3. (20 pts) Below is a set of sample data containing the ages and Depression sco

Question

3. (20 pts) Below is a set of sample data containing the ages and Depression scores (0 100, where 100 means most Depressed) of 8 people. Age DEP 18 97 51 44 34 75 22 89 60 38 76 54 28 80 49 60 a. Are these two variables related? If so, explain how they are related (to what extent and direction? HINT: use statistical evidence to support your answer) b. Predict the Depression score of a 42-year-old. c. Is this a good prediction? Why or why not (hint: use statistics to support your answer)? d. Predict the age of a person who has a Depression score of 100.

Explanation / Answer

Solution:

a) Correlation coefficient from excel output is given by

age

DEP

age

1

DEP

-0.86966

1

The variables are negatively related. This is because correlation coefficient is negative. That is -0.86966

The regression output for the variables from excel is

SUMMARY OUTPUT

Regression Statistics

Multiple R

0.869663

R Square

0.756315

Adjusted R Square

0.7157

Standard Error

11.40893

Observations

8

ANOVA

Df

SS

MS

F

Significance F

Regression

1

2423.894

2423.894

18.6219

0.005008

Residual

6

780.9815

130.1636

Total

7

3204.875

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Lower 95.0%

Upper 95.0%

Intercept

106.1196

9.895743

10.72376

3.88E-05

81.90556

130.3336

81.90556

130.3336

age

-0.92295

0.213878

-4.31531

0.005008

-1.44629

-0.39961

-1.44629

-0.39961

The slope is -0.92295

This means that an increase in one unit of age leads to a decrease in depression by 0.92295

b) Predict the Depression score of a 42-year-old.

The regression line is given by

DEP=106.119-0.92295AGE

= DEP=106.119-0.92295(42)

=106.119-38.7639

=67.3551

When age is 42 depression is predicted to be 67.3551

c) Is this a good prediction? Why or why not (hint: use statistics to support your answer)?

This is explained by the value of the r squared of the model. In this case r squared is 0.756315

This means 75.6315% of the depression is explained by the model.

d) Predict the age of a person who has a Depression score of 100.

DEP=106.119-0.92295AGE

= DEP=106.119-0.92295(100)

=106.119-92.295

=13.824

age

DEP

age

1

DEP

-0.86966

1

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