Question

The production of wine is a​ multibillion-dollar worldwide industry. In an attempt to develop a model of wine quality as judged by wine​ experts, data was collected from red wine variants. A sample of 20 wines is provided in the accompanying table. Develop a multiple linear regression model to predict wine​ quality, measured on a scale from 0​ (very bad) to 10​ (excellent) based on alcohol content​ (%) and the amount of chlorides. Complete parts a through g below. LOADING... Click the icon to view the table. a. State the multiple regression equation. Let Upper X Subscript 1 i represent the alcohol content​ (%) of wine i and let Upper X Subscript 2 i represent the number of chlorides for wine i. Quality Alcohol_Content(%) Chlorides 0 7.9 0.067 1 7.1 0.062 2 8.9 0.067 2 8.1 0.071 2 8.6 0.073 3 8.9 0.074 2 9.3 0.072 5 9.5 0.077 6 10.4 0.077 7 10.3 0.079 7 10.1 0.083 6 10.9 0.084 7 11.4 0.081 7 11.4 0.084 6 11.9 0.095 9 11.5 0.096 8 11.7 0.119 9 11.5 0.143 10 12.3 0.151 9 12.3 0.159 b. Interpret the meaning of the​ slopes, b 1 and b 2​, in this problem. c. Explain why the regression​ coefficient, b 0​, has no practical meaning in the context of this problem. c. Predict the mean quality rating for wines that have 8​% alcohol content and chlorides of 0.10. d. Construct a​ 95% confidence interval estimate for the mean quality rating for wines that have 8​% alcohol and 0.10 chlorides. e. Construct a​ 95% confidence interval estimate for the mean quality rating for wines that have 8​% alcohol and 0.10 chloride Construct a​ 95% prediction interval estimate for the quality rating for an individual wine that has 8​% alcohol and 0.10 chlorides. h. What conclusions can you reach concerning this regression​ model?

Answer 1

Answer:

The production of wine is a​ multibillion-dollar worldwide industry. In an attempt to develop a model of wine quality as judged by wine​ experts, data was collected from red wine variants. A sample of 20 wines is provided in the accompanying table. Develop a multiple linear regression model to predict wine​ quality, measured on a scale from 0​ (very bad) to 10​ (excellent) based on alcohol content​ (%) and the amount of chlorides. Complete parts a through g below. LOADING... Click the icon to view the table. a. State the multiple regression equation. Let Upper X Subscript 1 i represent the alcohol content​ (%) of wine i and let Upper X Subscript 2 i represent the number of chlorides for wine i.

Quality Alcohol_Content(%) Chlorides

0 7.9 0.067

1 7.1 0.062

2 8.9 0.067

2 8.1 0.071

2 8.6 0.073

3 8.9 0.074

2 9.3 0.072

5 9.5 0.077

6 10.4 0.077

7 10.3 0.079

7 10.1 0.083

6 10.9 0.084

7 11.4 0.081

7 11.4 0.084

6 11.9 0.095

9 11.5 0.096

8 11.7 0.119

9 11.5 0.143

10 12.3 0.151

9 12.3 0.159

Excel Addon Megastat used.

Menu used: correlation/Regression ---- Regression Analysis.

Regression Analysis
	R²	0.874
	Adjusted R²	0.859	n	20
	R	0.935	k	2
	Std. Error	1.153	Dep. Var.	Quality
ANOVA table
Source	SS	df	MS	F	p-value
Regression	156.2121	2	78.1060	58.78	2.31E-08
Residual	22.5879	17	1.3287
Total	178.8000	19
Regression output					confidence interval
variables	coefficients	std. error	t (df=17)	p-value	95% lower	95% upper
Intercept	-12.1027	1.8734	-6.460	5.87E-06	-16.0553	-8.1502
Alcohol_Content(%)	1.5593	0.2577	6.051	1.30E-05	1.0157	2.1030
Chlorides	17.6127	14.0507	1.254	.2270	-12.0316	47.2570
Predicted values for: Quality
			95% Confidence Interval		95% Prediction Interval
Alcohol_Content(%)	Chlorides	Predicted	lower	upper	lower	upper
8	0.1	2.133	0.616	3.651	-0.733	5.000

Quality = -12.1027+1.5593* Alcohol_Content(%)+17.6127* Chlorides

b. Interpret the meaning of the​ slopes, b 1 and b 2​, in this problem.

When Alcohol_Content(%) increases by 1 percent, the quality increases by 1.5593.

When Chlorides increases by 1 , the quality increases by 17.6127.

c. Explain why the regression​ coefficient, b 0​, has no practical meaning in the context of this problem.

When there is no Alcohol_Content(%) and Chlorides, the quality is negative value which has no meaning.

c. Predict the mean quality rating for wines that have 8​% alcohol content and chlorides of 0.10.

predicted value = 2.133

d. Construct a​ 95% confidence interval estimate for the mean quality rating for wines that have 8​% alcohol and 0.10 chlorides.

95% CI = (0.616, 3.651)

Construct a​ 95% prediction interval estimate for the quality rating for an individual wine that has 8​% alcohol and 0.10 chlorides.

95% PI =(-0.733, 5.000)

h. What conclusions can you reach concerning this regression​ model?

R square =0.874, 87.4% of variance in quality is explained by the model. The model is useful for predicting quality.