UNIT 1E DATA-DRIVEN INTRODUCTION TO LINEAR REGRESSION HOW TO FIND THE LINEAR REGRESSION LINE IN EXCEL: 1. Go to the second tab of our Excel sheet for today. 2. Right-click on one of the data points, and select “Add Trendline” 3. In the resulting dialog box, choose “linear,” and click the bottom two boxes Excel gives us back both the (least squares) best fit line and a measure of how well the line fit our data: 2 r is the square of the correlation coefficient . 2 r is always a number between 0 and 1. 2 If r is close to 1, the regression line is a pretty good fit for the data points. 2 If r is close to 0, the regression line is a pretty poor fit for the data points. HOW TO COMPARE TWO LINEAR MODELS FOR A GIVEN DATA SET: RESIDUAL SUM OF SQUARES (RSS) I’ll walk you through how to calculate the RSS for one data set. A small RSS means a small error in fitting the data, so if you have two models the one with the smallest RSS “wins.” 1. 2. 3. You should be given a table with input values 𝑥1 , 𝑥2 , … 𝑥 and corresponding output values 𝑦1 , 𝑦2 , … 𝑦 that are from the data. You should be given (or should have found) the equation for a linear model 𝑦 = (𝑥). You should use this model to compute output values for each given input value: 1 = (𝑥1 ), 2 = (𝑥2 ), … , = (𝑥 ). You should calculate the difference for each pair of output values (data from #1 versus calculated from #2), square those differences, and add them all up: = (𝑦1 − 1. 2 1) + (𝑦2 − 2 2) + + (𝑦 − )2 . Please see the “HW #1 Car Data” tab in today’s Excel file. Let’s explore the relationship between the weight of a car and its corresponding gas mileage. a. We want to create a scatter plot of mpg versus weight, since we suspect that heavier cars have worse mpg. (Imagine for a second what that plot will look like.) To do this: i. Copy Column C (mpg) by clicking the C at the top of that column to select it, and then select home>copy. ii. Insert that copy into Column H; do this by selecting column H, then selecting home>insert>insert copied cells. You now have weight in column G and mpg in column H. iii. Highlight those two columns of data, including the headings (weight and mpg). iv. Click the Insert tab, select Scatter, and choose the top-left option. v. Now format the chart to show the Axes labels, Chart Title, and to remove the Legend. See the instructions in the Day 1 Activity if you’ve forgotten how. b. 2 Find the linear regression line, and write its equation and r value here: Equation: _________________________________ 2 r value: __________________ c. Unit 1E p. 1 Do you think there’s a strong linear relationship between a car’s weight and its gas mileage? Explain. MA 151 Mathematical Methods for Business Marymount University 18 d. What gas mileage would you expect for a 3000-pound car? Use the regression line to answer. _______________________ e. 2. Attach a printout of your graph (AND NO spreadsheet