Question

Personal wealth tends to increase with age as older individuals have had more opportunities to earn and invest than younger individuals. The following data were obtained from a random sample of eight individuals and records their total wealth (Y) and their current age (X).

Person	Total wealth (‘000s of dollars) Y	Age (Years) X
A	280	36
B	450	72
C	250	48
D	320	51
E	470	80
F	250	40
G	330	55
H	430	72

A part of the output of a regression analysis of Y against X using Excel is given below:

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.954704
R Square	0.91146
Adjusted R Square	0.896703
Standard Error	28.98954
Observations	8
ANOVA
	df		SS		MS		F		Significance F
Regression	1		51907.64		51907.64
Residual	6		5042.361		840.3936
Total	7		56950
	Coefficients		Standard Error		t Stat		P-value
Intercept	45.2159		39.8049
Age	5.3265		0.6777

1. State the estimated regression line and interpret the slope coefficient.
2. What is the estimated total personal wealth when a person is 50 years old?
3. What is the value of the coefficient of determination? Interpret it.
4. Test whether there is a significant relationship between wealth and age at the 10% significance level. Perform the test using the following six steps.

Step 1. Statement of the hypotheses

Step 2. Standardised test statistic

Step 3. Level of significance

Step 4. Decision Rule

Step 5. Calculation of test statistic

Step 6. Conclusion

Answer 1

a. State the estimated regression line and interpret the slope coefficient

Total wealth=45.2159+5.3265*age

Here, the slope is 5.3265. It indicates that whenever the age is more by 1 year, the total wealth increases by 5.3265*1000=$5326.5

b.What is the estimated total personal wealth when a person is 50 years old?

When age=50,

Total wealth=45.2159+5.3265*50=45.2159+266.325=311.5409 thousand dollars.

c.What is the value of the coefficient of determination? Interpret it.

The coefficient of determination=0.9115.

This indicates that the model could explain 91.15% of the variation in the data.

d.step 1: Null hypothesis: The regression coefficients are zero(absent)

  

Step 2: Standardized test statistic:

Step 3: Level of significance :

Step 4: Decision rule:

Reject if the p-value of the respective t statistic is <0.1. The p-value will be calculated for a t-distribution at n-2=6 df.

Step 5: Calculation of test statistic:

  

, p-value=0.2993

,p-value=0.0002

step 6:Since the p-value for the intercept >0.1, we do not reject the null hypothesis.

p-value for the slope coefficient is <0.1, we reject the null hypothesis and conclude that the slope coefficient cannot be regarded as absent. Hence the linear regression is significant.