
4. D uniform ✓

Before chi-square, categories should be

1. A overlapping to boost n ✓
2. B without totals ✓
3. C mutually exclusive and collectively exhaustive ✓
4. D always continuous ✓

Phi coefficient is special case of association measure for

1. A 10×10 tables only ✓
2. B continuous variables ✓
3. C paired t tests ✓
4. D 2×2 tables ✓

Cramér's V measures

1. A only Type I error ✓
2. B only expected counts ✓
3. C strength of association in contingency tables ✓
4. D only regression R² ✓

Simpson's paradox can appear in

1. A only continuous regression ✓
2. B contingency tables when marginal and conditional associations differ ✓
3. C only one cell tables ✓
4. D only goodness of fit ✓

Likelihood-ratio chi-square G² is

1. A an alternative statistic that also approaches χ² distribution asymptotically ✓
2. B always negative ✓
3. C equal to t² always ✓
4. D unrelated to contingency tables ✓

If all Oi equal Ei exactly, χ² equals

1. A 1 ✓
2. B df ✓
3. C undefined ✓
4. D 0 ✓

For 4×3 contingency table, minimum df is

1. A 5 ✓
2. B 11 ✓
3. C 6 ✓
4. D 2 ✓

Chi-square test is classified as

1. A a test on paired normal means only ✓
2. B a test on categorical/count data ✓
3. C only for regression slopes ✓
4. D only for time series trend ✓

Expected counts under independence sum to

1. A zero ✓
2. B each Oi ✓
3. C df ✓
4. D grand total n in the full table ✓

Rejecting independence implies

1. A causation is proven ✓
2. B evidence of association between the two categorical variables ✓
3. C means are different ✓
4. D variances differ ✓

Goodness-of-fit to binomial with unknown p estimates p from data, reducing df by

1. A 0 ✓
2. B 1 ✓
3. C 2 ✓
4. D k ✓

When r = 1 or c = 1, independence test is

1. A df = 0 still valid ✓
2. B χ² = 0 always ✓
3. C use paired t ✓
4. D not applicable — need at least 2×2 ✓

Contribution of one cell to χ² is

1. A (Oi−Ei)²/Ei ✓
2. B Oi·Ei ✓
3. C (Oi−Ei) ✓
4. D √Ei only ✓

Pearson chi-square requires

1. A random sampling and mutually exclusive categories ✓
2. B normal distribution of each cell ✓
3. C equal variances of means ✓
4. D paired differences normal ✓

In 5×2 table, df for independence equals

1. A 9 ✓
2. B 5 ✓
3. C 4 ✓
4. D 1 ✓

Larger discrepancy between O and E produces

1. A smaller χ² ✓
2. B df = 0 ✓
3. C automatic acceptance ✓
4. D larger χ² value ✓

χ² statistic cannot be negative because

1. A Oi are always negative ✓
2. B it sums squared standardized differences ✓
3. C Ei are zero always ✓
4. D df is negative ✓

Pooling adjacent categories may be done when

1. A Ei are all large ✓
2. B expected counts are too small to meet chi-square rules ✓
3. C df is already maximum ✓
4. D H0 is certainly true ✓

Chi-square homogeneity test compares

1. A distribution of a categorical variable across several populations ✓
2. B means of normal variables only ✓
3. C variances only ✓
4. D correlation only ✓

McNemar's test applies to

1. A paired nominal data in 2×2 table (before–after) ✓
2. B independent two samples proportion z-test ✓
3. C r×c independence with large n ✓
4. D goodness of fit to normal ✓

Fisher's exact test is used for

1. A large r×c tables always ✓
2. B continuous normal data ✓
3. C 2×2 tables with small expected counts ✓
4. D paired t situations ✓

Standardized residual for cell (i,j) is approximately

1. A Oi−Eij without square root ✓
2. B (Oi − Eij)/√Eij ✓
3. C Eij/Oi ✓
4. D χ²/df ✓

Test of independence H0 states

1. A equal means across groups ✓
2. B variance ratio equals 1 ✓
3. C π = 0.5 ✓
4. D no association between row and column variables ✓

For goodness of fit to equally likely k categories, each Ei equals

1. A n·pi with unknown pi ✓
2. B Oi ✓
3. C n/k ✓
4. D k/n ✓

Chi-square distribution is

1. A symmetric like normal always ✓
2. B negative for large df ✓
3. C right-skewed and defined for positive values only ✓
4. D identical to t ✓

Marginal totals in contingency table are

1. A only diagonal cells ✓
2. B only χ² statistic ✓
3. C only expected counts ✓
4. D row sums and column sums ✓

Observed frequencies in contingency table are

1. A always equal to expected under H0 ✓
2. B the p-value ✓
3. C the actual sample counts in each cell ✓
4. D the degrees of freedom ✓

If χ² calculated exceeds χ²_{α, df}, decision at level α is

1. A fail to reject always ✓
2. B accept with certainty ✓
3. C use t-test instead ✓
4. D reject H0 of independence (or goodness of fit) ✓

Yates continuity correction for 2×2 tables

1. A adds 1 to every cell ✓
2. B doubles all df ✓
3. C replaces χ² with F ✓
4. D subtracts 0.5 from |Oi−Ei| before squaring (in standard formula) ✓

Chi-square approximation is unreliable when

1. A all Ei exceed 100 ✓
2. B many expected cell counts are less than 5 ✓
3. C n is large with adequate Ei ✓
4. D only r = 2 ✓

A 3×4 contingency table has df for independence test equal

1. A 6 ✓
2. B 11 ✓
3. C 7 ✓
4. D 3 ✓

A 2×2 contingency table for independence has

1. A 1 degree of freedom ✓
2. B 0 ✓
3. C 2 ✓
4. D 3 ✓

Expected frequency under independence in cell (i,j) is

1. A Oi only ✓
2. B (row i total × column j total) / grand total ✓
3. C row total only ✓
4. D column total − row total ✓

Chi-square test of independence in an r×c contingency table has df

1. A r + c − 1 ✓
2. B rc − 1 ✓
3. C n − 1 ✓
4. D (r − 1)(c − 1) ✓

In a goodness-of-fit test with k categories, df equal

1. A k − 1 − (number of parameters estimated from data) ✓
2. B k ✓
3. C k + 1 ✓
4. D n − 1 always ✓

Goodness-of-fit test evaluates whether

1. A two categorical variables are independent only ✓
2. B sample data conform to a specified theoretical distribution ✓
3. C means of two groups are equal ✓
4. D σ² is zero ✓