Relationship between lifespan and income from countries to countries over years

Relationship between lifespan and income from countries to countries over years

Significance Test for Linear Regression

http://www.r-tutor.com/elementary-statistics/simple-linear-regression/significance-test-linear-regression

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.

> eruption.lm = lm(eruptions ~ waiting, data=faithful)

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

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.

> help(summary.lm)