prints), formatted as suggested above, and showing the 2 regression line and r value. The way you can print only the graph is to right click the graph and ask Excel to put it in its own tab. Please find in the last tab of our Excel sheet for today information about the Standard and Poor stock values on every day for approximately the past year. We want to create a scatterplot that shows S&P value over time. a. Note there is a column for the date and then another how many days have passed since 1/10/2014. Because date values can be confusing to use in Excel, and because the values would otherwise be so large, we this “days” column. You may find the formula interesting – check it out. b. Use the days and the S&P value columns to create the scatterplot. Give the graph an appropriate title, appropriate gridlines, and label the axes. 2 c. Use Trendline to find the linear regression line, and display this on the graph. Write the equation and the r value here: Equation: ____________________ 2 r value: __________________ d. What does this regression line capture from this past year’s S&P values that you think is accurate? e. What do you think this regression line cannot do well? f. Assuming there are no shocks to our economy and the stock market continues to progress in a similar fashion, predict the value of the S&P index on February 1. Can you think of a way to also predict how accurate you will be? g. Please include a graph of the data with its regression line – please follow the instructions in number 2 and feel free to include both graphs on the same page. If you’d like to extend the graph line through the end of February, that would help you make your point in part f well. Unit 1E p. 2 MA 151 Mathematical Methods for Business Marymount University 19 3. Here’s a very simple example of some data that we want to fit with a regression line – we’ll ignore any potential application for now: Input 𝑥 1 2 3 4 5 Output 𝑦 1 1.8 3.1 3.8 3.5 a. Suppose you tried to estimate this data using the model 𝑦 = 𝑥. Without using Excel (so that you’re practicing for a test/quiz-type question), tell me the residual sum of squares (RSS). (Show your work. For your convenience, there are a couple blank rows on the table you can use if you choose.) b. Sketch a graph of the data versus this linear mode. Label your axes and scale clearly, and choose a scale that clearly shows the given data. c. By eyeballing this data, suggest a better equation for a linear model to fit the data, and demonstrate that this line does fit the data better by calculating the residual sum of squares (RSS). There are many possible answers. Show your work on the back of this sheet, but write the equation and sum here: Unit 1E p. 3 MA 151 Mathematical Methods for Business Marymount University 20 x 0 1 2 3 4 y 7 8 9 11 13 GUESSES m b 1 7 RESIDUAL CALCULATION yguess resid resid^2 7 0 0 8 0 0 9 0 0 10 1 1 11 2 4 residual