Question

2. Exercise 4.2

Dismiss All

Please Wait . . .

Please Wait...

The Pilot Pen Company has decided to use 15 test markets to examine the sensitivity of demand for its new product to various prices, as shown in the following table. Advertising effort was identical in each market. Each market had approximately the same level of business activity and population.

Complete the following worksheet and then estimate the demand function for Pilot's new pen using a linear regression model.

Test Market	Price Charged	Quantity Sold
	(cents)	(Thousands of Pens)
ii	xixi	yiyi	xixi yiyi	xi2xi2	yi2yi2
1	50	20	1,000	2,500	400
2	50	21	1,050	2,500	441
3	55	19	1,045	3,025	361
4	55	19.5	1,072.5	3,025	380.25
5	60	20	1,200	3,600	400
6	60	19	1,140	3,600	361
7	65	16	1,040	4,225	256
8	65	15	975	4,225	225
9	70	14.5	1,015	4,900	210.25
10	70	15.5	1,085	4,900	240.25
11	80	13	1,040	6,400	169
12	80	14	1,120	6,400	196
13	90	11.5	1,035	8,100	132.25
14	90	11	990	8,100	121
15	40	17	680	1,600	289
Total	980	246	selector 1    15,487.5 14,447.5 14,472.5	67,100	selector 2    4,326 4,515.75 4,182

Points:

Regression Parameters	Estimations
Slope (ββ )	selector 1    0.418 0.009 -0.190
Intercept (αα )	selector 2    -10.909 15.812 28.825

Points:

Close Explanation

Explanation:

What is the standard error of the estimate (sese )?

105.103

1.407

1.674

Points:

Close Explanation

Explanation:

What is the estimate of the standard deviation of the estimated slope (sbsb )?

0.025

0.030

1.896

Points:

Close Explanation

Explanation:

Can you reject the hypothesis (at the 0.05 level of significance) that there is no relationship (i.e., β=0β=0 ) between the price and quantity variables? (Hint: t0.025,13=2.16t0.025,13=2.16 )

No

Yes

Points:

Close Explanation

Explanation:

Complete the following worksheet and then use it to calculate the coefficient of determination.

ii	xixi	yiyi	yˆy^	(yˆi−y¯)2y^i−y¯2	(yi−y¯)2yi−y¯2
1	50	20	19.325	8.556	12.960
2	50	21	19.325	8.556	21.160
3	55	19	18.375	3.901	6.760
4	55	19.5	18.375	3.901	9.610
5	60	20	17.425	1.051	12.960
6	60	19	17.425	1.051	6.760
7	65	16	16.475	0.006	0.160
8	65	15	16.475	0.006	1.960
9	70	14.5	15.525	0.766	3.610
10	70	15.5	15.525	0.766	0.810
11	80	13	13.625	7.701	11.560
12	80	14	13.625	7.701	5.760
13	90	11.5	11.725	21.856	24.010
14	90	11	11.725	21.856	29.160
15	40	17	21.225	23.281	0.360
			Total	selector 1    110.949 111.721 110.955	selector 2    168.760 147.600 126.440

Points:

The coefficient of determination (r2r2 ) is selector 1   

• 0.657
• 0.757
• 0.752

Points:

Close Explanation

Explanation:

What is the price elasticity of demand at a price of 50 cents?

0.03

-0.02

-0.50

-0.46

Answer 1

a. The regression model expresses the demand (quantity sold) for pens as a function of price:

Qi= [\alpha] + [\beta] P1+ [\varepsilon] i

We can estimate the model to get predictions: Q i = [\alpha] + [\beta] P i.

We get a = 28 . 915 and b =-0.191

Hence, we predict that a one-cent increase in the price of a pen will lead to a decrease in sales of 191 units, on average.

C, In the model the Price elasticity of demand is given by

ED= ( [\delta] QP) / ( [\delta] PQ) = [\beta] (P/Q)

Which can be estimated by ED= [\beta] (P/Q)

At P = 50 , Q = a + b (50) = 19 . 363 .

Thus we have ED= (-0.191) 50/ 19.363 [\approx] -0.493

Therefore, we predict that a 1% increase in price will lead to a 0.493% decrease in sales, on average, at an initial price of $ 0.50.