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).
Answers
Given:
The complete data is,
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
To find:
a. State the estimated regression line and interpret the slope coefficient.
b. What is the estimated total personal wealth when a person is 50 years old?
c. What is the value of the coefficient of determination? Interpret it.
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
Step 2. Standardised test statistic
Step 3. Level of significance
Step 4. Decision Rule
Step 5. Calculation of test statistic
Step 6. Conclusion
Solution:
a.
The general form of the regression equation is given as,
y^ = bo + b1x
where,
bo = the intercept estimate
b1 = the slope estimate.
y = the dependent variable = wealth
x = independent variable = age.
From table3 and column coefficient,
bo=45.2159 and b1=5.3265
Therefore estimated regression equation is y^ = 45.2159 + 5.3265x.
The slope coefficient of 5.3265 indicates that with a 1 unit increase in x, the predicted y variable increases by 5.3265 units.
In terms of variables, as age increases by a year, the wealth will increase by 5326.5.
b.
The estimated total personal wealth when a person is 50 years old is obtained by substituting the value of x as 50 in an estimated regression equation.
y^ = 45.2159 + 5.3265(50) = 311.5409
Therefore the estimated personal wealth when a person is 50 years old is 311.5409 thousand dollars
c.
Coefficient of determination value is given in table1, as R-square which is 0.91146.
This implies 91.146% of the variation in the model is explained by the regression line, which implies the line fits well to the model.
d.
step1
To test whether there is a significant relationship between wealth and age at the 10% significance level, the hypothesis is given as,
Ho:β1=0
Ha:β1≠0
where β1 = the slope of the regression line.
Step 2.
The standardised test statistic is T-statistic given by the formula,
tn−2 = (b1−β1) / Sb1
Step 3.
The level of significance is given as 0.1
Step 4.
Decision Rule:
If the P-value is less than 0.1, the null hypothesis would be rejected and vice-versa.
Step 5.
The value of slope coefficient and standard error of the slope is given in table 3 and the number of observations(n) is given in table 1.
Substituting values,
t8−2 = (5.3265 – 0)/0.6777
t6 = 7.8597
Hence the standardized test statistic is 7.8597.
Step 6.
The P-value for the two-tailed test is obtained from T-table as.
P−value = 2P(t6 > 7.8597) = 2(0.00017) = 0.00034
Since the P-value is less than 0.1, the null hypothesis is rejected.
Hence it is concluded that at 0.1 significance level, then there is a significant relationship between wealth and age.