Finding the explained variation, the unexplained variation, and a prediction interval estimate

Intro

Howdy!  I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today we're going to learn how to find the explained variation, the unexplained variation, and a prediction interval estimate.  Here's our problem statement: Listed below are altitudes in thousands of feet and outside air temperatures in degrees Fahrenheit recorded during a flight.  Find the explained variation, unexplained variation, and indicated prediction interval.  There is sufficient evidence to support a claim of a linear correlation, so it is reasonable to use the regression equation when making predictions for the prediction interval.  Use a 95% confidence level with an altitude of 6327 feet.  And here we see our data set.

Part A

So Part A asks us first to find the explained variation.  To do this, I'm going to place my data into StatCrunch.  So here I have StatCrunch.  I'm going to resize this window a bit.  OK, now we're ready to go.

The explained variation as well as the unexplained variation will come from an ANOVA table.  So I need to create my ANOVA table, and to do that I'm just going to use the ANOVA table that comes out of the regression results.  So I go up here to Stat –> Regression –> Simple Linear (because the problem statement said we have a claim of linear correlation that is supported).  I'm going to select my x- and y-variables and press Compute!

Here we have our results window.  If I scroll down here, you can see my ANOVA table for the results.  The explained variation is the sum of the squares for the model, so that will be this number right here.  So I round that to two decimal places for my answer field.  Nice work!

Part B

The unexplained variation is the sum of squares for the error, so that's this number here.  Excellent!

Part C

And finally, for an indicated prediction interval, I go to the results window and click on this Options button, and in the drop down menu I click on Edit.  Then if I scroll back down here in my options window, I'm looking for this area.  This is Prediction of y.  So I put in the value of x for which I want to make a prediction.  Here we're making a prediction for 6327 feet, but since all the altitudes are expressed in thousand feet, I need to put that in as 6.327.

We're asked for a 95% confidence level.  That's the default selection here, so I'm just going to leave that alone.  I press Compute!, and out comes my results window.  When I go down to the bottom of that window, here's a table.  Here's the value that I put in for the prediction.  Here's the prediction that comes out of it.  And at the very end of that table at the bottom we see the limits for a prediction interval.  So I'm going to put those numbers in here.  Just round to four decimal places.  Nice work!

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