Finding the regression equation and best predicted value for earthquake data

Intro

Howdy!  I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today we're going to find the regression equation and best predictive value for earthquake data.  Here's our problem statement: Fifty matched pairs of magnitude-depth measurements are randomly selected from 10,594 earthquakes recorded in one year from a location in Southern California.  Find the best predicted depth of an earthquake with a magnitude of 1.3.  Use a significance level of 5%.

Part 1

OK, the first part of this problem asks us to find the regression equation.  To do this, I'm going to take the data and dump it into StatCrunch.  So here's my data.  I'm going to click on this icon and open my data in StatCrunch.  I'm going to resize this window so we can see everything.

Now, inside StatCrunch, I'm going to go to Stat --> Regression --> Simple Linear.  In my options window, I'm asked to identify the columns with my x- and y-variables.  Typically in these problems, the x-variable will be the one that's listed first; the y-variable will be the one that's listed second.  But you can also tell that this is the way that the variables are supposed to go because, when you look in the problem statement, you're asked to find a prediction for the depth.  The variable for which we find the prediction is always the Y, because it's what comes out of our regression equation.

So we see that we have the right variables selected.  And I just come down here and press Compute!, and here I get my results window.  Inside the results window, the regression equation appears here near the top.  Now, this regression equation is sandwiched in between a whole bunch of other stuff.  So I like to look down here at my parameter estimates table, and if you'll notice the numbers here are the same as the numbers up here.  So I like to just look down here to get the numbers to put into my answer fields.  I'm asked to round to one decimal place.  Well done!

Part 2

Now the second part of our problem asks us to find the best predicted depth given a magnitude of 1.3.  There are two options for making a prediction.  One is to use the regression model, and the second is to use the mean value of the y-values.  The regression equation is preferred; however, the regression model may not be suitable for use.   If it's a bad model because there's no correlation between the variables, then we don't want to use it to make predictions, and in that case, we'll use the mean value.

The way that we know whether or not the model is good is to test the R-value (our correlation coefficient).  So here from our data, we see our correlation coefficient is 0.19.  That's not very stellar.  In fact, it's probable that this is going to be a very bad model.  But there's an objective way to identify that.  We have to compare with our critical R-value.  Here in my problem statement, there's a link to a critical value table.  So I click on that link, and here the table I need to identify how many samples I have.  I have 50 pairs of sample data, so I'm going to go down to the line with 50, and we were told to use a significance level of 5%, so as you can see, that's this column right here.  So I go down to find the number next to 50, and here we go --- 0.279.

The R-value from our data (0.19) is less than this critical value, so we haven't exceeded this threshold.  That means we are in the Land of No Correlation.  And that means we should not be using this regression model to make predictions.  If this R value were greater than the critical R-value, then we could use it to make predictions.  But that's not the case here.  We have a bad model, so we're not going to use it to make predictions.

Instead, we're going to use the mean value of the y-values.  So to do that, I'm going to go up to Stat --> Summary Stats --> Columns, select the column with my y-values, select the mean is the statistic to be calculated, hit Compute!, and there's my best predicted value.  I'm asked to round to one decimal place. Good job!

And that's how we do it at Aspire Mountain Academy.  Be sure to leave your comments below and let us know how good a job we did or how we can improve.  And if your stats teacher is boring or just doesn't care to help you learn stats, go to aspiremountainacademy.com, where you can find out more about accessing our lecture videos or provide feedback on what you'd like to see.  Thanks for watching!  We'll see you in the next video.