Correlation and Regression Analysis MCQs 2026

Pearson product-moment correlation coefficient r measures

1. A only causation between variables ✓
2. B only curvilinear association ✓
3. C the strength and direction of linear association between two quantitative variables ✓
4. D only rank-order agreement ✓

The range of Pearson correlation coefficient r is

1. A from 0 to 1 only ✓
2. B from −1 to +1 inclusive ✓
3. C from −∞ to +∞ ✓
4. D from 0 to 100 only ✓

A Pearson r of +1 indicates

1. A no linear relationship ✓
2. B a perfect negative linear relationship ✓
3. C independence of variables ✓
4. D a perfect positive linear relationship ✓

A Pearson r near zero suggests

1. A perfect linear association ✓
2. B necessarily no relationship of any kind ✓
3. C equal variances only ✓
4. D little or no linear association ✓

The coefficient of determination r² represents

1. A the correlation sign only ✓
2. B the proportion of variance in Y explained by the linear regression on X ✓
3. C the intercept of the regression line ✓
4. D the standard error of estimate ✓

The method of least squares in regression chooses the line that

1. A minimizes the sum of absolute residuals only ✓
2. B minimizes the sum of squared residuals ✓
3. C maximizes correlation always ✓
4. D minimizes the intercept only ✓

In simple linear regression ŷ = a + bx, the slope b equals

1. A ȳ / x̄ always ✓
2. B Σxᵢyᵢ without centering ✓
3. C the correlation r alone without scaling ✓
4. D Σ(xᵢ − x̄)(yᵢ − ȳ) / Σ(xᵢ − x̄)² ✓

The least-squares intercept a in ŷ = a + bx is

1. A x̄ − b ȳ ✓
2. B ȳ − b x̄ ✓
3. C zero always ✓
4. D equal to the correlation r ✓

A residual in regression is defined as

1. A ŷᵢ − xᵢ ✓
2. B yᵢ + ŷᵢ ✓
3. C eᵢ = yᵢ − ŷᵢ ✓
4. D xᵢ − x̄ only ✓

The standard error of estimate in simple regression measures

1. A typical size of residuals around the regression line ✓
2. B only the slope sampling error ✓
3. C only the intercept ✓
4. D the correlation r directly ✓

In simple regression, testing H₀: β = 0 is equivalent to testing

1. A whether r² equals one ✓
2. B whether the intercept is zero ✓
3. C whether residuals are always positive ✓
4. D whether the population correlation ρ equals zero ✓

Multiple linear regression models Y as

1. A only one independent variable always ✓
2. B only polynomial terms without intercept ✓
3. C a linear function of two or more independent variables plus error ✓
4. D only ranks of Y ✓

The multiple correlation coefficient R is

1. A the correlation between observed Y and fitted Ŷ from multiple regression ✓
2. B always equal to each partial correlation ✓
3. C the square root of r in simple regression only ✓
4. D unrelated to R² ✓

R² in multiple regression is

1. A the proportion of total variation in Y explained by all regressors ✓
2. B always equal to Pearson r ✓
3. C the residual sum of squares ✓
4. D the partial correlation of one X only ✓

Adjusted R² differs from R² by

1. A always exceeding R² ✓
2. B penalizing the addition of extra predictors ✓
3. C ignoring sample size ✓
4. D equaling the intercept variance ✓

Partial correlation between Y and X₁ controlling for X₂ measures

1. A zero-order correlation only ✓
2. B multiple R directly ✓
3. C correlation of Y with X₁ after removing linear effects of X₂ from both ✓
4. D correlation of residuals with X₂ ✓

Multiple correlation R equals

1. A the negative square root of r² always ✓
2. B the sum of partial correlations ✓
3. C the intercept divided by slope ✓
4. D the positive square root of R² ✓

A positive Pearson correlation means

1. A Y always decreases as X increases ✓
2. B Y tends to increase as X increases linearly ✓
3. C variables are independent ✓
4. D the slope must be negative ✓

A negative Pearson correlation means

1. A Y always increases with X ✓
2. B Y tends to decrease as X increases linearly ✓
3. C r² is negative ✓
4. D the relationship must be nonlinear ✓

