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How to report linear regression results in a statistics paper

Linear Regression Overview and Example

Linear regression is a method for showing a linear relationship between an independent and a dependent variable. With any line, we have a model that is y equals an x plus b. In this case, a would-be the slope of your line and b would be your y-intercept.

In terms of where the line crosses the y-axis is your y-intercept, and the average rate of change is actually your slope. Now, getting back to linear regression. One method that we use is something called the least-squares regression.

What we do is we take our observed points and we try to find a linear relationship, and the way that we can do that is we can measure the vertical distances between our observed points and some line that we're trying to fit.

You could actually take this line and you can move it around, and while you're doing that, the vertical distances will change, and if you can find that sweet spot where the line will minimize the vertical distance and each vertical distance is going to result in a square if you can minimize the sum of all those squares, then you're able to find your best model.

When we refer to linear regression, our models would be slightly different than just a general linear model.

Our regression models would be y hat equals an x plus b. Y hat refers to our predicted value because linear regression is used for making predictions.

Our a is still our slope or our rate of change and our b is still going to be our y-intercept. Making predictions is useful, such as making a prediction on miles per gallon that we're using on a family trip where we go a certain speed.

When we predict within a valid range of x values, our predictions are fairly valid and that is called interpolation. When we predict outside of a valid range of x values, such as over here, that's called extrapolation. Predictions, in that case, may not be very reliable.

Let's suppose we took a trip and we kept track of our speed and miles per gallon that we achieved during our trip, and the information is in this table. Let's go ahead and find a linear model for this data and let's predict what our miles per gallon would be if we have an average speed of 75 miles per hour.

Whenever we do regression, it's a fairly complicated process so we would be using a calculator or computer software. When we put this information into a computer or the graphing calculator, we find out that our model is going to negative .4034 x plus 62.565. By using this model to figure out what our predicted miles per gallon will be.

We want to predict what our miles per gallon will be for a speed of 75 miles per hour as an average, so we'll throw our information in. We will replace x with 75, and when this is all said and done, we will get approximately 32.3. Now, what that means is, for an average speed of 75 miles per hour, we will get approximately 32.3 miles per gallon, and that should make sense. As your speed's going up, your miles per gallon should be going down.

### How to Write a Linear Regression Equation

**Summary**

Another Linear Regression Example and Conclusion

Let's consider an example where we have archaeopteryx fossil specimens for femur and humerus length. Let's go ahead and calculate a linear regression model and make a prediction for humerus length if we know the femur length is 57 centimeters. We take a look at our scatterplot here.

We can see that, as femur length is increasing, so is humerus length, so hopefully, we'll find that in the model as well when we make our prediction.

We're will go ahead and put this information into a graphing calculator or statistical software, and when we do, we're going to find that the model would be y hat equals 1.1969 x minus 3.6596. In order to make our prediction, we're will go ahead and put 57 in for x. We are going to replace the x with a 57.

When we do that, we will find out that we have a predicted length of approximately 64.56, so what does that mean? What that means is if we're going to use a femur length that is 57 centimeters to make a prediction, our predicted values will be 64.56 centimeters.

Linear regression describes a linear relationship between an independent and a dependent variable. Oftentimes, we use it to make predictions. Be careful to make predictions that are within the valid range of x values that you have.

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