Question

One of your aunts (after having several glasses of wine) insists that there is a linear relationship between a state's population and the wine consumed there. You look up the following data from 2013 regarding the population per state (and D.C.) in millions of people and wine consumed in millions of liters:

Variable n  $\sum x$$\sum x^2$

Population (51) (316.2049) (4437.7177)

Wine (51) (2931.4609) (539997.52)

Pop x Wine (51) (45983.94) N/A

(a)Test to see if a significant linear relationship exists between the population and wine consumed using $\alpha$ = 0.01

(b) Fit a linear regression and interpret the estimated slope and intercept

(c) Find and interpret the $R^2$ value for the regression obtained

Answer 1

a) The p-value for each term tests the null hypothesis that the coefficient is equal to zero (no effect). A low p-value (< 0.05) indicates that you can reject the null hypothesis. In other words, a predictor that has a low p-value is likely to be a meaningful addition to your model because changes in the predictor's value are related to changes in the response variable

With the given data

Analysis of Variance

Source	DF	SS	MS	F	P
Regression	1	1.92861E+11	1.92861E+11	410.71	0.031
Error	1	4.69583E+08	4.69583E+08
Total	2	1.93330E+11

Since p-value is less than the level of significance, we can say that a significant linear relationship exists between the population and wine consumed

Fitted linear regression equation is :

The regression equation is
Wine = - 21211 + 126.2 Population

Slope = - 21211

Intercept = 126.2

Model Summary

S	R-sq	R-sq(adj)
21669.9	99.76%	99.51%