Generally, when the number of data points is large, an F-statistic that is only a little bit larger than 1 is already sufficient to reject the null hypothesis (H0 : There is no relationship between speed and distance). However, how much larger the F-statistic needs to be depends on both the number of data points and the number of predictors. The further the F-statistic is from 1 the better it is. That’s why the adjusted $R^2$ is the preferred measure as it adjusts for the number of variables considered.į-statistic is a good indicator of whether there is a relationship between our predictor and the response variables. Essentially, it will vary with the application and the domain studied.Ī side note: In multiple regression settings, the $R^2$ will always increase as more variables are included in the model. Nevertheless, it’s hard to define what level of $R^2$ is appropriate to claim the model fits well. That why we get a relatively strong $R^2$. Step back and think: If you were able to choose any metric to predict distance required for a car to stop, would speed be one and would it be an important one that could help explain how distance would vary based on speed? I guess it’s easy to see that the answer would almost certainly be a yes. Or roughly 65% of the variance found in the response variable (dist) can be explained by the predictor variable (speed). It always lies between 0 and 1 (i.e.: a number near 0 represents a regression that does not explain the variance in the response variable well and a number close to 1 does explain the observed variance in the response variable). $R^2$ is a measure of the linear relationship between our predictor variable (speed) and our response / target variable (dist). It takes the form of a proportion of variance. The R-squared ($R^2$) statistic provides a measure of how well the model is fitting the actual data. In our case, we had 50 data points and two parameters (intercept and slope). Simplistically, degrees of freedom are the number of data points that went into the estimation of the parameters used after taking into account these parameters (restriction). It’s also worth noting that the Residual Standard Error was calculated with 48 degrees of freedom. In other words, given that the mean distance for all cars to stop is 42.98 and that the Residual Standard Error is 15.3795867, we can say that the percentage error is (any prediction would still be off by) 35.78%. In our example, the actual distance required to stop can deviate from the true regression line by approximately 15.3795867 feet, on average. The Residual Standard Error is the average amount that the response (dist) will deviate from the true regression line. Due to the presence of this error term, we are not capable of perfectly predicting our response variable (dist) from the predictor (speed) one. Theoretically, every linear model is assumed to contain an error term E. Residual Standard Error is measure of the quality of a linear regression fit. Consequently, a small p-value for the intercept and the slope indicates that we can reject the null hypothesis which allows us to conclude that there is a relationship between speed and distance.

Three stars (or asterisks) represent a highly significant p-value. In our model example, the p-values are very close to zero. Typically, a p-value of 5% or less is a good cut-off point. A small p-value indicates that it is unlikely we will observe a relationship between the predictor (speed) and response (dist) variables due to chance. The Pr(>t) acronym found in the model output relates to the probability of observing any value equal or larger than t. # lm(formula = dist ~ speed.c, data = cars)

R studio regression output