Econ Stats Exam 2

What is the population regression model?

Y = β₀ + β₁X + ε.

What does Y represent in the population regression model?

The population response (dependent) variable.

What does X represent in the population regression model?

The explanatory (independent) variable.

What does β₀ represent?

The population y-intercept.

What does β₁ represent?

The population slope; the change in the mean of Y for a one-unit increase in X.

What does ε represent?

The random error term for an individual observation.

What is the sample regression equation (least squares regression line)?

ŷ = b₀ + b₁x.

What does ŷ represent?

The predicted value of the response variable.

What does b₀ represent?

The estimated y-intercept.

What does b₁ represent?

The estimated slope.

What does x represent in the sample regression equation?

The observed or chosen value of the explanatory variable.

What is the difference between β₀ and b₀?

β₀ is the true population intercept (parameter); b₀ is its sample estimate (estimator).

What is the difference between β₁ and b₁?

β₁ is the true population slope (parameter); b₁ is its sample estimate (estimator).

What are parameters?

Fixed, usually unknown numerical characteristics of a population, such as β₀ and β₁.

What are estimators?

Random variables calculated from a sample to estimate population parameters, such as b₀ and b₁.

What is an estimate?

The specific numerical value an estimator produces from a particular sample.

What is the difference between Y and ŷ?

Y is the actual observed response; ŷ is the response predicted by the regression line.

What is a residual?

The difference between an actual value and its predicted value: e = y - ŷ.

What is the goal of least squares regression?

To find the line that minimizes the sum of squared residuals.

What does SSR stand for?

Sum of Squares Regression.

What does SSR measure conceptually?

The variation in Y explained by the regression model.

What does SSE stand for?

Sum of Squares Error.

What does SSE measure conceptually?

The variation in Y that remains unexplained by the regression model.

What does SST stand for?

Total Sum of Squares.

What does SST measure conceptually?

The total variation in the observed Y values around their mean.

What is the relationship between SST, SSR, and SSE in simple linear regression with an intercept?

SST = SSR + SSE.

What does R² measure?

The proportion of the total variation in Y explained by the regression model.

How do you interpret R² = 0.80?

The model explains 80% of the variation in Y.

What is the range of R² in ordinary least squares regression with an intercept?

0 to 1, or 0% to 100%.

What does an R² close to 1 indicate?

The model explains a large proportion of the variation in Y.

What does an R² close to 0 indicate?

The model explains little of the variation in Y.

What is the sampling distribution of b₁?

The probability distribution of the estimated slope b₁ across all possible random samples of the same size.

What is the standard error of b₁?

The estimated standard deviation of the sampling distribution of b₁; it measures how much the estimated slope varies from sample to sample.

What is the formula for the standard error of b₁?

SE(b₁) = s / √[Σ(xᵢ - x̄)²], where s = √[SSE/(n - 2)].

What does a small standard error of b₁ indicate?

The estimated slope is relatively precise and varies less from sample to sample.

What does a large standard error of b₁ indicate?

The estimated slope is less precise and varies more from sample to sample.

What is the t-statistic for testing the population slope?

t = (b₁ - β₁,₀) / SE(b₁), where β₁,₀ is the hypothesized slope.

What is the usual null hypothesis when testing whether a linear relationship exists?

H₀: β₁ = 0.

What is the alternative hypothesis for a two-sided slope test?

Hₐ: β₁ ≠ 0.

What does H₀: β₁ = 0 mean?

There is no linear relationship between X and the population mean of Y.

What does Hₐ: β₁ ≠ 0 mean?

There is a statistically significant linear relationship between X and the population mean of Y.

What degrees of freedom are used for the t-test of the slope in simple linear regression?

n - 2.

When do you reject the null hypothesis using the p-value method?

Reject H₀ if the p-value is less than α.

When do you fail to reject the null hypothesis?

When the p-value is greater than or equal to α.

What does a statistically significant slope NOT prove?

It does not prove causation or that the relationship is practically important.