sum of squares (RSS) 5 14 12 Profit ($billion) DATA x y 0 7 1 8 2 9 3 11 4 13 10 8 6 4 2 0 0 1 2 3 YEAR (0=1996) 3 (0=1996) 4 5 0 4 7 11 0 1 2 3 4 Actual Data 7 8 9 11 13 GE Profits Profit (in billions) Year 18 16 14 12 10 8 6 4 2 0 y = 1,5x + 6,6 R² = 0,9698 0 1 2 3 Years Since 1996 4 y = 1,5x + 6,6 R² = 0,9698 5 DATA Area BTU's 150 4900 180 5425 240 6375 250 6400 290 7025 315 7400 350 7900 370 8275 410 8750 450 9400 x 150 180 240 250 290 315 350 370 410 450 GUESSES m b 14,80 2736,54 RESIDUAL SUM 19769,33 RESIDUAL CALCULATION y yguess resid resid^2 4900 4957,07 -57,07 3256,60 5425 5401,17 23,83 567,74 6375 6289,38 85,62 7329,96 6400 6437,42 -37,42 1400,27 7025 7029,56 -4,56 20,81 7400 7399,65 0,35 0,12 7900 7917,77 -17,77 315,90 8275 8213,84 61,16 3740,01 8750 8805,99 -55,99 3134,41 9400 9398,13 1,87 3,51 residual sum 19769,33 While shopping for an air conditioner, a buyer consulted the following table giving room area and the number of BTUs it takes to cool that area. What are the slope and intercept of the best linear model? energy (BTUs) 10000 8000 6000 4000 2000 0 0 Area 150 180 240 250 290 315 350 370 410 450 BTU's 4900 5425 6375 6400 7025 7400 7900 8275 8750 9400 100 200 300 area (sq feet) 400 500 BTU's 10000 9000 y = 14,804x + 2736,5 R² = 0,9989 8000 7000 6000 5000 4000 3000 2000 1000 0 0 50 100 150 200 250 300 350 400 450 500 150 4957 450 9398 make AMC Concord AMC Pacer AMC Spirit Buick Century Buick Electra Buick LeSabre Buick Opel Buick Regal Buick Riviera Buick Skylark Cad. Deville Cad. Eldorado Cad. Seville Chev. Chevette Chev. Impala Chev. Malibu Chev. Monte Carlo Chev. Monza Chev. Nova Dodge Colt Dodge Diplomat Dodge Magnum Dodge St. Regis Ford Fiesta Ford Mustang Linc. Continental Linc. Mark V Linc. Versailles Merc. Bobcat Merc. Cougar Merc. Marquis Merc. Monarch Merc. XR-7 Merc. Zephyr Olds 98 Olds Cutl Supr Olds Cutlass Olds Delta 88 Olds Omega Olds Starfire Olds Toronado Plym. Arrow Plym. Champ Plym. Horizon Plym. Sapporo Plym. Volare price 4099 4749 3799 4816 7827 5788 4453 5189 10372 4082 11385 14500 15906 3299 5705 4504 5104 3667 3955 3984 4010 5886 6342 4389 4187 11497 13594 13466 3829 5379 6165 4516 6303 3291 8814 5172 4733 4890 4181 4195 10371 4647 4425 4482 6486 4060 mpg rep78 22 17 22 20 15 18 26 20 16 19 14 14 21 29 16 22 22 24 19 30 18 16 17 28 21 12 12 14 22 14 15 18 14 20 21 19 19 18 19 24 16 28 34 25 26 18 3 3 3 4 3 3 3 3 3 2 3 3 4 3 2 2 3 5 2 2 2 4 3 3 3 3 4 4 3 3 4 3 4 3 3 4 3 1 3 3 5 3 2 headroom trunk weight mpg 2,5 11 2930 3 11 3350 3 12 2640 4,5 16 3250 4 20 4080 4 21 3670 3 10 2230 2 16 3280 3,5 17 3880 3,5 13 3400 4 20 4330 3,5 16 3900 3 13 4290 2,5 9 2110 4 20 3690 3,5 17 3180 2 16 3220 2 7 2750 3,5 13 3430 2 8 2120 4 17 3600 4 17 3600 4,5 21 3740 1,5 9 1800 2 10 2650 3,5 22 4840 2,5 18 4720 3,5 15 3830 3 9 2580 3,5 16 4060 3,5 23 3720 3 15 3370 3 16 4130 3,5 17 2830 4 20 4060 2 16 3310 4,5 16 3300 4 20 3690 4,5 14 3370 