Question

Questions 1-6 are based on the following scenario:
Designers of backpacks use exotic material such as supernylon Delrin, high-density
ethylene, aircraft aluminum, and thermo molded floam to make packs that fit comfortably
and distribute weight to eliminate pressure points. The following data show the capacity
(cubic inches), comfort rating, and price (dollars) for 10 backpacks tested by Outside
Magazine. Comfort was measured using a rating from 1 to 5, with a rating of 1 denoting
average comfort and a rating of 5 denoting excellent comfort.  Regress price on comfort
and capacity using excel and answer the following questions.  (Place your answers in the highlighted cells below.)
Data for this scenario is as follows:
Manufacturer & Model	Capacity	Comfort	Price($)
Camp Trails Paragon	4330	2	190
EMS 5500	5500	3	219
Lowe Almayo	5500	4	249
Marmot Muir	4700	3	249
Kelly Bigfoot	5200	4	250
Gregory Whitney	5500	4	340
Osprey 75	4700	4	389
Arc' Teryx Bora	5500	5	395
Dana Design Terraplane	5800	5	439
The Works @ Mystery	5000	5	525
1. What is the interpretation of adjusted R-square for this problem?
2. What price would you predict for a backpack with a mean capacity of 5700 (cubic inches) and comfort rating of 4?
3. Provide the appropriate test statistic and p-value for assessing whether there is evidence that comfort aids in predicting price.
4. Provide an interpretation for the sample regression coefficient of capacity.
5. Is the overall model significant at 0.05 level of significance? Why or Why not.
6. Does capacity aid in predicting price at 0.05 level of significance? Why or Why not.

Answer 1

Regression Statistics
Multiple R	0.912053596
R Square	0.831841762
Adjusted R Square	0.783796551
Standard Error	51.13629498
Observations	10
ANOVA
	df	SS	MS	F	Significance F
Regression	2	90548.05535	45274.02768	17.31372898	0.001949905
Residual	7	18304.44465	2614.920664
Total	9	108852.5
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	356.1208346	197.1740072	1.806124649	0.113859072	-110.1216045	822.3632737	-110.1216045	822.3632737
Capacity	-0.098744049	0.045876543	-2.15238646	0.068372268	-0.207224835	0.009736736	-0.207224835	0.009736736
Comfort	122.8672136	21.79975287	5.63617461	0.000785707	71.31898932	174.4154379	71.31898932	174.4154379

Answer 1. R² adjusted =0.7837. R² adjusted tells the predictive capability of the model. R2 adjusted tells the variability in the response variable explained by only those regressors which are actually effecting the response variable. If you see, the p-value of the regression coefficient of Capacity is 0.0683 >0.05. Hence at a 5% level of significance, we do not reject H0. And p-value of the regression coefficient of Comfort is 0.0007 <0.05. Hence at a 5% level of significance, we reject H0.

Test of Hypothesis (Individual Coefficients)

H0: The regression coefficient is not significantly different from 0.(i.e.,bi=0)against H1: the Regression coefficient is significantly different from 0.

Hence Comfort is significantly contributing to the model and Capacity is not significant. R² adjusted tells the variability in the price explained by the regressor Comfort. Hence, 78.37% variability in the price explained by the regressor Comfort.

Answer 2.

Fitted Model: price_cap= 356.1208 -0.09874Capacity + 122.8672 Comfort

price_cap=

284.7486073

Answer 3

Test of Hypothesis

H0: The regression coefficient of comfort is not significantly different from 0.against H1: the Regression coefficient of comfort is significantly different from 0.

Let b₂ be the regression coefficient of comfort

Test statistics: t =(b₂_cap- b2)/s.e(b₂_cap) where s.e(b₂_cap)= standard error of the b₂_cap

t= 5.6361 and p-value=0.000785.

p-value of regression coefficient of Comfort is 0.0007 <0.05. Hence at 5% level of significance we reject H0. Hence Comfort is significantly contributing to the model.  

Answer 4

The sample regression coefficient of capacity is actually the rate of change of price with respect to capacity which is -0.098744. It means as the capacity increases the price decreases. The unit increase in capacity decreases the price by 0.0987.

Answer 5

Test of Hypothesis

H0: The regression coefficient is not significantly different from 0.(i.e.,b1=b2=0)

against H1: the Regression coefficient is significantly different from 0.

F= 17.31372 is greater than F critical value (2,7)=4.74 at a 5% level f significance. Hence We reject H0. Hence, the overall model is significant.

Answer 6

Test of Hypothesis

H0: The regression coefficient of capacity is not significantly different from 0.against H1: the Regression coefficient of capacity is significantly different from 0.

Note that, the p-value of the regression coefficient of Capacity is 0.0683 >0.05. Hence at a 5% level of significance, we do not reject H0. Hence, the Regression coefficient of Capacity is not significantly different from 0.