Identified Question

Question

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B. Zero covamano between erron term
12 As the sumple saxe momnses the variance of the sumple noon
A. Increases
B. Decreases
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13 Crost-sectional data are dation
A. True
B. false
14. An R-sount clase to I imphe that Xanc invition
A. True
Suppose we hout enimoted the following rogression
years meat (40) (2) his whereY,Annual production of whea was and the lake B. the bracket are sundard errors A. Find LSS and RSS 1 point B. Esumite variance of the residuils ad noint C. Construct ustatistics value for x oneffroment (2 points)
Answer
Let’s go through each question one by one:
As the sample size increases, the variance of the sample mean: A. Increases B. Decreases
Answer: B. Decreases
Explanation: As the sample size increases, the variance of the sample mean decreases. This is because the sample mean is an unbiased estimator of the population mean, and as the sample size increases, the sample mean becomes a more accurate estimate of the population mean, resulting in a smaller variance.
Cross-sectional data are data: A. True B. False
Answer: A. True
Explanation: Cross-sectional data refers to data collected at a single point in time or over a short period of time. This type of data is used to analyze and compare different subjects or entities at a specific point in time.
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 variable(s) explain a large proportion of the variance in the dependent variable. In other words, the regression model fits the data well, and the independent variable(s) have a strong relationship with the dependent variable.
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)
B. Estimate variance of the residuals (1 point)
C. Construct test statistics value for X (2 points)
Answer:
A. To find the least squares sum (LSS) and the residual sum of squares (RSS), we need to fit the regression model to the data. The LSS is the sum of the squared differences between the observed values and the predicted values, while the RSS is the sum of the squared differences between the observed values and the mean of the observed values.
B. The variance of the residuals can be estimated using the formula:
, where n is the number of observations.
C. To construct the test statistic value for X, we need to calculate the t-statistic, which measures the significance of the independent variable in the regression model. The t-statistic is calculated as:
, where b is the coefficient of the independent variable and SE(b) is the standard error of the coefficient.

Answer 1

Let's clarify and formalize the answers to your questions based on the information provided.

Identified Questions:

1. 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.

2. 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.

3. 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.

4. 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!