2 10 2730 3,5 17 4030 2 11 3260 2,5 11 1800 4 17 2200 1,5 8 2520 5 16 3330 22 17 22 20 15 18 26 20 16 19 14 14 21 29 16 22 22 24 19 30 18 16 17 28 21 12 12 14 22 14 15 18 14 20 21 19 19 18 19 24 16 28 34 25 26 18 length 186 173 168 196 222 218 170 200 207 200 221 204 204 163 212 193 200 179 197 163 206 206 220 147 179 233 230 201 169 221 212 198 217 195 220 198 198 218 200 180 206 170 157 165 182 201 Pont. Catalina Pont. Firebird Pont. Grand Prix Pont. Le Mans Pont. Phoenix Pont. Sunbird Audi 5000 Audi Fox BMW 320i Datsun 200 Datsun 210 Datsun 510 Datsun 810 Fiat Strada Honda Accord Honda Civic Mazda GLC Peugeot 604 Renault Le Car Subaru Toyota Celica Toyota Corolla Toyota Corona VW Dasher VW Diesel VW Rabbit VW Scirocco Volvo 260 5798 4934 5222 4723 4424 4172 9690 6295 9735 6229 4589 5079 8129 4296 5799 4499 3995 12990 3895 3798 5899 3748 5719 7140 5397 4697 6850 11995 18 18 19 19 19 24 17 23 25 23 35 24 21 21 25 28 30 14 26 35 18 31 18 23 41 25 25 17 4 1 3 3 2 5 3 4 4 5 4 4 3 5 4 4 3 5 5 5 5 4 5 4 4 5 4 1,5 2 3,5 3,5 2 3 2,5 2,5 1,5 2 2,5 2,5 2,5 3 2,5 3,5 3,5 3 2,5 2,5 3 2 2,5 3 3 2 2,5 20 7 16 17 13 7 15 11 12 6 8 8 8 16 10 5 11 14 10 11 14 9 11 12 15 15 16 14 3700 3470 3210 3200 3420 2690 2830 2070 2650 2370 2020 2280 2750 2130 2240 1760 1980 3420 1830 2050 2410 2200 2670 2160 2040 1930 1990 3170 18 18 19 19 19 24 17 23 25 23 35 24 21 21 25 28 30 14 26 35 18 31 18 23 41 25 25 17 214 198 201 199 203 179 189 174 177 170 165 170 184 161 172 149 154 192 142 164 174 165 175 172 155 155 156 193 turn displacement 40 40 35 40 43 43 34 42 43 42 44 43 45 34 43 31 41 40 43 35 46 46 46 33 43 51 48 41 39 48 44 41 45 43 43 42 42 42 43 40 43 37 37 36 38 44 gear_ratio 121 258 121 196 350 231 304 196 231 231 425 350 350 231 250 200 200 151 250 98 318 318 225 98 140 400 400 302 140 302 302 250 302 140 350 231 231 231 231 151 350 156 86 105 119 225 3,58 2,53 3,08 2,93 2,41 2,73 2,87 2,93 2,93 3,08 2,28 2,19 2,24 2,93 2,56 2,73 2,73 2,73 2,56 3,54 2,47 2,47 2,94 3,15 3,08 2,47 2,47 2,47 2,73 2,75 2,26 2,43 2,75 3,08 2,41 2,93 2,93 2,73 3,08 2,73 2,41 3,05 2,97 3,37 3,54 3,23 foreign Domestic Domestic Domestic Domestic Domestic Domestic Domestic Domestic Domestic Domestic Domestic Domestic Domestic Domestic Domestic Domestic Domestic Domestic Domestic Domestic Domestic Domestic Domestic Domestic Domestic Domestic Domestic Domestic Domestic Domestic Domestic Domestic Domestic Domestic Domestic Domestic Domestic Domestic Domestic Domestic Domestic Domestic Domestic Domestic Domestic Domestic 45 40 35 30 25 20 15 10 5 0 0 1000 2000 42 42 45 40 43 41 37 36 34 35 32 34 38 36 36 34 33 38 34 36 36 35 36 36 35 35 36 37 231 231 231 231 231 151 131 97 121 119 85 119 146 105 107 91 86 163 79 97 134 97 134 97 90 89 97 163 2,73 3,08 2,93 2,93 3,08 2,73 3,2 3,7 3,64 3,89 3,7 3,54 3,55 3,37 3,05 3,3 3,73 3,58 3,72 3,81 3,06 3,21 3,05 3,74 3,78 3,78 3,78 2,98 Domestic Domestic Domestic