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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?

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    $\begingroup$ A decent approximation of the 95 % confidence interval is Estimate -+ 2 * SE. (The 2.5 % and 97.5 % quantiles of the standard normal distribution are -1.96 and +1.96.) There is not much guessing needed here. $\endgroup$
    – Roland
    Nov 16, 2020 at 7:13

1 Answer 1

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The p value is for a test of the null hypothesis that the estimate is equal to zero. It literally means the probability of observing these data (or data even further from zero), if the parameter for this estimate IS actually zero. Since this probability is extremely small (0.00000003) this is very strong evidence that the parameter is not zero and if we constructed a 95% confidence interval, this would not span zero. If the confidence interval did span zero then we would be able to say that zero is a plausible value for this estimate. With such a small p-value, this is not plausible.

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