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 a. State the estimated regression line and interpret the slope coefficient. (1.5 marks) b. What is the estimated total personal wealth when a person is 50 years old? (1 mark) c. What is the value of the coefficient of determination? Interpret it. (1.5 marks) d. 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 (1 mark) Step 2. Standardised test statistic (0.5 mark) Step 3. Level of significance (0.5 mark) Step 4. Decision Rule (1.5 marks) Step 5. Calculation of test statistic (1.5 marks) Step 6. Conclusion

Explanation:

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)

 H01 : Bo = 0 H11 : Bo +0

H02 : B1 = 0 H12 : B1 +0

Step 2: Standardized test statistic:

ti SE(B)

Step 3: Level of significance :a = 0.1

Step 4: Decision rule:

Reject Но 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:

ti = 45.2159 39.8049 1.1359, p-value=0.2993

t2 = 5.3265 0.6777 7.8567,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.

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