Finding ŷ from a Regression on TI-83/84
with a Note on Finding Residuals

Summary: The regression line represents the model that best fits the data. One important reason for doing the regression in the first place is to answer the question, what average y value does the model predict for a given x? This page shows you two methods of answering that question.

Method 1: Trace on the Regression Line Graph (preferred)

You can make predictions while examining the graph of the regression line on the TI-83/84 or TI-89.

Advantages to this method: aside from being pretty cool, it avoids rounding errors, and it’s very fast for multiple predictions.

One time only, verify the format settings.	[2nd ZOOM makes FORMAT] Verify that you have CoordOn and ExprOn; the other settings aren’t important.
Activate tracing on the regression line.	[TRACE]
Look in the upper left corner to make sure that the regression equation is displayed.	If you see P:L1,L2, press [▲] to display the regression equation.

Enter the x value.

Press the black-on-white numeric keys including [(−)] and decimal point if needed.   As soon as you press the first number, you’ll see a large X= appear at the bottom left of the screen. Enter any additional digits and press [ENTER].   The TI-83/84 displays the predicted average y value (ŷ) at the bottom right and puts a blinking cursor at that point on the regression line.

Caution: ŷ = 267.1 yd is the predicted or expected average distance for a club-head speed of 102 mph. But that does not mean any particular golf ball hit at that speed will travel that exact distance. You can think of ŷ as the average travel distance that we would expect for a whole lot of golf balls hit at that speed.

Caution: A regression equation is valid only within the range of actual measured x values, and a little way left and right of that range. If you try to go too far outside the valid range, the calculator will display ERR:INVALID.

Method 2: Use Calculated Regression Equation (if necessary)

But what if you don’t still have the regression line on your calculator, for instance if you’ve done a different regression? In that case, you can go back to your written-down regression equation and plug in the desired x value.

Advantage of this method: You already know how to substitute into equations.  Disadvantages: depending on the specific numbers involved, you may introduce rounding errors. Also, since you’re entering more numbers there’s an increased chance of entering a number wrong.

Example To find the predicted average y value for x = 102, go back to the regression equation that you wrote down, and substitute 102 for x:

ŷ = 3.1661x − 55.7966

ŷ = 3.1661*102 − 55.7966

ŷ = 267.1456 → 267.1

In this example, the rounding error was very small, and it disappeared when you rounded ŷ to one decimal place. But there will be problems where the rounding error is large enough to affect the final answer, so always use the trace method if you can.

Again, please observe the Cautions above. With this method, the calculator won’t tell you when your x value is outside a reasonable range, so you need to be aware of that issue yourself.

Finding Residuals

Each measured data point has an associated residual, defined as y−ŷ, the distance of the point above or below the line. To find a residual, the actual y comes from the original data, and the predicted ŷ comes from one of the methods above.

Example: Find the residual for x = 102.
Solution: From the original data, y = 264. From either of the methods above, ŷ = 267.1. Therefore the residual is y−ŷ = 264−267.1 = −3.1 yards.

If a given x value occurs in more than one data point, you have multiple residuals for that x value.

What’s New

• 9 Jun 2013: Add a cross reference to the new Excel version of these instructions.
• (intervening changes suppressed)
• 29 May 2002: new document