statistics Project

Part A

(a) The simple linear regression equation
= β0 + β1 Xi, β0 = -74.5926, β1 =0.0954
= -74.5926+ 0.0954Xi
Where is personal consumption expenditure in billion of dollars Xi is disposable personal income (PCE) in billions of dollars
(*PCE is used as a short form of disposable personal income in the following report)
(b) Interpret the coefficient
The slope of disposable income (β1= 0.0953) tell us for each additional billion of dollars in disposable personal income, the PCE increase by 0.0953 billion dollars

(c) Comment on the significance of model (α = 0.05)
Hypotheses:
H0: β1 =0 H1 : β1 ≠ 0
Decision rule: reject H0, if |tcalc|> |t(α/2, n-k-1)|
Where tcrit = t (0.025, 98) =1.9845
Test statistic: t = = = 48.368
Decision: Reject H0 because t calc > t crit
Conclusion: There is sufficient evidence to conclude that there is significant relationship between disposable personal income and PCE at 5% level of significance. (d) The coefficient of determination

519797.8574/541572.1763=0.9598
It implies 95.98% of the variation in PCE can be explained by the variation in disposable personal income. The rest 4.02% of the variation in PCE is due to factors other than disposable income. (e) Test the assumptions

1. Linear test

We can find the assumption of linearity is not violated because this plot is approximately a straight line in this diagram.

2. Independent test

Hypotheses:
H0:=0 H1: ≠ 0
Decision rule: Reject H0 if D< dL (α,k,n)
Do not reject H0 if D > du
No decision if dL < D < du
Where dL (0.05,1,100)=1.65 and du(0.05,1,100)=1.69
Test statistic: Dcalc=6192.4602748/21774.3187301= 0.2844
Decision: Reject H0 because D < dL
Conclusion: There is sufficient evidence to conclude that there is positive autocorrelation at 5% level of significance.
We can evaluate the assumption of independence is violated.

3. Normality test

We can