Question

The accompanying data are x = advertising share and y = market share for a particular brand of cigarettes during 10 randomly selected years. x 0.101 0.073 0.072 0.077 0.086 0.047 0.060 0.050 0.070 0.052 y 0.138 0.126 0.120 0.086 0.079 0.076 0.065 0.059 0.051 0.039

(a) Calculate the equation of the estimated regression line. (Round your answers to six decimal places.) Obtain the predicted market share when the advertising share is 0.09. (Round your answer to five decimal places.)

(b) Compute r2. (Round your answer to three decimal places.)

(c) Calculate a point estimate of σ. (Round your answer to four decimal places.) On how many degrees of freedom is your estimate based?

Answer 1

X	Y	XY	X²	Y²
0.101	0.138	0.013938	0.010201	0.019044
0.073	0.126	0.009198	0.005329	0.015876
0.072	0.120	0.00864	0.005184	0.0144
0.077	0.086	0.006622	0.005929	0.007396
0.086	0.079	0.006794	0.007396	0.006241
0.047	0.076	0.003572	0.002209	0.005776
0.060	0.065	0.0039	0.0036	0.004225
0.050	0.059	0.00295	0.0025	0.003481
0.070	0.051	0.00357	0.0049	0.002601
0.052	0.039	0.002028	0.002704	0.001521

Sample size, n =	10
Ʃ x =	0.688
Ʃ y =	0.839
Ʃ xy =	0.061212
Ʃ x² =	0.049952
Ʃ y² =	0.080561
x̅ =	0.0688
y̅ =	0.0839
SSxx = Ʃx² - (Ʃx)²/n =	0.0026176
SSyy = Ʃy² - (Ʃy)²/n =	0.0101689
SSxy = Ʃxy - (Ʃx)(Ʃy)/n =	0.0034888

a) Slope, b = SSxy/SSxx = 1.332823961
Intercept, a = y̅ -b* x̅ = -0.007798289

Regression equation :               
ŷ = -0.007798 + 1.332824 x

Predicted value at X =   0.09          
ŷ = -0.007798 + 1.332824 * 0.09 = 0.11216   

b) Coefficient of determination, r² = (SSxy)²/(SSxx*SSyy) = 0.457

c) Point estimate of σ, se = √( (SSyy - b*SSxy)/(n-2) ) = 0.0263

Degree of freedom = n-2 = 10-2 = 8