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QUANTIFYING PREDICTABILITY Exercise #1: Make sure that your calculator has its r value on. 2
Exercise #2: In the following exercises four data sets with equal x values are given to illustrate different types of positive correlations. For each, enter the data, observe the scatter plot, and record the r value, known as the correlation coefficient, for a linear fit to the nearest thousandth. (e) How does the correlation coefficient quantify the fit of a positive correlation? 3
Exercise #3: The following data set is that of two variables that have a negative correlation. Enter the data, produce the scatter plot, and record the r value. How is the negative correlation reflected in the r value? Exercise #4: Given the scatter plot shown below, which of the r values would most likely represent the correlation between the two variables? Explain your choice. 4
Exercise #5: Which of the following scatter plots would have a correlation coefficient closest to 1? 5
Exercise #6: There are two primary types of crude oil sold in the world, West Texas Intermediate (WTI) and Brent Crude. Each is priced differently on a daily basis and each has a correlation with the average price per gallon for unleaded gasoline. The two linear regression models, along with their r values, are shown below. Give a prediction for the price per gallon of unleaded gasoline, y, on a day when the price for WTI is $103 and the price for Brent is $109, x. Which model did you choose and why? Brent Crude: y = 0.028x + 0.71, r = 0.973 WTI Crude: y = 0.031x + 0.67, r = 0.924 6
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RESIDUALS Exercise #7: A skydiver jumps from an airplane and an attached micro computer records the time and speed of the diver for the first 12 seconds of the diver s freefall. The data is shown in the table below. (a) Find the equation for the line of best fit for this data set. Round both coefficients to the nearest tenth. As well, determine the correlation coefficient and round it to the nearest hundredth. Based on the correlation coefficient, characterize the fit as positive or negative and how strong of a fit it is. Create a plot with both the data and line of best fit shown on it. You do not need to reproduce the plot below. 8
Exercise #7: A skydiver jumps from an airplane and an attached micro computer records the time and speed of the diver for the first 12 seconds of the diver s freefall. The data is shown in the table below. (b) The residual of a data point is defined as the difference between the observed y value and the predicted y value. Using tables on your calculator fill in the table below with the predicted values (rounded to the nearest integer) and the residuals for each data point. 9
(c) Sketch a plot of the residuals below. Your teacher will need to show you how to do this on your graphing calculator. Make sure all other scatter plots and equations are off. Do the residuals show any distinct pattern? 10
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Exercise #8: A school district was attempting to correlate the number of hours a student studies in a given week with their grade point average. They surveyed 8 students and found the following data. (a) Find the equation for the line of best fit and the associated r value. Round the linear coefficients to the nearest tenth and the r value to the nearest hundredth. Then create a scatter plot with both the data and the line graphed. You do not need to reproduce that graph here unless your teacher asks you to. (b) What is the value of the residual associated with the data point (11, 94)? Show the calculation that leads to your answer. 12
Exercise #8: A school district was attempting to correlate the number of hours a student studies in a given week with their grade point average. They surveyed 8 students and found the following data. (c) Produce, using your calculator, the residual graph. It does not need to be exact, but show your WINDOW and the correct general location of the residuals. (d) Why does this residual plot show a more appropriate linear model than the one in Exercise #1, even though the r value is worse? 13
QUANTIFYING PREDICTABILITY RESIDUALS Daily Learning Outcome: I know how to interpret the correlation coefficient of a scatter plot. I know how to make a residual plot. https://www.youtube.com/watch?v=bs4zaio7_mm https://www.youtube.com/watch?v=y3hix7mt0gw 14