Assume that the error term ϵ in the linear regression model is independent of x, and is normally distributed, with zero mean and constant variance. We can decide whether there is anysignificant relationship between x and y by testing the null hypothesis that β = 0.
Problem
Decide whether there is a significant relationship between the variables in the linear regression model of the data set faithful at .05 significance level.
Solution
We apply the lm function to a formula that describes the variable eruptions by the variablewaiting, and save the linear regression model in a new variable eruption.lm.
Then we print out the F-statistics of the significance test with the summary function.
> summary(eruption.lm)
Call:
lm(formula = eruptions ~ waiting, data = faithful)
Residuals:
Min 1Q Median 3Q Max
-1.2992 -0.3769 0.0351 0.3491 1.1933
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.87402 0.16014 -11.7 <2e-16 ***
waiting 0.07563 0.00222 34.1 <2e-16 ***
---
Signif. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’.’ 0.1 ’ ’ 1
Residual standard error: 0.497 on 270 degrees of freedom
Multiple R-squared: 0.811, Adjusted R-squared: 0.811
F-statistic: 1.16e+03 on 1 and 270 DF, p-value: <2e-16
Call:
lm(formula = eruptions ~ waiting, data = faithful)
Residuals:
Min 1Q Median 3Q Max
-1.2992 -0.3769 0.0351 0.3491 1.1933
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.87402 0.16014 -11.7 <2e-16 ***
waiting 0.07563 0.00222 34.1 <2e-16 ***
---
Signif. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’.’ 0.1 ’ ’ 1
Residual standard error: 0.497 on 270 degrees of freedom
Multiple R-squared: 0.811, Adjusted R-squared: 0.811
F-statistic: 1.16e+03 on 1 and 270 DF, p-value: <2e-16
Answer
As the p-value is much less than 0.05, we reject the null hypothesis that β = 0. Hence there is a significant relationship between the variables in the linear regression model of the data setfaithful.
Note
Further detail of the summary function for linear regression model can be found in the R documentation.
