Question

PoolVac, Inc. manufactures and sells a single product called the “Sting Ray,” which is a patent-protected automatic cleaning device for swimming pools. PoolVac’s Sting Ray faces its closest competitor, Howard Industries, also selling a competing pool cleaner. Using the last 30 quarters of production and cost data, PoolVac wishes to estimate its average variable costs using the following quadratic specification:

AVC = a + bQ + cQ 2

The quarterly data on average variable cost (AVC), and the quantity of Sting Rays produced and sold each quarter (Q) are presented in the data file. PoolVac also wishes to use its sales data for the last 30 quarters to estimate demand for its Sting Ray. Demand for Sting Rays is specified to be a linear function as the following:

Qd = d + eP + fM + gPH in which its price (P), average income for households in the U.S. that have swimming pools (M), and the price of the competing pool cleaner sold by Howard Industries (PH).

1.       Run the appropriate regression to estimate the average variable coast function (avc) for sting rays. Evaluate the statistical significance of the three estimated parameters using a significance level of 5 percent. Be sure to comment on the algebraic signs of the three parameter estimates.

Sting Ray-PoolVac, Inc.
Quarter/Year	Period (t)	AVC	Q	P	M	PH
1st/2006	1	109	1647	275	58000	175
2nd/2006	2	118	1664	275	58000	175
3rd/2006	3	121	1295	300	58000	200
4th/2006	4	102	1331	300	56300	200
1st/2007	5	121	1413	300	56300	200
2nd/2007	6	102	1378	300	56300	200
3rd/2007	7	105	1371	300	57850	200
4th/2007	8	101	1312	300	57850	200
1st/2008	9	108	1301	325	57850	250
2nd/2008	10	113	854	350	57600	250
3rd/2008	11	114	963	350	57600	250
4th/2008	12	105	1238	325	57600	225
1st/2009	13	107	1076	325	58250	225
2nd/2009	14	104	1092	325	58250	225
3rd/2009	15	104	1222	325	58250	225
4th/2009	16	102	1308	325	58985	250
1st/2010	17	116	1259	325	58985	250
2nd/2010	18	126	711	375	58985	250
3rd/2010	19	116	1118	350	59600	250
4th/2010	20	139	91	475	59600	375
1st/2011	21	152	137	475	59600	375
2nd/2011	22	116	857	375	60800	250
3rd/2011	23	127	1003	350	60800	250
4th/2011	24	123	1328	320	60800	220
1st/2012	25	104	1376	320	62350	220
2nd/2012	26	114	1219	320	62350	220
3rd/2012	27	133	1321	312	62845	233
4th/2012	28	131	1354	312	62950	233
1st/2013	29	127	1307	312	62950	233
2nd/2013	30	121	1299	307	63025	221

Answer 1

Question 1:

The regression output is:

	R²	0.431
	Adjusted R²	0.389
	R	0.656
	Std. Error	9.748
	n	30
	k	2
	Dep. Var.	AVC
ANOVA table
Source	SS	df	MS	F	p-value
Regression	1,941.4398	2	970.7199	10.22	.0005
Residual	2,565.5269	27	95.0195
Total	4,506.9667	29
Regression output					confidence interval
variables	coefficients	std. error	t (df=27)	p-value	95% lower	95% upper
Intercept	151.6131
Q	-0.0585	0.0184	-3.177	.0037	-0.0962	-0.0207
Q²	0.00002200	0.00001021	2.156	.0402	0.00000106	0.00004294

The regression equation form is:

AVC = a + bQ + cQ²

The regression equation is:

AVC = 151.6131 -0.0585*Q + 0.000022*Q²

The hypothesis being tested is:

H₀: β₁ = β₂ = 0

H₁: At least β_i ≠ 0

The p-value from the output is 0.0005.

Since the p-value (0.0005) is less than the significance level (0.05), we can reject the null hypothesis.

Therefore, we can conclude that the regression model is significant.

Question 2:

The regression output is:

	R²	0.959
	Adjusted R²	0.955
	R	0.979
	Std. Error	75.172
	n	30
	k	3
	Dep. Var.	Q
ANOVA table
Source	SS	df	MS	F	p-value
Regression	34,68,419.3076	3	11,56,139.7692	204.60	3.38E-18
Residual	1,46,922.1924	26	5,650.8536
Total	36,15,341.5000	29
Regression output					confidence interval
variables	coefficients	std. error	t (df=26)	p-value	95% lower	95% upper
Intercept	3,407.9578
P	-9.4006	1.1914	-7.890	2.29E-08	-11.8497	-6.9516
M	0.0067	0.0070	0.956	.3481	-0.0077	0.0211
PH	1.9972	1.2483	1.600	.1217	-0.5687	4.5632

The regression equation form is:

Qd = d + eP + fM + gPH

The regression equation is:

Qd = 3,407.9578 -9.4006*P + 0.0067*M + 1.9972*PH

The hypothesis being tested is:

H₀: β₁ = β₂ = β₃ = 0

H₁: At least β_i ≠ 0

The p-value from the output is 0.0000.

Since the p-value (0.0000) is less than the significance level (0.05), we can reject the null hypothesis.

Therefore, we can conclude that the regression model is significant.