Let's clarify and formalize the answers to your questions based on the information provided.
Identified Questions:
-
As the sample size increases, the variance of the sample mean:
- A. Increases
- B. Decreases
Answer: B. Decreases
Explanation: The variance of the sample mean actually decreases with an increase in sample size. This is because the sample mean is a more accurate estimate of the population mean as we include more data points, leading to a smaller standard error of the mean. -
Cross-sectional data are data:
- A. True
- B. False
Answer: A. True
Explanation: Cross-sectional data pertains to data collected at a single point in time, allowing for comparison among various subjects (individuals, firms, etc.) simultaneously. -
An R-squared value close to 1 implies that:
- A. True
- B. False
Answer: A. True
Explanation: An R-squared value close to 1 indicates that the independent variables in a regression model explain a large portion of the variance in the dependent variable. It implies a good fit of the model to the data.
Regression Analysis Questions
Now, let's address the regression-related questions.
-
Suppose we have not enumerated the following regression where Y is the annual production of wheat, X is the independent variable, and u is the error term.
A. Find LSS and RSS (1 point)
- To find the Least Squares Sum (LSS) we need to refer to the residuals. LSS typically isn’t a standard term; you might be referring to the sum of squared residuals (which is the Residual Sum of Squares or RSS).
- The Residual Sum of Squares (RSS) is calculated as: \[ \text{RSS} = \sum (Y_i - \hat{Y}_i)^2 \] where \( Y_i \) is the observed value and \( \hat{Y}_i \) is the predicted value from the regression model.
B. Estimate variance of the residuals (1 point)
- The variance of the residuals can be calculated using the formula: \[ \text{Var}(u) = \frac{\text{RSS}}{n - k} \] where \( n \) is the number of observations and \( k \) is the number of independent variables in the model (including the intercept).
C. Construct test statistics value for X (2 points)
- To construct the test statistic for the coefficient of the independent variable \( X \), you can calculate the t-statistic:
\[
t = \frac{b}{SE(b)}
\]
where:
- \( b \) is the estimated coefficient for \( X \),
- \( SE(b) \) is the standard error of the estimated coefficient \( b \).
To complete these calculations, you'll need the results from the regression analysis, including coefficient estimates and residuals for your specific dataset.
If you have further details or a dataset, we can delve deeper into these calculations!