Here is a summary data from a texbook
##
## Call:
## lm(formula = mpg ~ weight, data = automobile)
##
## Residuals:
## Min 1Q Median 3Q Max
## -12.012 -2.801 -0.351 2.114 16.480
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 46.3173644 0.7952452 58.24 <2e-16 ***
## weight -0.0076766 0.0002575 -29.81 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.345 on 396 degrees of freedom
## Multiple R-squared: 0.6918, Adjusted R-squared: 0.691
## F-statistic: 888.9 on 1 and 396 DF, p-value: < 2.2e-16
The author then goes on to say that
A 95% confidence interval for β1, as we learned how to calculate last week, would also verify the strength of the linear representation of weight in this SLR model
Now, here is another data (quadratic equation)
##
## Call:
## lm(formula = mpg ~ weight + I(weight^2), data = automobile)
##
## Residuals:
## Min 1Q Median 3Q Max
## -12.6643 -2.7111 -0.3293 1.8185 16.0863
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.252e+01 2.971e+00 21.042 < 2e-16 ***
## weight -1.864e-02 1.959e-03 -9.517 < 2e-16 ***
## I(weight^2) 1.716e-06 3.042e-07 5.643 3.19e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.185 on 395 degrees of freedom
## Multiple R-squared: 0.7148, Adjusted R-squared: 0.7133
## F-statistic: 495 on 2 and 395 DF, p-value: < 2.2e-16
And author says here
We can say here that by looking at the summary output, we can guess that a confidence interval (say, 95%) for β2 would not cover 0, suggesting that the quadratic term for weight is appropriate to add in the model.
Can someone explain that how from this summary data author is able to interpret this?