Domestic Domestic Domestic Foreign Foreign Foreign Foreign Foreign Foreign Foreign Foreign Foreign Foreign Foreign Foreign Foreign Foreign Foreign Foreign Foreign Foreign Foreign Foreign Foreign Foreign mpg 3000 4000 5000 6000 Effective date 1/10/2013 1/11/2013 1/14/2013 1/15/2013 1/16/2013 1/17/2013 1/18/2013 1/22/2013 1/23/2013 1/24/2013 1/25/2013 1/28/2013 1/29/2013 1/30/2013 1/31/2013 2/1/2013 2/4/2013 2/5/2013 2/6/2013 2/7/2013 2/8/2013 2/11/2013 2/12/2013 2/13/2013 2/14/2013 2/15/2013 2/19/2013 2/20/2013 2/21/2013 2/22/2013 2/25/2013 2/26/2013 2/27/2013 2/28/2013 3/1/2013 3/4/2013 3/5/2013 3/6/2013 3/7/2013 3/8/2013 3/11/2013 3/12/2013 3/13/2013 3/14/2013 3/15/2013 3/18/2013 3/19/2013 3/20/2013 Days 0 1 4 5 6 7 8 12 13 14 15 18 19 20 21 21 24 25 26 27 28 31 32 33 34 35 39 40 41 42 45 46 47 48 51 54 55 56 57 58 61 62 63 64 65 68 69 70 S&P 500 1472,12 1472,05 1470,68 1472,34 1472,63 1480,94 1485,98 1492,56 1494,81 1494,82 1502,96 1500,18 1507,84 1501,96 1498,11 1513,17 1495,71 1511,29 1512,12 1509,39 1517,93 1517,01 1519,43 1520,33 1521,38 1519,79 1530,94 1511,95 1502,42 1515,6 1487,85 1496,94 1515,99 1514,68 1518,2 1525,2 1539,79 1541,46 1544,26 1551,18 1556,22 1552,48 1554,52 1563,23 1560,7 1552,1 1548,34 1558,71 3/21/2013 3/22/2013 3/25/2013 3/26/2013 3/27/2013 3/28/2013 4/1/2013 4/2/2013 4/3/2013 4/4/2013 4/5/2013 4/8/2013 4/9/2013 4/10/2013 4/11/2013 4/12/2013 4/15/2013 4/16/2013 4/17/2013 4/18/2013 4/19/2013 4/22/2013 4/23/2013 4/24/2013 4/25/2013 4/26/2013 4/29/2013 4/30/2013 5/1/2013 5/2/2013 5/3/2013 5/6/2013 5/7/2013 5/8/2013 5/9/2013 5/10/2013 5/13/2013 5/14/2013 5/15/2013 5/16/2013 5/17/2013 5/20/2013 5/21/2013 5/22/2013 5/23/2013 5/24/2013 5/28/2013 5/29/2013 5/30/2013 5/31/2013 6/3/2013 6/4/2013 6/5/2013 6/6/2013 6/7/2013 71 72 75 76 77 78 81 82 83 84 85 88 89 90 91 92 95 96 97 98 99 102 103 104 105 106 109 110 111 112 113 116 117 118 119 120 123 124 125 126 127 130 131 132 133 134 138 139 140 141 143 144 145 146 147 1545,8 1556,89 1551,69 1563,77 1562,85 1569,19 1562,17 1570,25 1553,69 1559,98 1553,28 1563,07 1568,61 1587,73 1593,37 1588,85 1552,36 1574,57 1552,01 1541,61 1555,25 1562,5 1578,78 1578,79 1585,16 1582,24 1593,61 1597,57 1582,7 1597,59 1614,42 1617,5 1625,96 1632,69 1626,67 1633,7 1633,77 1650,34 1658,78 1650,47 1667,47 1666,29 1669,16 1655,35 1650,51 1649,6 1660,06 1648,36 1654,41 1630,74 1640,42 1631,38 1608,9 1622,56 1643,38 6/10/2013 6/11/2013 6/12/2013 6/13/2013 6/14/2013 6/17/2013 6/18/2013 6/19/2013 6/20/2013 6/21/2013 6/24/2013 6/25/2013 6/26/2013 6/27/2013 6/28/2013 7/1/2013 7/2/2013 7/3/2013 7/5/2013 7/8/2013 7/9/2013 7/10/2013 7/11/2013 7/12/2013 7/15/2013 7/16/2013 7/17/2013 7/18/2013 7/19/2013 7/22/2013 7/23/2013 7/24/2013 7/25/2013 7/26/2013 7/29/2013 7/30/2013 7/31/2013 8/1/2013 8/2/2013 8/5/2013 8/6/2013 8/7/2013 8/8/2013 8/9/2013 8/12/2013 8/13/2013 8/14/2013 8/15/2013 8/16/2013 8/19/2013 8/20/2013 8/21/2013 