Correlation is symmetric in the sense that

1. A corr(X, Y) equals corr(Y, X) ✓
2. B corr(X,Y) equals −corr(Y,X) ✓
3. C only Y may be regressed on X ✓
4. D r depends on which variable is response ✓

Pearson r is inappropriate when the relationship is

1. A strongly curvilinear ✓
2. B perfectly linear ✓
3. C moderately linear with outliers removed ✓
4. D based on interval data ✓

Outliers can

1. A never affect least squares ✓
2. B only change the intercept never the slope ✓
3. C eliminate the need for residuals ✓
4. D distort Pearson r and regression coefficients ✓

The regression line always passes through

1. A the point (x̄, ȳ) ✓
2. B the origin unless forced ✓
3. C the minimum observed X only ✓
4. D any arbitrary point ✓

Extrapolation beyond the observed X range is risky because

1. A the linear model may not hold outside the data cloud ✓
2. B r² becomes exactly one ✓
3. C residuals vanish ✓
4. D slope always reverses sign ✓

Homoscedasticity in regression means

1. A residual variance increases linearly always ✓
2. B errors are perfectly correlated ✓
3. C residual variance is constant across levels of X ✓
4. D Y is measured in ranks only ✓

Heteroscedasticity refers to

1. A zero residuals ✓
2. B perfect multicollinearity ✓
3. C non-constant variance of residuals across X or ŷ ✓
4. D negative R² ✓

Multicollinearity among predictors causes

1. A unstable and inflated standard errors of regression coefficients ✓
2. B R² to be exactly zero ✓
3. C residuals to sum to n ✓
4. D perfect prediction always ✓

The variance inflation factor (VIF) is used to detect

1. A heteroscedasticity only ✓
2. B non-normality only ✓
3. C autocorrelation only ✓
4. D multicollinearity ✓

Durbin–Watson statistic helps detect

1. A autocorrelation in regression residuals ✓
2. B multicollinearity ✓
3. C heteroscedasticity only ✓
4. D zero correlation ✓

In regression ANOVA, the F-test for overall significance tests

1. A H₀: intercept equals zero only ✓
2. B H₀: all slope coefficients except intercept are zero ✓
3. C H₀: r equals one ✓
4. D H₀: residuals are all equal ✓

Total sum of squares SST in regression equals

1. A Σ(yᵢ − ŷᵢ)² only ✓
2. B Σ(yᵢ − ȳ)² ✓
3. C Σ(xᵢ − x̄)² only ✓
4. D Σŷᵢ² only ✓

Explained sum of squares SSR equals

1. A Σ(yᵢ − ŷᵢ)² ✓
2. B Σ(xᵢ − x̄)² ✓
3. C Σ(ŷᵢ − ȳ)² ✓
4. D Σeᵢ² with e signed ✓

Error sum of squares SSE equals

1. A Σ(ŷᵢ − ȳ)² ✓
2. B Σ(yᵢ − ȳ)² − Σxᵢ ✓
3. C Σxᵢyᵢ ✓
4. D Σ(yᵢ − ŷᵢ)² ✓

The Gauss–Markov theorem states OLS estimators are BLUE under

1. A classical linear regression assumptions ✓
2. B any nonlinear model ✓
3. C rank data only ✓
4. D zero sample size ✓

A standardized regression coefficient compares

1. A the effect of a one standard deviation change in X on Y ✓
2. B raw units only without scaling ✓
3. C only the intercept ✓
4. D only residual variance ✓

A dummy (indicator) variable in regression takes values

1. A 0 and 1 to represent category membership ✓
2. B only continuous measurements ✓
3. C only ranks ✓
4. D only squared terms ✓

The partial regression coefficient β₁ in multiple regression represents

1. A the zero-order slope ignoring other variables ✓
2. B always equal to Pearson r ✓
3. C the intercept change only ✓
4. D the change in expected Y per unit change in X₁ holding other Xs fixed ✓

Stepwise regression

1. A always includes every candidate variable ✓
2. B enters or removes predictors based on preset criteria ✓
3. C never uses R² ✓
4. D eliminates the need for diagnostics ✓

A prediction interval for a new Y differs from a confidence interval for E(Y) because it includes

1. A only slope uncertainty ✓
2. B individual observation variability (error term) ✓
3. C only the mean always ✓
4. D zero residual variance ✓

