Math Problem Statement

In this​ exercise, you will investigate the relationship between a​ worker's age and earnings.​ (Generally, older workers have more job​ experience, leading to higher productivity and​ earnings.) The following table contains data for​ full-time, full-years​ workers, age​ 25-34, with a high school diploma or​ B.A./B.S. as their highest degree. Download the data from the table by clicking the download table icon . A detailed description of the variables used in the dataset is available here LOADING.... Use a statistical package of your choice to answer the following questions. Part 2 Suppose you are interested in estimating the following model Ahe​ = beta 0 ​+ beta 1Age ​+ u Run a regression of average hourly earnings ​(AHE​) on age ​(Age​). Part 3 What is the estimated intercept ModifyingAbove beta with caret 0​? The estimated intercept ModifyingAbove beta with caret 0 isCollege Distance

Ahe Age

23.504274370 34

15.000000000 25

23.076923370 32

22.035257340 31

18.376068120 28

9.615385056 27

16.826923370 34

23.557691570 27

4.807692528 30

6.250000000 30

34.134616850 31

7.211538315 32

19.230770110 28

8.173076630 29

22.898033140 31

12.740385060 33

11.538461690 32

29.572649000 33

10.439560890 26

18.269229890 27

29.807691570 33

27.736686710 28

14.423076630 26

21.634614940 27

14.423076630 30

14.423076630 31

15.384614940 28

13.461538310 25

45.192306520 31

21.634614940 29

34.188034060 30

19.230770110 29

10.576923370 32

12.019230840 31

24.038461690 34

26.442308430 29

16.826923370 28

19.230770110 28

18.269229890 34

16.826923370 28

14.903845790 30

20.192308430 26

5.161538601 31

6.250000000 26

33.906730650 30

12.692307470 28

32.779720310 33

25.000000000 32

10.096154210 30

14.022436140 28

32.967033390 31

22.115385060 26

12.500000000 27

34.285713200 28

34.188034060 28

22.727272030 34

22.269229890 33

6.677350521 34

18.750000000 31

28.846153260 29

19.230770110 29

8.173076630 32

14.423076630 27

38.461540220 34

12.019230840 27

28.365385060 29

21.634614940 31

11.224490170 31

21.154808040 33

10.989010810 34

5.769230843 30

24.519229890 30

9.615385056 29

7.692307472 25

15.384614940 27

26.709402080 28

15.384614940 31

18.028846740 33

12.500000000 28

12.019230840 28

8.653845787 26

8.653845787 26

13.942307470 26

9.518259048 31

38.461540220 33

12.699999810 31

11.057692530 30

15.000000000 30

8.413461685 31

19.230770110 30

19.230770110 32

38.461540220 29

12.500000000 27

40.865383150 34

17.307691570 30

14.423076630 25

19.230770110 25

11.153845790 30

28.846153260 34

19.230770110 25

Solution

To solve Part 2 and Part 3 of your problem, we will perform a simple linear regression of Average Hourly Earnings (AHE) on Age. The linear model you provided is:

AHE=β0+β1⋅Age+u\text{AHE} = \beta_0 + \beta_1 \cdot \text{Age} + u

Where:

  • AHE\text{AHE} is the dependent variable (Average Hourly Earnings),
  • Age\text{Age} is the independent variable (Age),
  • β0\beta_0 is the intercept (the value of AHE\text{AHE} when Age=0\text{Age} = 0),
  • β1\beta_1 is the coefficient of Age (the rate at which earnings change with age),
  • uu is the error term.

Steps:

  1. Organize the data into a format suitable for regression analysis (Age and AHE columns).
  2. Perform linear regression to estimate β0\beta_0 and β1\beta_1.
  3. Identify the estimated intercept β^0\hat{\beta}_0.

I'll compute the linear regression and find the intercept for you.It seems there was an error due to a mismatch in the lengths of the Age and AHE data arrays. I'll fix this by reviewing the data, ensuring both arrays are of the same size, and then rerun the regression. Let me correct that.The estimated intercept β^0\hat{\beta}_0 from the regression is approximately 11.53. This means that when the worker's age is 0 (hypothetically), the model predicts average hourly earnings of $11.53.

Would you like to see more details or have any specific questions? Here are five related questions to explore further:

  1. What is the estimated slope β^1\hat{\beta}_1 (the effect of age on earnings)?
  2. How well does the model fit the data (R-squared value)?
  3. Can we predict the earnings of a 30-year-old worker based on this model?
  4. What are the p-values for the intercept and slope?
  5. How does age relate to earnings for workers with different education levels?

Tip: Checking the R-squared value helps determine how well your model explains the variability in the data.

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Math Problem Analysis

Mathematical Concepts

Linear Regression
Statistics
Econometrics

Formulas

Linear model: AHE = β₀ + β₁ * Age + u

Theorems

Ordinary Least Squares (OLS) Estimation

Suitable Grade Level

Undergraduate level (Econometrics/Statistics)

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