8/22/2013 8/23/2013 8/26/2013 150 151 152 153 154 157 158 159 160 161 164 165 166 167 168 171 172 173 175 178 179 180 181 182 185 186 187 188 189 192 193 194 195 196 199 200 201 201 202 205 206 207 208 209 212 213 214 215 216 219 220 221 222 223 226 1642,81 1626,13 1612,52 1636,36 1626,73 1639,04 1651,81 1628,93 1588,19 1592,43 1573,09 1588,03 1603,26 1613,2 1606,28 1614,96 1614,08 1615,41 1631,89 1640,46 1652,32 1652,62 1675,02 1680,19 1682,5 1676,26 1680,91 1689,37 1692,09 1695,53 1692,39 1685,94 1690,25 1691,65 1685,33 1685,96 1685,73 1706,87 1709,67 1707,14 1697,37 1690,91 1697,48 1691,42 1689,47 1694,16 1685,39 1661,32 1655,83 1646,06 1652,35 1642,8 1656,96 1663,5 1656,78 8/27/2013 8/28/2013 8/29/2013 8/30/2013 9/3/2013 9/4/2013 9/5/2013 9/6/2013 9/9/2013 9/10/2013 9/11/2013 9/12/2013 9/13/2013 9/16/2013 9/17/2013 9/18/2013 9/19/2013 9/20/2013 9/23/2013 9/24/2013 9/25/2013 9/26/2013 9/27/2013 9/30/2013 10/1/2013 10/2/2013 10/3/2013 10/4/2013 10/7/2013 10/8/2013 10/9/2013 10/10/2013 10/11/2013 10/14/2013 10/15/2013 10/16/2013 10/17/2013 10/18/2013 10/21/2013 10/22/2013 10/23/2013 10/24/2013 10/25/2013 10/28/2013 10/29/2013 10/30/2013 10/31/2013 11/1/2013 11/4/2013 11/5/2013 11/6/2013 11/7/2013 11/8/2013 11/11/2013 11/12/2013 227 228 229 230 233 234 235 236 239 240 241 242 243 246 247 248 249 250 253 254 255 256 257 260 261 262 263 264 267 268 269 270 271 274 275 276 277 278 281 282 283 284 285 288 289 290 291 291 294 295 296 297 298 301 302 1630,48 1634,96 1638,17 1632,97 1639,77 1653,08 1655,08 1655,17 1671,71 1683,99 1689,13 1683,42 1687,99 1697,6 1704,76 1725,52 1722,34 1709,91 1701,84 1697,42 1692,77 1698,67 1691,75 1681,55 1695 1693,87 1678,66 1690,5 1676,12 1655,45 1656,4 1692,56 1703,2 1710,14 1698,06 1721,54 1733,15 1744,5 1744,66 1754,67 1746,38 1752,07 1759,77 1762,11 1771,95 1763,31 1756,54 1761,64 1767,93 1762,97 1770,49 1747,15 1770,61 1771,89 1767,69 11/13/2013 11/14/2013 11/15/2013 11/18/2013 11/19/2013 11/20/2013 11/21/2013 11/22/2013 11/25/2013 11/26/2013 11/27/2013 11/29/2013 12/2/2013 12/3/2013 12/4/2013 12/5/2013 12/6/2013 12/9/2013 12/10/2013 12/11/2013 12/12/2013 12/13/2013 12/16/2013 12/17/2013 12/18/2013 12/19/2013 12/20/2013 12/23/2013 12/24/2013 12/26/2013 12/27/2013 12/30/2013 12/31/2013 1/2/2014 1/3/2014 1/6/2014 1/7/2014 1/8/2014 1/9/2014 1/10/2014 303 304 305 308 309 310 311 312 315 316 317 319 322 323 324 325 326 329 330 331 332 333 336 337 338 339 340 343 344 346 347 350 351 357 358 361 362 363 364 365 1782 1790,62 1798,18 1791,53 1787,87 1781,37 1795,85 1804,76 1802,48 1802,75 1807,23 1805,81 1800,9 1795,15 1792,81 1785,03 1805,09 1808,37 1802,62 1782,22 1775,5 1775,32 1786,54 1781 1810,65 1809,6 1818,32 1827,99 1833,32 1842,02 1841,4 1841,07 1848,36 1831,98 1831,37 1826,77 1837,88 1837,49 1838,13 1842,37 Source: S&P Dow Jones Indices. 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Purchase answer to see full attachment