Regression toward the mean describes

1. A perfect correlation over time ✓
2. B slope always increasing ✓
3. C extreme values tending to be followed by less extreme values on average ✓
4. D R² decreasing to zero ✓

Spurious correlation occurs when

1. A r equals exactly one ✓
2. B residuals are homoscedastic ✓
3. C two variables correlate because of a lurking variable or coincidence ✓
4. D the slope is zero ✓

Correlation does not imply

1. A linear association ✓
2. B a symmetric measure ✓
3. C computation from paired data ✓
4. D causation ✓

Covariance and correlation are related because

1. A r equals cov without scaling ✓
2. B r = cov(X,Y) / (sₓ sᵧ) ✓
3. C cov is always between −1 and 1 ✓
4. D correlation equals variance ✓

Pearson r is unitless because

1. A it uses ranks only ✓
2. B it ignores sample size ✓
3. C it equals the slope always ✓
4. D standardization removes measurement units ✓

In a scatter diagram, a tight upward cloud suggests

1. A negative correlation ✓
2. B zero correlation only ✓
3. C heteroscedasticity proof ✓
4. D positive correlation ✓

The least-squares principle was applied to regression by

1. A Fisher only in ANOVA tables ✓
2. B Kolmogorov in non-parametrics ✓
3. C Legendre and Gauss in the early 19th century ✓
4. D Laspeyres in index theory ✓

If all residuals are zero, then

1. A r must be zero ✓
2. B SSE equals SST ✓
3. C R² equals 1 and all points lie on the fitted line ✓
4. D the slope is undefined always ✓

In simple regression, b equals r times

1. A sₓ / sᵧ ✓
2. B sₓ sᵧ ✓
3. C 1 − r² ✓
4. D sᵧ / sₓ ✓

The normal equations in OLS lead to

1. A only iterative numerical methods always ✓
2. B closed-form solutions for a and b in simple regression ✓
3. C zero degrees of freedom always ✓
4. D rank-based estimates ✓

Serial correlation in residuals violates the assumption of

1. A linearity only ✓
2. B independence of errors ✓
3. C existence of Y only ✓
4. D positive r only ✓

Leverage of an observation depends mainly on

1. A only its Y value ✓
2. B only the residual sign ✓
3. C only sample size below 5 ✓
4. D how unusual its predictor values are ✓

An influential observation

1. A substantially changes fitted regression results if removed ✓
2. B always has small residual ✓
3. C must have zero leverage ✓
4. D cannot be an outlier ✓

A fitted value ŷᵢ is

1. A the value of Y predicted by the regression for given Xᵢ ✓
2. B always equal to yᵢ ✓
3. C the residual eᵢ ✓
4. D the mean x̄ only ✓

Regression assumes errors are

1. A always positive ✓
2. B equal to xᵢ ✓
3. C random variables with mean zero ✓
4. D identical to yᵢ ✓

Multiple regression R² never decreases when

1. A an additional predictor is added ✓
2. B an irrelevant predictor is removed ✓
3. C multicollinearity is absent ✓
4. D sample size increases ✓

Partial correlation r₁₂·₃ equals zero when

1. A r₁₂ equals one ✓
2. B the partial regression coefficient of X₁ is zero given X₃ ✓
3. C R² equals zero always ✓
4. D residuals are heteroscedastic ✓

A scatter plot with heteroscedastic fan shape suggests

1. A perfect homoscedasticity ✓
2. B r must equal 1 ✓
3. C regression standard errors may be unreliable without remedy ✓
4. D no need to examine residuals ✓

Standardized residuals help identify

1. A only leverage without magnitude ✓
2. B only the intercept ✓
3. C unusually large deviations relative to estimated error scale ✓
4. D only index numbers ✓

Testing ρ = 0 using t with n − 2 df relies on

1. A only ordinal measurement ✓
2. B only χ² distribution always ✓
3. C bivariate normality for exact inference ✓
4. D no distributional assumptions ever ✓

In interpreting β = 2.5 for income predicting spending, expected spending rises

1. A 2.5 percent always ✓
2. B by the correlation only ✓
3. C only when r is negative ✓
4. D 2.5 units per one-unit increase in income holding other variables fixed ✓