Precalculus Honors 4.01-4.06 Handouts PDF

4.01 Intro to Linear Regression Name Precalculus Honors Period Date Tom raises turkeys. He records the weight, 𝒚, measured in pounds, of one of his x 2 5 8 11...

4.01 Intro to Linear Regression Name Precalculus Honors Period Date Tom raises turkeys. He records the weight, 𝒚, measured in pounds, of one of his x 2 5 8 11 14 turkeys (named Tori) over several months, 𝒙. The data is shown in the table. y 7 9 18 17 24 1. Plot the points from the table. Then, using a ruler, draw the line that you think best fits the data. 2. Find the equa on of your line. Label the 𝑦 in your equa on as 𝑦. This is how we show what is predicted by an equa on rather than the actual 𝑦-value measured. Use the table below for problems 3-6. 3. Make a table using your equa on, and put it in 𝑦. 4. Find each residual, which is the difference between the actual and predicted y-values. (𝑟𝑒𝑠𝑖𝑑 = 𝑦 − 𝑦) 5. Square each residual (𝑟𝑒𝑠𝑖𝑑 ). 𝒙 𝒚 𝑦 𝑟𝑒𝑠𝑖𝑑 𝑟𝑒𝑠𝑖𝑑 2 7 5 9 8 18 11 17 14 24 6. What is the sum of the squared residuals? 7. Who in your group has the lowest sum? How does their line look compared to the others in the group? 8. Enter the original data into a calculator. To do this, go [stat]Edit. Put the data into L1 and L2. Then do [2nd] [mode] to quit. Then go to the [Y=] screen and delete anything there. Then do [2nd][Y=]. Hit [enter] and make your screen look like  Then do [zoom] 9: statplot. Did it make a scatterplot? 9. Do [stat]->Calc 4: LinReg. Write the equation with the values given. Again, use 𝑦, since this is for a predicted 𝑦-value. Carefully draw this line (in another color) on your graph. 10. Recreate the residuals table, this time using your equation from #9. How does the sum of the squared residuals compare to yours and the sums of your group members? 𝒙 𝒚 𝑦 𝑟𝑒𝑠𝑖𝑑 𝑟𝑒𝑠𝑖𝑑 2 7 5 9 8 18 11 17 14 24 Save this space for some class notes: 4.02 The Correlation Coefficient Name Precalculus Honors Period Date Back to the turkeys again. The table shows the weight, 𝑦, of a turkey at mes 𝑥 months a er it was hatched. The graph shows a sca er plot of the data. 1. Find the average of the y-values, 𝑦 (see the table below). Plot a horizontal line on the graph at 𝑦 = 𝑦. 2. The devia on of each point is its directed distance from the line 𝑦 = 𝑦 to the point. Show that you understand the meaning of devia on by drawing the devia on of each point on the graph. 3. Calculate the deviation and the square of the deviation for each data point. Record these in the table to the left. Then calculate the sum of the squares of the deviations, 𝑆𝑆. 𝒙 𝒚 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 𝑑𝑒𝑣 4. Run linear regression on the data. Write the regression equa on, 𝑦, and 2 7 record the value of 𝑟 and the value of the correla on coefficient 𝑟. 5 9 8 18 11 17 5. On the sca erplot, plot the graph of the regression equa on. You recall that the residual of each data point is the directed ver cal distance from the 14 24 regression line to the point. Draw on the graph the residual for each point. 6. Find the sum of the squared residuals, 𝑆𝑆. Use the table lower le. 7. 𝑆𝑆 is the amount of 𝑆𝑆 that remains a er regression has taken out as much of it as possible. The frac on of 𝑆𝑆 that is removed by the linear regression is called the coefficent of determina on. Calculate 8. Write where the coefficient of determina on shows up in the linear 𝒙 𝒚 𝒚 𝑟𝑒𝑠𝑖𝑑 𝑟𝑒𝑠𝑖𝑑 regression on the calculator. Explain how the correla on coefficient 𝑟 is 2 7 calculated from the coefficient of determina on. 5 9 8 18 11 17 14 24 9. Draw a hand turkey 4.03 Linear Regression Name Precalculus Honors Period Date 1. Suppose that these data have been measure for two related variables 𝑥 and 𝑦. a. Enter the data into two lists on your calculator. Show by linear regression that the best-fi ng linear x y func on is 𝑦 = 1.4𝑥 + 3.8. Record the correla on coefficient. 5 11 8 16 11 19 14 27 17 25 20 29 b. Make a sca erplot of the data in your calculator. Also graph 𝑦. How well does the linear func on fit the 23 33 data? 26 42 29 44 32 51 c. Calculate 𝑥̅ and 𝑦, the means of 𝑥 and 𝑦. Show algebraically that the mean-mean point (𝑥̅ , 𝑦) is on the regression line. (To calculate the means, do [stat] calc 1:1-var stats with L1 to get 𝑥̅. Repeat with L2 for 𝑦) d. The line 𝑦 = 1.5𝑥 + 1.95 also contains the mean-mean point (𝑥̅ , 𝑦), but it has a slope of 1.5 instead of 1.4. Plot the line on the same screen as part b. Can you tell from the graphs which line fits the data be er? Explain. 