Question

Shown below is a portion of a computer output for a regression analysis relating Y (demand) and X (unit price). ANOVA df SS Regression 1 Error 3132.661 Total 47 8181.479 Coefficients Standard Error Intercept 80.390 3.102 X -2.137 0.248 (a) Compute the coefficient of determination and fully interpret its meaning. Be very specific. (b) Find the standard error for b1 (Sb1). (c) Perform a t test and determine whether or not demand and unit price are related. Let  = 0.05. (d) Perform an F test and determine whether or not demand and unit price are related. Let  = 0.05

Answer 1

Solution

Preparatory Work

Let us first complete the ANOVA Table: [given figures in bold, others derived.]

Source of Variation	df	SS	MS	F	Fcrit
Regression	1	5048.818	5048.818	74.137	4.04
Error	46	3132.661	68.101
Total	47	8181.479

Explanations

df for Error = df for Total – df for Regression.

SS for Regression = SS for Total - SS for Error

MS = SS/df

F = MSR/MSE

Fcrit = upper 5% point of F_{1, 46}

Other Details given

Intercept = 80.390 with standard error = 3.102 ......................................................................................... (1)

Slope coefficient, b = - 2.137 with standard error = 0.248 ........................................................................ (2)

Now, to work out the solution,

Part (a)

Coefficient of determination = SSR/SST

= 0.6171 Answer 1

Interpretation

Coefficient of determination, r² represents the proportion of the variation in the response variable that is explained by the variation in the predictor variable. In the present scenario, 62% of the variation in demand is accounted for by unit price. Answer 2

Part (b)

Standard error for b1 = 0.248 [vide (2)] Answer 3

Part (c)

t- test to determine whether or not demand and unit price are related at a = 0.05. i.e., to test population correlation coefficient, ρ is zero or not.

Hypotheses:

Null: H₀: ρ = 0 Vs Alternative H₁: ρ ≠ 0

Test Statistic:

t = r√{(n - 2)/(1 – r²)}

= √(0.6171 x 46/0.3829) [df for SST in ANOVA = n - 1]

= 8.795

Distribution, Significance Level, α, Critical Value,

t ~ t_{n – 2}

So, Critical Value = upper (α/2) % point of t_{n – 2} = t_{46, 0.025} = 2.015

Decision

H₀ is rejected since | t_cal | > t_crit

Conclusion:

There is sufficient evidence to suggest that the linear correlation between demand and price is significant and hence we conclude that demand and unit price are related. Answer 4

Part (d)

F test to determine whether or not demand and unit price are related, i.e., ANOVA

Referring to the ANOVA Table at the top, F > Fcrit.

So, we conclude demand and unit price are related. Answer 5

DONE