Question

In: Statistics and Probability

PoolVac, Inc. manufactures and sells a single product called the “Sting Ray,” which is a patent-protected...

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

Solutions

Expert Solution

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 βi ≠ 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 βi ≠ 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.

orchestra answered 1 year ago
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