2. Lisa Carr fills up her car’s gas tank and drives off. The table shows the number of gallons of gas le Distance in the tank at various numbers of miles driven. Driven Gas Left a. Run a linear regression on the data. Write down the linear regression equa on, along with 𝑟 and 𝑟. (mi) (gal) How do these numbers tell you that the regression line fits the data perfectly? Why is 𝑟 nega ve? 6 16.7 22 15.9 44 14.8 50 14.5 b. Plot the data and the regression equa on on the same screen. How can you tell graphically that the 60 14.0 regression line fits the data perfectly? c. Show that your mathema cal model predicts that the tank is empty a er 340 miles. Because this number is found by extrapola on, how confident are you that the car will actually run out of gas a er 340 miles? 3. Data cloud problem. Below are shown “clouds” of data points and their correla ons. 𝑟 = 0.96 𝑟 = 0.68 𝑟 = −0.77 𝑟 = −0.33 a. Based on the graphs above, match what you think each cloud to their correla ons. A. –0.93 B. –0.86 C. –0.25 D. 0.47 E. 0.64 b. Sketch the cloud you would expect for the data with the following correla on coefficients. 𝑟 = 0.99 𝑟 = 0.92 𝑟 = 0.31 𝑟 = 0.04 4.04 Violent Crimes Problem (wuut?) Name Precalculus Honors Period Date An aspiring reporter is attempting to draw some conclusions about the crime rate in the United States. Here are actual numbers for arrests for violent crimes reported for the years 1971 to 1997. Let 𝒙 be the number of years that have passed since 1970, and let 𝒚 be the number of violent crime arrests reported in each year 𝒙. Year Total Year Total Year Total 1971 323,060 1980 475,160 1989 685,500 1972 350,410 1981 490,460 1990 705,500 1973 380,560 1982 526,200 1991 718,890 1974 429,350 1983 499,390 1992 742,130 1975 451,310 1984 493,960 1993 754,110 1976 441,630 1985 497,560 1994 778,730 1977 435,720 1986 553,900 1995 796,250 1978 469,700 1987 546,300 1996 729,900 1979 467,700 1988 625,900 1997 717,750 1. To make things simpler, the reporter does a linear regression with the eight years 1974 and 1989 through 1995 (in bold). The year 1974 was included to give a wider range of data and therefore, perhaps, a more accurate mathematical model. Show that the correlation is very good and that interpolating to 1975 gives a highly accurate result. 2. Show numerically that interpolating to 1985 and extrapolating to 1997 both produce quite inaccurate results. 3. Enter all the data into your calculator. Then make a scatterplot. Does the result look like the one given here? 850000 4. Find the particular equation of the linear function Total violent crime arrests 750000 that best fits the entire data set. Record the correlation 650000 coefficient, 𝑟. Plot this function as y2 and the linear function from problem 1 as y1. Sketch the results on the 550000 figure. 450000 350000 250000 1970 1975 1980 1985 1990 1995 2000 Year 5. A reporter looking only at the data from 1971 through 1985 would observe that the number of crime arrests was leveling off. Suppose the reporter assumed that a quadratic function was a reasonable mathematical model. What quadratic function would best fit this range of data? What is the value of 𝑟 ? Plot this quadratic function on your calculator and sketch the result on the previous figure. How far off would the quadratic function’s prediction be for the number of crime arrests in 1997? 6. From context (or from looking it up in the precalc books in the back of the room or google), what do interpolate and extrapolate mean? What is the danger of extrapolating? 7. Look back to the turkey problem in 4.02. What does the model predict a 7-year-old turkey to weigh? Is this realistic? Are you interpolating or extrapolating? 8. What did you learn as a result of this exploration that you did not know before? 4.05 Nonlinear Regression Name Precalculus Honors Period Date 1. Enter the data into your calculator and make a sca erplot (zoom 9 to create your window!). 