Call: lm(formula = SALES ~ POP + MEDHVL, data = WiscLottery) Residuals: Min 1Q M

Question

Call:

lm(formula = SALES ~ POP + MEDHVL, data = WiscLottery)

Residuals:

    Min      1Q Median      3Q     Max

-8540.5 -1451.5 -454.0   794.8 15745.2

Coefficients:

              Estimate Std. Error t value Pr(>|t|)   

(Intercept) -2.681e+03 1.789e+03 -1.498   0.1407   

POP          5.911e-01 5.587e-02 10.581     5.02e-14 ***

MEDHVL       6.431e+01 3.375e+01 1.906   0.0628 .

---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3692 on 47 degrees of freedom

Multiple R-squared: 0.8009,    Adjusted R-squared: 0.7924

F-statistic: 94.52 on 2 and 47 DF, p-value: < 2.2e-16

1.Perform t-test for each regression Coecient in your model.

2.Perform F-test to test the overall signi cance of the regression. Comment on the adequacy of the model.

Explanation / Answer

1. t-test

In the t-test for regression model, we try to decide whether there is any significant relationship between x and y, i.e. our independent and dependent variables, by testing the null hypothesis.

In the question we are asked to perform t test for each coefficient. I will try to put it in simplest words.

In your output look at the table under coefficient heading.

This table shows various details about the coefficients of variables used in our regression model. With t-test, we try to figure out which variables significantly changes our output value of dependent variable and which ones not.

A larger (insignificant) p-value suggests that changes in the predictor are not associated with changes in the response.

In the above output we can see that p value(last column in POP row) of POP is 5.02 x 10-14 which is lesser than 0.05 significance level. So, POP is a statistically significant variable.

Let us do same for MEDHVL , its p value is 0.0628 which is greater than 0.05, which means that it is not a significantly significant variable for our model.

2. F-test

F – Test for overall significance compares a regression model having intercept only with the current model. And then tries to decide on whether addition of these variables together is significant enough for them to be there or not.

we have the following NULL hypothesis:

H0 :The fit of intercept only model and the current model is same.

P Value of F Statistic 94.52 for DF 47 is extremely small (<2.2 x 10-16), i.e smaller that 0.001 so we can reject H0.

Which means that our regression model is significant. So, it is adequate.

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