Question

Is the number of calories in a beer related to the number of carbohydrates​ and/or the percentage of alcohol in the​ beer? The accompanying table has data for 35 beers. The values for three variables are​ included: the number of calories per 12​ ounces, the alcohol​ percentage, and the number of carbohydrates​ (in grams) per 12 ounces. Complete parts a through d.

Calories	Alcohol%	Carbohydrates
110	4.2	6.6
70	0.4	13.3
110	4.2	7
55	2.4	1.9
110	4.2	7
64	2.8	2.4
116	4.2	8
114	3.8	8.3
135	4.2	11.5
110	3.9	8.3
94	4.1	2.6
95	4.1	2.6
95	4.1	3.2
103	4.1	5
103	4.1	5.5
110	4.1	7
124	4.1	9.7
99	4.3	3.1
110	4.1	7.4
113	4.3	6.7
105	4.2	5.7
128	4.3	11.4
95	4.2	3.2
138	4.3	12.5
96	4.2	3.2
111	4.4	5.8
174	6.1	11.1
179	6.4	10.6
181	6.5	10.6
188	6.5	12.7
200	6.6	17
215	6.7	19.3
214	6.8	19.4
231	6.9	20
200	7	18

Perform a multiple linear regression​ analysis, using calories as the dependent variable and percentage alcohol and number of carbohydrates as the independent variables.

Let

Upper X 1X1

represent alcohol percentage and let

Upper X 2X2

represent the number of carbohydrates.

ModifyingAbove Upper Y with caret equals nothing plus left parenthesis nothing right parenthesis Upper X 1 plus left parenthesis nothing right parenthesis Upper X 2

Answer 1

We will do this problem with the help of Excel.

Load the data into Excel.

Go to Data>Megastat.

Select the option Correlation/Regression and go to Regression.

Select percentage alcohol and number of carbohydrates​ as the independent variable(s), x.

Select calories as the dependent variable, y.

Click OK.

The output will be as follows:

	R²	0.991
	Adjusted R²	0.990	n	35
	R	0.995	k	2
	Std. Error	4.507	Dep. Var.	Calories
ANOVA table
Source	SS	df	MS	F	p-value
Regression	70,230.6558	2	35,115.3279	1728.98	2.50E-33
Residual	649.9156	32	20.3099
Total	70,880.5714	34
Regression output					confidence interval
variables	coefficients	std. error	t (df=32)	p-value	95% lower	95% upper
Intercept	-0.1426
Alcohol%	19.8187	0.7047	28.125	3.89E-24	18.3834	21.2541
Carbohydrates	4.2690	0.1895	22.526	3.39E-21	3.8830	4.6550

Therefore, our regression equation is:

Calories = -0.1426 + 19.8187*Alcohol(%) + 4.2690*Carbohydrates

Or

y = -0.1426 + 19.8187*x₁ + 4.2690*x₂

where x₁ = Alcohol(%)

x₂ = Carbohydrates

Thus, the above output is the result of the multiple linear regression.