How to Find ŷ from a Regression on TI-89
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-89.

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

Display the graph, if it’s not already on screen.	[◆] [GRAPH]
Trace the regression line, not the data points.	[F3] brings up the trace cursor. But note the P1 in the upper right corner of the screen. That tells you that you’re tracing the data points and not the regression line.   Press [▼] until the upper right displays 1 rather than P1.
The current x and y coordinates are displayed at the bottom of the screen. You can type your desired x, and the calculator will figure the corresponding ŷ.	Press the white-on-gray numeric keys including [(-)] and decimal point if needed. The x coordinate changes to match your typing.   After entering your number, press [ENTER]. The calculator moves the cursor and displays the corresponding ŷ value.

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.