QCM : Fundamentals of Regression and Hypothesis Testing — 16 questions

Question 1

1. Which statement best describes the normality assumption in simple linear regression?

The independent variable must be normally distributed for valid inference

The dependent variable must be normally distributed for valid inference

Both variables must be exactly normal in every sample

The residuals should be normally distributed for valid inference

Explication

In simple linear regression, normality is assumed for the residuals, not for the raw X and Y variables. This assumption supports inference about the regression coefficients.

Answer

The residuals should be normally distributed for valid inference

Question 2

2. What does a non-random pattern in a residuals-versus-X plot most strongly suggest?

That the regression slope must be zero

That homoskedasticity is satisfied

That the sample size is too large for inference

A violation such as heteroskedasticity or non-independence

Explication

A systematic residual pattern indicates the residuals are not behaving randomly across X, which points to problems like heteroskedasticity or non-independence. A random pattern is what supports the homoskedasticity assumption.

Answer

A violation such as heteroskedasticity or non-independence

Question 3

3. How is the estimated slope in simple linear regression computed?

By dividing the centered cross-product of X and Y by the centered sum of squares of X

By dividing the centered sum of squares of Y by the centered sum of squares of X

By subtracting the mean of Y from the mean of X

By multiplying the sample correlation by the mean of X

Explication

The slope estimate is hat b1 = sum((Xi - Xbar)(Yi - Ybar)) / sum((Xi - Xbar)^2). This uses the centered cross-product over the centered spread in X.

Answer

By dividing the centered cross-product of X and Y by the centered sum of squares of X

Question 4

4. If the estimated slope is known, how is the intercept found in simple linear regression?

By dividing the mean of Y by the slope

By subtracting slope times the mean of X from the mean of Y

By subtracting the mean of Y from the slope

By adding slope times the mean of X to the mean of Y

Explication

The intercept is hat b0 = Ybar - hat b1 Xbar. Adding instead of subtracting would make the intercept too large.

Answer

By subtracting slope times the mean of X from the mean of Y

Question 5

5. What does the slope t-test in simple linear regression test?

Whether the correlation is negative using a chi-square statistic

Whether the intercept differs from 0 using z with n degrees of freedom

Whether the residual variance equals 0 using an F statistic

Whether the slope differs from 0 using t with n minus 2 degrees of freedom

Explication

The slope t-test evaluates H0: b1 = 0 with df = n - 2. It uses the estimated slope divided by its standard error.

Answer

Whether the slope differs from 0 using t with n minus 2 degrees of freedom

Question 6

6. What change makes a prediction interval for a future Y value wider?

Using an Xf farther from Xbar

Using an Xf equal to Xbar

Replacing the critical t value with 1

Dropping the forecast-uncertainty term inside the standard error

Explication

The forecast standard error includes a term involving (Xf - Xbar)^2, so the interval gets wider as Xf moves farther from Xbar. The widest intervals occur away from the center of the data.

Answer

Using an Xf farther from Xbar

Question 7

7. In a regression with a 0/1 dummy variable, what does the intercept represent?

The mean of the group coded 1

The mean of the group coded 0

The overall sample mean regardless of group

The difference between the two group means

Explication

With a 0/1 indicator, the intercept is the predicted mean when the dummy equals 0. The slope then represents the difference between the 1-group and the 0-group.

Answer

The mean of the group coded 0

Question 8

8. In a 0/1 dummy-variable regression, what does the dummy coefficient represent?

The correlation between group membership and Y

The mean of the group coded 0

The difference in means between the group coded 1 and the group coded 0

The median difference between the two groups

Explication

The dummy coefficient measures how much the mean of Y changes when the indicator moves from 0 to 1. It is an estimated mean difference, not a correlation.

Answer

The difference in means between the group coded 1 and the group coded 0

Question 9

9. Which test is most appropriate for two approximately normal continuous variables when testing whether the population correlation is zero?

F-test for equal variances

Pearson correlation t-test

Spearman rank-correlation test

Chi-square test of independence

Explication

For approximately normal continuous variables, the parametric Pearson correlation t-test is used to test whether the population correlation is zero. Spearman is preferred when normality is badly violated.