𝒙 𝒇(𝒙) From its appearance, what kind of func on do you think it might be? Why? 3 4.1 7 24 8 37 11 136 2. By regression, find the par cular equa on of the best-fi ng linear, quadra c, exponen al, 12 212 and power func on. Include the value of 𝑟. (Reminder: to do a regression, go to [stat]  calc, then choose the regression you want. Linear: 𝑟 = Quadra c: 𝑟 = Exponen al: 𝑟 = Power: 𝑟 = Which is best? How do you know? 3. Enter the data into your calculator and make a sca erplot. From its appearance, what kind of 𝒙 𝒇(𝒙) func on do you think it might be? Why? 3 70 7 1360 8 2180 11 6600 12 8980 4. By regression, find the par cular equa on of the best-fi ng linear, quadra c, exponen al, and power func on. Include the value of 𝑟. (Reminder: to do a regression, go to [stat]  calc, then choose the regression you want. Linear: 𝑟 = Quadra c: 𝑟 = Exponen al: 𝑟 = Power: 𝑟 = Which is best? How do you know?  5. Enter the data into your calculator and make a sca erplot. From its appearance, what kind of 𝒙 𝒇(𝒙) func on do you think it might be? Why? 1 1 3.5 7 5.5 11 20 18 72 6. By regression, find the par cular equa on of the best-fi ng linear, quadra c, exponen al, and power func on. Include the value of 𝑟. (Reminder: to do a regression, go to [stat]  calc, then choose the regression you want. Linear: 𝑟 = Quadra c: 𝑟 = Exponen al: 𝑟 = Power: 𝑟 = Which is best? How do you know? Last unit, you did a LOT of algebra to find equa ons for func ons. Do you have any new tricks now? A. Iden fy the func on type. B. Quickly find equa ons for each of the func ons shown. (no algebra needed!) 7. 8. y 8 (32,8) 6 4 2 (1,2) x 5 10 15 20 25 30 35 9. 10. y 10 y (5,9) 16 8 6 12 4 (2,3) 8 2 x (2,5.58) (11,1) 4 2 4 6 8 10 12 14 (4,2.0088) -2 x -4 2 4 6 8 4.06 Regression in Context Name Precalculus Honors Period Date 1. Ann A. Student types a paragraph on her computer. Then she adjusts the width of the 𝒙 𝒚 paragraph by changing the margins. The graph and table show the widths and number of (inches (number lines the paragraph takes. wide) of lines) 6.5 5 a. Would it be possible to make the width equal zero? Why, then, would a decreasing 5.5 6 power func on be more reasonable than an exponen al func on for number of lines as 5 7 a func on of paragraph width? Confirm that a power func on fits be er by running both 3.75 9 power and exponen al regressions and comparing the correla on coefficients. 3 11 1.5 24 1 38 40 number of lines 30 20 10 b. Plot the power func on and the exponen al func on from part a on the same 0 screen as a sca er plot of the data. How does the result confirm that the 0 1 2 3 4 5 6 7 power func on fits be er? line width c. Because each paragraph contains the same words, you might expect the area of the page taken by the paragraph to be constant. In a list on your calculator, calculate the area of each paragraph. Use the fact that there are 7 lines per inch. Make a sca er plot of the areas as a func on of paragraph width. By linear regression, show that there is a downward trend in the areas but that the correla on is not very strong. Sketch the graph and points. 2. An ad for a bank lists the numbers of years it will take for the balance in your account to reach a certain level if you invest $1000 in one of their accounts. a. Make a sca er plot of the data. Which way is the graph concave? How can you tell Years to from the shape of the graph that a logarithmic func on would fit the data? Dollars Reach $ 1,000 0 $ 1,100 1.94 b. By logarithmic regression, find the par cular equa on of the best-fi ng logarithmic $ 1,500 8.11 func on. How does the correla on coefficient confirm that a logarithmic func on fits $ 2,000 13.86 well? (note: you’ll need to use LNREG, there is no LOGREG) $ 3,000 21.97  c. Plot the equa on from part b on the sca er plot of part a. Sketch the result. d. Interpolate using your mathema cal model to find out how long it takes for $2500 (the average of $2000 and $3000) to be in the account. Is this number of years equal to the average of 13.86 and 21.97? e. If you wanted to leave your money in the account un l it had reached $5000, how many years would you have to leave it there? Which method do you use, interpola on or extrapola on? Explain. 3. Farmer MacDonald researched what happens with compe on of wild oats (a weed) with Percent various crops. The more wild oats that are growing with a crop, the greater the loss in yield Wild Oats lost of of that crop. The table and graph show the results of that research. (plants/m2) crop a. From the sca er plot, tell what kind of func on is most likely to fit the data. By 1 3 regression, find the par cular equa on of this kind of func on. Plot it and the sca er 5 8 plot on the same screen. Sketch the result. 10 10 20 17 50 25 100 34 150 40 200 48 b. Based on your mathema cal model, what percent of crop loss would be expected for 500 wild oat plants per square meter? Do you find this number by extrapola on or by interpola on? c. How many wild oat plants per square meter do you predict it would take to choke out the wheat crop completely? Do you find this number by extrapola on or by interpola on?