Question

A liquor wholesaler is interested in assessing the effect of the price of a premium scotch whiskey on the quantity sold. The results in the accompanying table on price, in dollars, and sales, in cases, were obtained from a sample of 8 weeks of sales records.

Price 19.3 20.6 19.8 21.3 20.8 19.7 17.9 17.4

Sales 25.2 15.1 18.9 11.8 10.7 16.1 29.6 35.4

1. Estimate by least squares the linear regression of increase in sales on increase in sales on increase in price.

Yi=___+___xi

2. Estimate the std of the regression slope cofficient Sb=
3. Estimate the std of the population model
4. Find 90% confidence interval for the slope of the population regression line.
5. Next week he decides to sett price to 20$ which sales can he expect?
6. Find 90% confidence interval for the expected sales of next week?

Answer 1

	X	Y	X * Y	X2	Ŷ	(Y - Ŷ)2	Sxx =Σ (Xi - X̅ )2	Syy = Σ( Yi - Y̅ )2	Sxy = Σ (Xi - X̅ ) * (Yi - Y̅)
	19.3	25.2	486.36	372.49	22.2220	8.9	0.0900	23.5225	-1.4550
	20.6	15.1	311.06	424.36	14.1098	1.0	1.0000	27.5625	-5.2500
	19.8	18.9	374.22	392.04	19.1020	0.0	0.0400	2.1025	-0.2900
	21.3	11.8	251.34	453.69	9.7417	4.2	2.8900	73.1025	-14.5350
	20.8	10.7	222.56	432.64	12.8618	4.7	1.4400	93.1225	-11.5800
	19.7	16.1	317.17	388.09	19.7260	13.1	0.0100	18.0625	-0.4250
	17.9	29.6	529.84	320.41	30.9583	1.8449	2.8900	85.5625	-15.7250
	17.4	35.4	615.96	302.76	34.0783	1.7468	4.8400	226.5025	-33.1100
Total	156.8	162.8	3108.51	3086.48	15991.8998	35.5387	13.2000	549.5400	-82.3700

X̅ = Σ (Xi / n ) = 156.8/8 = 19.6
Y̅ = Σ (Yi / n ) = 162.8/8 = 20.35

Estimated Error Variance (σ̂²) =
S² = ( 549.54 - -6.2402 * -82.37 ) / 8 - 2
S² = 5.9225
S = 2.4336

Part a)

Equation of regression line is Ŷ = a + bX
b = ( n Σ(XY) - (ΣX* ΣY) ) / ( n Σ X² - (ΣX)² )
b = ( 8 * 3108.51 - 156.8 * 162.8 ) / ( 8 * 3086.48 - ( 156.8 )²)
b = -6.2402

a =( ΣY - ( b * ΣX ) ) / n
a =( 162.8 - ( -6.2402 * 156.8 ) ) / 8
a = 142.657
Equation of regression line becomes Ŷ = 142.657 + -6.2402 X

Part b)

Part c)

Unbiased Estimator of σ2 = S² = Σ (Y - Ŷ )² / n - 2 = 35.5387 / 6 = 5.9231
Standard Error of Estimate S = √ ( Σ (Y - Ŷ ) / n - 2) = √(35.5387 / 6) = 2.4337

Part d)

Confidence Interval

Critical value   = 1.943 ( From t table )

90% confidence interval is -7.5418 < < -4.9386

Part e)

= 142.657 + -6.2402 X
= 17.853

Part f)

Confidence Interval of

Critical value = 1.943 ( From t table )
X̅ = (Xi / n ) = 156.8/8 = 19.6
= 17.853

90% confidence interval is ( 16.102 < < 19.604 )