Finding the variations and a prediction interval for diamond weights and price data

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 variations and a prediction interval for diamond weights and price data. Here's our problem statement: The table below lists weights in carats and prices in dollars of randomly selected diamonds.  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 a diamond that weighs 0.8 carats.

Part A

OK, Part A wants us to find the explained variation.  This is a quantity that we find in the ANOVA table.  And to get the ANOVA table, we're going to have to perform linear regression analysis.  So the first step to do that is to get my data here and dump it into StatCrunch.  Here's my data in StatCrunch, and now I'm going to resize this window so we can see everything a little bit better.  Great.

Now to get the linear regression analysis, I go to Stat --> Regression --> Simple Linear.  In the options window, I'm going to select my x- and y-variables.  Typically the x-variable is the variable that's mentioned first.  The y-variable is the one that's mentioned next.  The ANOVA table comes out of the hypothesis test, so I want to leave this radio button for hypothesis test selected.  And then down here is an area for Prediction of Y, which we'll use in a moment when we get to that part of the problem.

Well, I have everything I need for my ANOVA table, so I come down here and hit Compute!, and here is my results window.  If I scroll down here to the bottom, here's my analysis of variance or ANOVA table.  The explained variation is going to be the sum of the squares --- that's this SS that you see here --- for the model.  The reason why this is the explained portion of the variation is that we understand where the model comes from.  We can explain why the model is what it is because we have the data, we know what the data is and we know how the model is created from the data.  So the sum of the squares for the model is going to be the portion of the variation that is explained.  We're instructed to round to the nearest whole number, so I'm just going to type this number in because it's already rounded to the nearest whole number.  Excellent!

Part B

Now Part B asks us to find the unexplained variation, which is the sum of the squares for the error.  The error is what we can't explain.  We don't understand why the model is so different from reality.  Why do we observe something in reality?  This is not predicted by the model.  There's a certain random nature to real life that we can't explain with our model.  And so the sum of the squares connected with the error is the unexplained variation.  Once again, were asked around to the nearest whole number.  Nice work!

Part C

And now Part C asks us to find the indicated prediction interval.  The prediction interval we need to find is explained here in the problem statement.  "For the prediction interval, use a 95% confidence level with a diamond that weighs 0.8 carats."  So to get that I need to come back here to my options window and scroll down to this area labeled Prediction of Y.

In this field for x-value, I'm going to stick in the 0.8 carats, because the weight of the diamond, notice, is the x-variable.  So that's going to be my x value.  For my prediction, the prediction level is a default 95%.  That's what we were asked to actually calculate our prediction interval for.  So we have everything we need. I press Compute!, and then here is my results window.  If I scroll down to the bottom, here's an area entitled Predicted values.  And in that table, there at the very end is my prediction interval.  So I'm going to go in, and again we're asked to round to the nearest whole number as needed.  Excellent!

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