Answer

Pearson correlation t-test

Question 10

10. When continuous variables show marked skewness or outliers, which correlation measure is most appropriate?

Chi-square test of independence

Spearman rank correlation

Paired comparisons t-test

Pearson correlation

Explication

Spearman rank correlation is a nonparametric rank-based measure that is well suited to skewed data and outliers. Pearson correlation is the parametric alternative for roughly normal continuous variables.

Answer

Spearman rank correlation

Question 11

11. What test statistic is used to test a single normal population variance?

A z statistic with n degrees of freedom

An F statistic with r × c degrees of freedom

A t statistic with n − 2 degrees of freedom

A chi-square statistic with n − 1 degrees of freedom

Explication

A single population variance is tested with chi-square using (n − 1)s²/σ₀² and n − 1 degrees of freedom. An F test is used for comparing two variances, not one.

Answer

A chi-square statistic with n − 1 degrees of freedom

Question 12

12. Which test is appropriate for comparing the variances of two independent normal samples?

A chi-square test of independence

An F-test for equal variances

A Pearson correlation t-test

A paired comparisons t-test

Explication

Comparing two independent variances requires an F-test based on the ratio of the sample variances. Chi-square is used for a single variance or contingency-table independence, not for two-sample variance comparison.

Answer

An F-test for equal variances

Question 13

13. When the data consist of mean returns measured on the same days for two indexes, which hypothesis test should be used?

A chi-square test of independence

A one-sample variance test

A paired comparisons test

A one-sample z test for proportions

Explication

The observations are matched by day, so the samples are dependent and should be analyzed with a paired comparisons test. The other choices do not address within-subject or matched-pair differences.

Answer

A paired comparisons test

Question 14

14. If a question asks whether two means are different, what alternative hypothesis is appropriate?

A one-tailed alternative with >

A one-tailed alternative with <

A two-tailed alternative with e

A two-tailed alternative with c0

Explication

The word “different” calls for a two-tailed alternative, written as not equal to. One-tailed alternatives are used only when the direction of the difference is specified.

Answer

A two-tailed alternative with e

Question 15

15. What is the standard error of the sample mean when the population standard deviation is known?

σ multiplied by the square root of n

σ divided by the square root of n

n divided by σ

s divided by n

Explication

The standard error of the sample mean is σ/√n when the population standard deviation is known. This shows why larger samples reduce sampling variability.

Answer

σ divided by the square root of n

Question 16

16. Which method estimates sampling variability by repeatedly leaving out one observation at a time?

Central limit theorem

Stratified sampling

Bootstrap resampling

Jackknife

Explication

The jackknife recomputes the statistic after systematically omitting one observation at a time. Bootstrap instead resamples with replacement from the full data set.

QCM : Fundamentals of Regression and Hypothesis Testing — 16 questions

Questions et réponses du QCM

1. Which statement best describes the normality assumption in simple linear regression?

2. What does a non-random pattern in a residuals-versus-X plot most strongly suggest?

3. How is the estimated slope in simple linear regression computed?

4. If the estimated slope is known, how is the intercept found in simple linear regression?

5. What does the slope t-test in simple linear regression test?

6. What change makes a prediction interval for a future Y value wider?

7. In a regression with a 0/1 dummy variable, what does the intercept represent?

8. In a 0/1 dummy-variable regression, what does the dummy coefficient represent?

9. Which test is most appropriate for two approximately normal continuous variables when testing whether the population correlation is zero?

10. When continuous variables show marked skewness or outliers, which correlation measure is most appropriate?

11. What test statistic is used to test a single normal population variance?

12. Which test is appropriate for comparing the variances of two independent normal samples?

13. When the data consist of mean returns measured on the same days for two indexes, which hypothesis test should be used?

14. If a question asks whether two means are different, what alternative hypothesis is appropriate?

15. What is the standard error of the sample mean when the population standard deviation is known?

16. Which method estimates sampling variability by repeatedly leaving out one observation at a time?

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