Question

URGENT PLEASE HELP

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.

Wine Quality Content (%) 7.6 Chlorides 0.061 0.061 0.073 0.071 0.071 0.071 0.074 0.079 0.083 0.075 0.082 0.081 0.083 0.084 0.098 0.091 0.137 0.113 0.154 7.4 8.9 7.5 8.9 9.1 8.6 9.4 10.3 10.3 10.7 10.9 11.4 11.5 11.9 11.6 11.6 11.6 4 10 12.1 12.3 0.158
b. Interpret the meaning of the slopes B1 and B2 in this problem

c. Explain why the regression​ coefficient, b0​, has no practical meaning in the context of this problem.

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

e. Construct a​ 95% confidence interval estimate for the mean quality rating for wines that have 8​%alcohol and .10 chlorides.

f. Construct a​ 95% prediction interval estimate for the quality rating for an individual wine that has 8​% alcohol and .10

chlorides.

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

Answer 1

Let's carry out regression in Excel (go to Data tab -> Data Analysis -> Regression, choose Quality as Y-column, and Alcohol Content, Chlorides as Y-columns). The output is:

Regression Statistics
Multiple R	0.93164272
R Square	0.867958158
Adjusted R Square	0.852423824
Standard Error	1.172348171
Observations	20

	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	-11.90786112	1.808850665	-6.583109017	4.64988E-06	-15.72420243	-8.091519812
Alcohol Content (%)	1.625394856	0.24835144	6.544736977	5.00049E-06	1.101419118	2.149370594
Chlorides	9.014905454	14.17419472	0.636008298	0.533240795	-20.89003137	38.91984228

a)  $\hat{Y_i}$ = -11.91 + 1.63 * $\hat{X_1_i}$ + 9.01 * $\hat{X_2_i}$

b) Slope b1 for X1 is 1.63, which means the quality rating increases by 1.63 every % increase in alcohol content.

Slope b2 for X2 is 9.01, which indicates quality rating increases by 9.01 per unit increase in amount of chlorides.

c) The -ve value -11.91 of b0 tells us that the rating is -11.91 in cases where both the alcohol content and chlorides content are 0. But, given a quality rating below 0 is meaningless, the intercept is meaningless.

d) For 8% alcohol content and chlorides of 0.1, predicted quality rating,

$\hat{Y}$ = -11.91 + 1.63 * 8 + 9.01 * 0.1 = 2.031 ~ 2

e) 95% confidence interval for this prediction is given by:

1(r- 7)2 n-2, (1-a/2) * yY n(n-1) *s2

= (0.596, 3.466)

f) 95% prediction interval for this prediction is given by:

n- 1)*s prediction interval, Pl-y+-tn-2, (a-a/2) Sy

= (-4.54, 8.6)

g) Given the high p-value of chloride amount (0.533), chloride isn't a significant predictor of quality rating. However, given the low p-value of alcohol content % (5e-06), alcohol content is a very significant predictor of wine quality rating. Also, given the R-squared value ~ 0.87, nearly 87% of the variation in quality ratings is explained by these 2 predictors (which should hopefully improved if we remove the less significant predictor, chloride content).