Use the data you collected and plot the points to create scattergrams or scatter plots.

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1 Key terms: bivariate data, scatterplot (also called scattergram), correlation (positive, negative, or none as well as strong or weak), regression equation, interpolation, extrapolation, and correlation coefficient. Bivariate data may or may not be related by some criteria. Consider: height and shoe size weight and average daily caloric intake height and salary phone number and savings account balance Use the data you collected and plot the points to create scattergrams or scatter plots.

2 y x

3 Correlation You can see the type of correlation between the variables by using the scatter plot. POSITIVE if y increases as x increases (reading the graph from left to right) and NEGATIVE if y decreases as x increases. STRONG if the points are closely grouped. WEAKER the more the points are spread. Which scattergram below shows positive correlation? Which is stronger?

4 Regression equation, interpolation, extrapolation These equations are called MODELS. We use these mathematical models to make predictions. If our prediction is for an input value that lies within the domain of our collected data it is called interpolation. If our prediction is for an input value that lies outside of the domain of our collected data it is called extrapolation. You can estimate a "line of fit" by choosing two points on the plane which in your opinion will create a line that is close to as many points as possible. Use your scattergrams and draw in your estimate of a line of fit. Is it positive, negative, or no correlation? Is it strong or weak? Now use some graph paper and a ruler to create a scattergram and line of fit for this data from pg 96 #5 Write the equation for the line you drew and use this model to predict the value for September. Is this interpolation or extrapolation? You can enter the data in your calculator and have it calculate the "best fit line" by finding the linear regression. As we do this, you should enter the steps in the calculator portion of your notebook. We will solve # 5 again as well as #8

5 To create a scattergram on your calculator: choose STAT and EDIT then enter you should see list names at the top of the columns > if you do not have L 1 and then L 2 etc you should choose STAT and SETUP EDITOR then enter and it should say DONE > if the lists have values in them you must clear them by putting your cursor on top of the name of the list and pressing CLEAR not DEL but you must also press enter to move the cursor back down into the list in order to empty out all the values now you enter your data > put the x values in L 1 and the y values in L 2 now you must set up your graph by choosing STATPLOT it is above the Y= button so you need to use the 2nd key to get there > any plot will do so hit enter on one of the plots such as PLOT 1 > you must set it to ON > choose the first type of graph which is a scatter plot > make sure the x list is L 1 If it is not correct your must enter the correct list name by using the 2nd button and the number 1 (Look above the number 1 and you should see L 1 ) > make sure the y list is L 2 > choose a mark to be plotted the box, the +, or one tiny pixel (I only use this if I have many many points plotted in the same window) you do not want anything else on your graph so go to your Y= and make sure no equations are turned on you need a good window so choose ZOOM STAT You should now have a scatterplot.

6 To calculate the regression equation after you have entered your data and created a scattergram. use the STAT CALC menu choose the type of equation you wish to use > we will choose either LinReg(ax+b) or QuadReg press enter and the cursor will be waiting next to the command so you can enter more information > if you are using a newer model the regression name is at the top and it shows what list is being used for x and for y it should be L 1 for x and L 2 for y If it is not correct your must enter the correct list name by using the 2nd button and the number of the list you need. (Look above the number 1 and you should see L 1 ) > you then have to enter the name of the location in which you want to store the regression equation We ususally put it in y 1 unless something is already stored there then choose another location. To get that name you must enter VARS then Y VARS then FUNCTION ENTER then choose the correct y name from the list and press ENTER > now you are ready to calculate > if you are using an older model you should see the name of your regression equation and the cursor next to it still blinking. > you then have to enter the name of the location in which you want to store the regression equation We ususally put it in y 1 unless something is already stored there then choose another location. To get that name you must enter VARS then Y VARS then FUNCTION ENTER then choose the correct y name from the list and press ENTER > now you press enter and the calculations should be on your screen

7 Calculating the correlation coefficient, r of the coefficient of determination R 2 When you run the regression, the coefficients should be listed along with the equation and other values. If you do not see the corellation coefficient or coefficient of determination, you must turn them on as follows Choose the CATALOG command which is found above the number 0 but you must use the 2nd key to get it. You will be in ALPHA mode so press the key with the letter D which is above the x -1 now scroll down until you find DiagnosticOn and press enter. The command should be on your screen the blinking cursor after it. You must hit enter again and it must say done in order to turn it on. Now run the regression again and you should see the coefficients.

8 Correlation coefficient When you create a regression equation on your calculator, it also calculates the correlation coefficient. The closer the absolute value of r is to 1 the better the fit (more accurate predictions). The closer it is to 0 the worse the fit (poor predictions not useful in approximating true values). Estimate the values on your worksheet now. Use your calculator to solve pg 97 #9 and 10

9 Find both a linear and quadratic regression. Use each model to predict the height at 1 second and 1.5 seconds. Discuss the results and the "better" fit model

10 We will link and share a program called ScatSim. You will be asked to guess the value of r for randomly generated scattergrams. We will practice a few together and then you can do it on your own. Bonus points will be given if you can score.98 or higher on a round which includes at least 10 scatterplots. You must have a witness verify the number of plots before you begin and the score at the end. The witness can be someone from home or school. This is due by December 4th.

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