2.15: Finding the Equation of Best Fit Using a Graphing Calculator
The calorie requirements for males is shown in the table below. Use your calculator to find the line of best fit for the data.
Calorie Requirements (Male), 1-59 years
Age Range, \begin{align*}x\end{align*} | 1-3 | 4-6 | 7-10 | 11-14 | 15-18 | 19-59 |
---|---|---|---|---|---|---|
Calorie Needs, \begin{align*}y\end{align*} | 1230 | 1715 | 1970 | 2220 | 2755 | 2550 |
The age is measured in years. Source: www.fatfreekitchen.com
Watch This
Watch the first part of this video, until about 3:30.
Keep in mind that a line of best fit can also be called a linear regression.
James Sousa: Linear Regression on the Graphing Calculator
Guidance
In the previous lesson, we learned how to find the linear equation of best fit by hand. This entire process can also be done by your graphing calculator. It is recommended that you use a graphing calculator for two main reasons: accuracy and consistency. The graphing calculator will be more accurate than a calculation by hand and it will also be more consistent between students, or a greater likelihood that everyone will get the same answer.
Example A
Below is a table for the total number of home runs hit from 1990-2000. Make a scatterplot using your graphing calculator.
1990 | 1991 | 1992 | 1993 | 1994 | 1995 | 1996 | 1997 | 1998 | 1999 | 2000 |
---|---|---|---|---|---|---|---|---|---|---|
3317 | 3383 | 3038 | 4030 | 3306 | 4081 | 4962 | 4640 | 5064 | 5528 | 5693 |
Solution: Recall pair is a point; (1990, 3317), for example. In the graphing calculator, we need to enter these as \begin{align*}x-\end{align*}values and \begin{align*}y-\end{align*}values. To do this, we need to create lists. The \begin{align*}x-\end{align*}values will be List1, or L1 and the \begin{align*}y-\end{align*}values will be List2, or L2. Instructions below are for the TI 83/84.
To create a list:
1. Press STAT.
2. In EDIT, select 1:Edit…. Press ENTER.
3. The List table appears. If there are any current lists, you will need to clear them. To do this, arrow up to L1 so that it is highlighted (black). Press CLEAR, then ENTER. Repeat with L2, if necessary.
4. Now, enter the data into the lists. Enter all the entries into L1 (years) first and press enter between each entry. Then, repeat with L2 and the total home run numbers. Your screen should look something like this when you are done.
5. Press \begin{align*}2^{nd}\end{align*} MODE (QUIT).
6. Press \begin{align*}Y=\end{align*}.
7. Clear any equations that are in the \begin{align*}Y=\end{align*}. To do this, arrow down to the equation and press CLEAR. Press \begin{align*}2^{nd}\end{align*} MODE (QUIT).
8. Press \begin{align*}2^{nd} Y=\end{align*} (STAT PLOT). Turn Plot1 on by highlighting On and pressing ENTER. Then, select the first option for the Type of stat plot. Make sure the \begin{align*}X\end{align*}list is L1 and the \begin{align*}Y\end{align*}list is L2. To change these, scroll down to \begin{align*}X\end{align*}list (for example), and press \begin{align*}2^{nd}\end{align*} 1 (L1). \begin{align*}2^{nd}\end{align*} 2 is L2. Mark should also be on the first option.
9. Press GRAPH. Nothing may show up. If this is the case, press ZOOM and scroll down to 9:ZoomStat. Press ENTER. Your plot should look something like this.
Example B
Find the equation of best fit for the data from Example A. Use a graphing calculator.
Solution: Here are the directions for finding the equation of best fit. In the TI 83/84, it is also called Linear Regression, or LinReg.
1. After completing steps 1-5 above, press STAT and then arrow over to the CALC menu.
2. Select 4:LinReg\begin{align*}(ax+b)\end{align*}. Press ENTER.
3. You will be taken back to the main screen. Type (L1,L2) and press ENTER. L1 is \begin{align*}2^{nd}\end{align*} 1, L2 is \begin{align*}2^{nd}\end{align*} 2 and the comma is the button above the 7.
4. The following screen shows up. To the calculator, \begin{align*}a\end{align*} is the slope and \begin{align*}b\end{align*} is still the \begin{align*}y-\end{align*}intercept. Therefore, the equation of the line is \begin{align*}y = 267.4x + 2939.55\end{align*}.
5. If you would like to plot the line on the scatterplot from Example A above, (after Step 9 in Example A) press \begin{align*}Y=\end{align*} and enter in the equation from Step 4. Press GRAPH.
Example C
Use the equation found in Exmaple B to predict the number of home runs hit in 2015.
Solution: Use \begin{align*}x=25\end{align*} because \begin{align*}x=0\end{align*} is 1990. Plug into the linear regression equation, \begin{align*}y=267.4x+2939.55\end{align*}
\begin{align*}y=267.4(25)+2939.55 \\ y=9624.55\end{align*}
Because we cannot have a fraction of a home run, round up to 9625.
Intro Problem Revisit Following the steps outlined in this lesson on your calculator, the line of best fit is \begin{align*}y=1076.3333333333 + 284.85714285714x\end{align*}.
Guided Practice
Round all answers to the nearest hundredth.
1. Use the data set from Examples A and B from the previous concept and find the equation of best fit with the graphing calculator. Compare this to the answer we found in Example B. Use both equations to find the sales for 2010.
2. Use the data set from the Guided Practice in the previous concept and find the equation of best fit with the graphing calculator. Compare this to the answer we found.
Answers
1. From Examples A and B in the previous concept, we did not have a table. You need to estimate the values from the scatterplot. Here is a sample table for the scatterplot. Remember, this is years vs. money, in billions.
1999 (0) | 2000 (1) | 2001 (2) | 2002 (3) | 2003 (4) | 2004 (5) | 2005 (6) | 2006 (7) | 2007 (8) | 2008 (9) | 2009 (10) |
---|---|---|---|---|---|---|---|---|---|---|
14.8 | 14.1 | 13.8 | 13 | 12 | 12.8 | 13 | 12.3 | 11 | 9 | 6.2 |
Now, using Examples A (Steps 1-5) and B (Steps 1-4) from this concept, determine the equation of best fit. If you used the data set above, you should get \begin{align*}y = -0.66x+15.28\end{align*}. The equation we got in the previous concept was \begin{align*}y = - \frac{4}{7}x+14.57\end{align*} or \begin{align*}y = -0.57x+14.57\end{align*}. Using the calculator’s equation for 2010 (11), we get \begin{align*}y = -0.66(11)+15.28 = 8.02\end{align*} billion. Using the equation from the previous concept, we get \begin{align*}y = -0.57(11)+14.57 = 8.3\end{align*} billion. As you can see, the answers are pretty close.
2. Using the calculator, we get \begin{align*}y = -0.096x+15.25\end{align*}. By hand, we got \begin{align*}y = - \frac{2}{21}x+15.29\end{align*} or \begin{align*}y = -0.095x+15.29\end{align*}. Comparing answers from 2.5, or 30 months, with the calculator, we get 12.37 hours and we got 12.4 when we did it by hand. All in all, the answers were quite close.
Practice
Round all decimal answers to the nearest hundredth.
- Using the Apple data from the previous concept (repeated below), find the equation of best fit with your calculator. Set Oct. 2009 as \begin{align*}x = 0\end{align*}.
The price of Apple stock from Oct 2009 - Sept 2011 source: Yahoo! Finance
10/09 | 11/09 | 12/09 | 1/10 | 2/10 | 3/10 | 4/10 | 5/10 | 6/10 | 7/10 | 8/10 | 9/10 |
---|---|---|---|---|---|---|---|---|---|---|---|
$181 | $189 | $198 | $214 | $195 | $208 | $236 | $249 | $266 | $248 | $261 | $258 |
10/10 | 11/10 | 12/10 | 1/11 | 2/11 | 3/11 | 4/11 | 5/11 | 6/11 | 7/11 | 8/11 | 9/11 |
$282 | $309 | $316 | $331 | $345 | $352 | $344 | $349 | $346 | $349 | $389 | $379 |
- Predict the sales for January 2012.
- Using the Home Run data from the previous concept (repeated below), find the equation of best fit with your calculator. Set 2000 as \begin{align*}x = 0\end{align*}.
Total Number of Home Runs Hit in Major League Baseball, 2000-2010 source: www.baseball-almanac.com
2000 | 2001 | 2002 | 2003 | 2004 | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 |
---|---|---|---|---|---|---|---|---|---|---|
5693 | 5458 | 5059 | 5207 | 5451 | 5017 | 5386 | 4957 | 4878 | 4655 | 4613 |
- Predict the total number of home runs hit in 2015. Compare this to the answer from Example B.
- The table below shows the temperature for various elevations, taken at the same time of day in roughly the same location. Using your calculator, find the equation of best fit.
Elevation, ft. | 0 | 1000 | 5000 | 10,000 | 15,000 | 20,000 | 30,000 |
---|---|---|---|---|---|---|---|
Temperature, \begin{align*}^\circ{F}\end{align*} | 60 | 56 | 41 | 27 | 9 | -8 | -40 |
- Using your answer from #5, what would be the estimated temperature at 50,000 feet?
- The table below shows the average life expectancy (in years) of the average male in relation to the year they were born. Using your calculator, find the equation of best fit. Set 1930 as \begin{align*}x = 30\end{align*}.
Year of birth | 1930 | 1940 | 1950 | 1960 | 1970 | 1980 | 1990 | 2000 | 2007 |
---|---|---|---|---|---|---|---|---|---|
Life expectancy, Males | 58.1 | 60.8 | 65.6 | 66.6 | 67.1 | 70 | 71.8 | 74.3 | 75.4 |
Source: National Center for Health Statistics
- Based on your equation from #7, what would you predict the life expectancy of a male born in 2012 to be?
- The table below shows the average life expectancy (in years) of the average female in relation to the year they were born. Using your calculator, find the equation of best fit. Set 1930 as \begin{align*}x = 30\end{align*}.
Year of birth | 1930 | 1940 | 1950 | 1960 | 1970 | 1980 | 1990 | 2000 | 2007 |
---|---|---|---|---|---|---|---|---|---|
Life expectancy, Males | 61.6 | 65.2 | 71.1 | 73.1 | 74.7 | 77.4 | 78.8 | 79.7 | 80.4 |
Source: National Center for Health Statistics
- Based on your equation from #9, what would you predict the life expectancy of a male born in 2012 to be?
- Science Connection Why do you think the life expectancy for both men and women has increased over the last 70 years?
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Image Attributions
Here you'll learn how to use the graphing calculator to find the linear equation of best fit for a scatterplot.