Question

Brand	Calories	Price($)	%Alcohol Content	% sq	Type
BrooklynBrand	159	6.24	5.2	27.04	Other
Leinenkugel'sRed	160	4.79	5	25	Other
SamuelAdamsBoston	160	5.96	4.9	24.01	Other
GeorgeKillian'sIrishRed	162	4.7	4.9	24.01	Other
RedWolf	157	4.11	5.5	30.25	Other
HenryWeinhard'sPrivateRes.	151	3.85	4.9	24.01	Other
Sterling	155	2.52	4.7	22.09	Other
Legacy	135	5.46	5.1	26.01	Other
Dominion	162	6	5.4	29.16	Other
LoneStar	142	3.71	4.8	23.04	Other
AbitaAmber	146	6.7	4.4	19.36	Other
YuenglingPremium	148	4.99	4.3	18.49	Other
BerghoffOriginal	170	4.1	5.1	26.01	Other
SamuelAdamsBoston	160	5.96	5	25	Other
SierraNevadaPale	172	6.31	5.8	33.64	Other
FullSailAmber	170	6.42	5.9	34.81	Other
Liberty	184	7.79	6	36	Other
ElkMountainAmber	201	5.05	5.6	31.36	Other
CelisPaleBock	155	5.26	4.7	22.09	Other
Pete'sWicked	170	5.84	5.3	28.09	Other
AnchorSteam	158	7.22	4.9	24.01	Other
DockStreetAmber	159	6.12	5.4	29.16	Other
Bass	150	7.37	5.1	26.01	Other
RedhookESB	177	6.47	5.6	31.36	Other
NewAmsterdamNewYork	146	6.72	3.7	13.69	Other
CatamountAmber	151	7.59	4.9	24.01	Other
RedNectar	163	6.36	5.3	28.09	Other
OldDetroitAmber	186	6.52	5.9	34.81	Other
BridgePortBlueHeronPale	168	6.34	5.9	34.81	Other
Geary'sPale	142	7.1	4.7	22.09	Other
MolsonGolden	148	4.78	5	25	Other
LabattBlue	150	4.63	5	25	Other
Foster's	140	5.41	5	25	Other
Kirin	150	6.39	5	25	Other
DosEquis	160	5.52	4.8	23.04	Other
Heineken	160	6.38	5	25	Other
CoronaExtra	148	5.68	4.6	21.16	Other
St.PauliGirl	148	5.82	4.9	24.01	Other
Beck's	148	5.83	4.3	18.49	Other
PilsnerUrquell	160	7.8	4.1	16.81	Other
OldMilwaukee	145	2.82	4.5	20.25	4
Stroh's	142	3.2	4.4	19.36	4
RedDog	147	3.83	5	25	4
Budweiser	148	4.02	4.9	24.01	4
Icehouse	149	3.88	5.5	30.25	4
MolsonIce	155	4.79	5.6	31.36	4
Michelob	159	4	5	25	4
BudIce	148	3.95	5.5	30.25	4
Busch	143	3.27	4.9	24.01	4
CoorsOriginal	137	4.02	4.6	21.16	4
GeneseeCreamAle	153	3.26	4.6	21.16	4
MillerHighLife	143	3.19	5	25	4
PabstBlueRibbon	144	2.9	4.7	22.09	4
Milwaukee'sBest	133	2.36	4.6	21.16	4
MillerGenuineDraft	143	3.93	5	25	4
RollingRock	143	4.25	4.6	21.16	4

Can the number of calories in a bottle of beer be explained by the percentage of alcohol, and by the price per bottle?

a. Build a linear regression model and comment on the model significance,
    accuracy of predictions, and linear relationships of the independent
    variables and the dependent variable.

b. Draw a scatter plot between %alcohol and calories. Any pattern?

c. Add a quadratic % alcohol to the model. A justifiable move? You can plot a
    scatter plot between %alcohol and calories to see the pattern.

d. Add beer type4 to to your quadratic model. What can you argue at 1%
    significance level about the amount of calories per bottle in a type 1 beer?

f. Which bottle is expected to contain more calories: (i) A $3 type 4 beer with
    4.35% alcohol (ii) A $3 type 4 beer, with 3% alcohol. Show your work.

g. At 95% confidence, would you change your definite answer of part 'f'? Explain.

Answer 1

When we regress this in Excel, we get the following results..

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.694994465
R Square	0.483017306
Adjusted R Square	0.463508525
Standard Error	9.553036658
Observations	56
ANOVA
	df	SS	MS	F	Significance F
Regression	2	4519.032288	2259.516144	24.75896923	2.55337E-08
Residual	53	4836.806998	91.26050939
Total	55	9355.839286
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	62.08747246	13.84804961	4.48348137	3.96748E-05	34.3118024	89.86314252	34.3118024	89.86314252
Price($)	2.75637582	0.912194885	3.021696203	0.003865005	0.926744583	4.586007058	0.926744583	4.586007058
%Alcohol Content	15.67209239	2.786637611	5.624015239	7.12962E-07	10.08280516	21.26137962	10.08280516	21.26137962

Here, the model seems to be significant, as we find the Intercept, Alcohol content seem to impact the no of calories in the bottle positively.

The actual and predicted values are given as below.

The model is unable to predict some abrupt peaks as highlighted in the graph. Also, the R square and Adjusted R square are on the lower side owing to the aforementioned reason. Considering all these, we can say that the model is moderately accurate and is not able to predict , when there is a sudden spike in the calories.

The linear relationship between independent and dependent variables is shown below.

There seems to be a decent correlation between Calories and Alcohol content and Calories and Price. Also, the R² is moderate between the two sets of variables.

B.

Scatter plot between Alcohol content and calories.

C. Scatter plot between Alcohol % and Calories.

There seems to be a linear relationship between these two variables

When we regress % Alcohol Sq along with other variables, we get the following results

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.72640782
R Square	0.527668322
Adjusted R Square	0.500418417
Standard Error	9.218563289
Observations	56
ANOVA
	df	SS	MS	F	Significance F
Regression	3	4936.780012	1645.593337	19.36404294	1.46138E-08
Residual	52	4419.059274	84.98190911
Total	55	9355.839286
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	289.7535348	103.5502879	2.798191495	0.00719103	81.96468046	497.5423891	81.96468046	497.5423891
Price($)	2.004994825	0.943240509	2.125645373	0.038301719	0.11224427	3.897745379	0.11224427	3.897745379
%Alcohol Content	-74.87455593	40.92776217	-1.82943195	0.073070951	-157.0021191	7.253007266	-157.002119	7.253007266
% sq	9.078677712	4.094763227	2.217143509	0.031009284	0.861934165	17.29542126	0.861934165	17.29542126

It doesnt seem to be justifiable move as it made %Alcohol Content negative .