Question

You have obtained a sub-sample of 1744 individuals from the Current Population Survey (CPS) and are interested in the relationship between weekly earnings and age. The regression, using heteroskedasticity-robust standard errors, yielded the following result:_____.

Earn = 239.16 + 5.20×Age , R = 0.05, SER = 287.21.,
(20.24) (0.57)
where Earn and Age are measured in dollars and years respectively.
(a) Is the relationship between Age and Earn statistically significant?
(b) The variance of the error term and the variance of the dependent variable are related. Given the distribution of earnings, do you think it is plausible that the distribution of errors is normal?
(c) Construct a 95% confidence interval for both the slope and the intercept.

Answer 1

Final answer:

The relationship between Age and Earn is likely statistically significant, the normality of error distribution may be questioned but is less of a concern with a large sample size, and confidence intervals require t-distribution critical values alongside the standard errors provided.

Step-by-step explanation:

The student has asked about the statistical significance of the relationship between weekly earnings (Earn) and age in a regression model, the plausibility of the distribution of errors being normal, and how to construct a 95% confidence interval for the slope and intercept of the regression line.

(a) The t-statistic for the slope can be calculated by dividing the slope coefficient by its standard error. The given slope is 5.20 with a standard error of 0.57. Thus, t = 5.20 / 0.57 which equals approximately 9.12. This value is high, and typically for a large sample size, a t-value greater than 2 in absolute value would suggest statistical significance. Therefore, the relationship between Age and Earn is likely statistically significant.

(b) The normality of the distribution of errors in regression analysis is an assumption for conducting certain hypothesis tests and constructing confidence intervals. Given the SER (standard error of regression) is 287.21, implying substantial variability, one might question the normality assumption, especially if the distribution of earnings is skewed or has outliers. However, with a large sample size, the Central Limit Theorem helps in that the sampling distribution of the estimate tends to be normal.

(c) To construct a 95% confidence interval for the slope and the intercept, we would need the specific heteroskedasticity-robust standard errors for each. Assuming they are the ones provided (20.24 for intercept, 0.57 for slope), we could use these to calculate the intervals. However, without knowing the appropriate critical value from the t-distribution, we cannot complete this task. Typically, you would calculate the intervals as 'estimate ± (critical value) * (standard error)', for both the slope and the intercept coefficients.

Answer 2

Let me give you an example of a segment addition problem that uses three points that asks the student to solve for x but has a solution x = 20.

First, I assumed values for each x, y and z and then manipulated their coefficients to get the total at the end of each equation.

20 + 10 +30 = 60

40 + 0 + 40 = 80

40 + 10 = 50

Then exchangeing these numbers into values and we have the following equation.

x + 2y + 3z = 60

2x + 4z = 80

2x + z = 50 so its easy

If you will solve them manually by substituting their variables into these equations, you can get

x = 20

y = 5

z = 10

Step-by-step explanation: