Q: Given two random variables $X$ and $Y$, which formula correctly expresses the variance of their sum?

$V(X+Y) = V(X) + V(Y) + 2\text{Cov}(X,Y)$

Q: Which formula gives the population coefficient of correlation $\rho$ between $X$ and $Y$?

$\rho = \frac{\text{Cov}(X,Y)}{\sqrt{V(X)} \sqrt{V(Y)}}$

Question 1

Which of the following is the correct formula for the sample variance of a dataset $x_1, x_2, \ldots, x_n$?

Accepted Answer

$s^2_x = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2$

Question 2

Suppose you toss a fair coin three times. Which property of a random experiment does this scenario best illustrate?

Accepted Answer

A collection of every possible outcome can be described prior to its performance.

Question 3

Given two random variables $X$ and $Y$, which formula correctly expresses the variance of their sum?

Accepted Answer

$V(X+Y) = V(X) + V(Y) + 2	ext{Cov}(X,Y)$

Question 4

Which of the following statements about the population mean $\mu$ is correct for a discrete random variable $X$?

Accepted Answer

$E(X) = \sum_x x P(x)$

Question 5

A researcher is analyzing the relationship between hours studied ($X$) and exam score ($Y$). If $	ext{Cov}(X,Y) = 0$, which of the following must be true?

Accepted Answer

There is no linear relationship between $X$ and $Y$.

Question 6

Which formula gives the population coefficient of correlation $\rho$ between $X$ and $Y$?

Accepted Answer

$ho = \frac{	ext{Cov}(X,Y)}{\sqrt{V(X)} \sqrt{V(Y)}}$

Question 7

In the context of linear regression, what does the OLS estimator minimize?

Accepted Answer

The sum of squared residuals between observed $Y_i$ and predicted $\hat{Y}_i$.

Question 8

Which formula correctly defines the $R^2$ statistic in a regression model?

Accepted Answer

$R^2 = 1 - \frac{SSR}{TSS}$

Question 9

If $X$ and $Y$ are independent random variables, which property of conditional expectation holds?

Accepted Answer

$E(Y|X) = E(Y)$

Question 10

A medical researcher wants to estimate the effect of a new drug ($X_1$) on blood pressure ($Y$), controlling for age ($X_2$) and weight ($X_3$). Which regression model should be used?

Accepted Answer

$Y_i = \beta_0 + \beta_1 X_{1i} + \beta_2 X_{2i} + \beta_3 X_{3i} + u_i$

Question 11

Which of the following is the correct formula for the standard error of the regression (SER)?

Accepted Answer

$SER = \sqrt{s^2_u}$

Question 12

Which property of variance states that $V(c) = 0$ for any constant $c$?

Accepted Answer

Property 1

Question 13

In hypothesis testing for a regression coefficient, what is the formula for the t-statistic?

Accepted Answer

$t = \frac{\hat{\beta}_j - \beta_{j,0}}{SE(\hat{\beta}_j)}$

Question 14

Which of the following is a correct interpretation of the adjusted $R^2$ statistic in regression analysis?

Accepted Answer

It penalizes the addition of unnecessary regressors.

Question 15

Which law of expectation allows you to write $E(aX + b) = aE(X) + b$ for constants $a$ and $b$?

Accepted Answer

Property 3

Question 16

A company wants to predict sales ($Y$) based on advertising spend ($X$). If the variance of the error term $u$ depends on $X$, what is this called?

Accepted Answer

Heteroskedasticity

Question 17

Which formula represents the 95% confidence interval for a regression coefficient $\hat{\beta}_j$?

Accepted Answer

$[\hat{\beta}_j - 1.96SE(\hat{\beta}_j), \hat{\beta}_j + 1.96SE(\hat{\beta}_j)]$

Question 18

Why is 'consistency' of an estimator important in econometric analysis?

Accepted Answer

It guarantees the estimator converges to the true parameter value as sample size increases.

Question 19

Which property of covariance states that $	ext{Cov}(X,Y) = 	ext{Cov}(Y,X)$?

Accepted Answer

Property 1

Comprehensive Review of Statistics and Regression Methods

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