1 00:00:11,020 --> 00:00:13,920 The last chapter we are going to talk about in this 2 00:00:13,920 --> 00:00:17,820 semester is correlation and simple linear regression. 3 00:00:18,380 --> 00:00:23,300 So we are going to explain two types in chapter 4 00:00:23,300 --> 00:00:29,280 12. One is called correlation. And the other type 5 00:00:29,280 --> 00:00:33,500 is simple linear regression. Maybe this chapter 6 00:00:33,500 --> 00:00:40,020 I'm going to spend about two lectures in order to 7 00:00:40,020 --> 00:00:45,000 cover these objectives. The first objective is to 8 00:00:45,000 --> 00:00:48,810 calculate the coefficient of correlation. The 9 00:00:48,810 --> 00:00:51,210 second objective, the meaning of the regression 10 00:00:51,210 --> 00:00:55,590 coefficients beta 0 and beta 1. And the last 11 00:00:55,590 --> 00:00:58,710 objective is how to use regression analysis to 12 00:00:58,710 --> 00:01:03,030 predict the value of dependent variable based on 13 00:01:03,030 --> 00:01:06,010 an independent variable. It looks like that we 14 00:01:06,010 --> 00:01:10,590 have discussed objective number one in chapter 15 00:01:10,590 --> 00:01:16,470 three. So calculation of the correlation 16 00:01:16,470 --> 00:01:20,740 coefficient is done in chapter three, but here 17 00:01:20,740 --> 00:01:26,060 we'll give some details about correlation also. A 18 00:01:26,060 --> 00:01:28,480 scatter plot can be used to show the relationship 19 00:01:28,480 --> 00:01:31,540 between two variables. For example, imagine that 20 00:01:31,540 --> 00:01:35,400 we have a random sample of 10 children. 21 00:01:37,800 --> 00:01:47,940 And we have data on their weights and ages. And we 22 00:01:47,940 --> 00:01:51,640 are interested to examine the relationship between 23 00:01:51,640 --> 00:01:58,400 weights and age. For example, suppose child number 24 00:01:58,400 --> 00:02:06,260 one, his 25 00:02:06,260 --> 00:02:12,060 or her age is two years with weight, for example, 26 00:02:12,200 --> 00:02:12,880 eight kilograms. 27 00:02:17,680 --> 00:02:21,880 His weight or her weight is four years, and his or 28 00:02:21,880 --> 00:02:24,500 her weight is, for example, 15 kilograms, and so 29 00:02:24,500 --> 00:02:29,680 on. And again, we are interested to examine the 30 00:02:29,680 --> 00:02:32,640 relationship between age and weight. Maybe they 31 00:02:32,640 --> 00:02:37,400 exist sometimes. positive relationship between the 32 00:02:37,400 --> 00:02:41,100 two variables that means if one variable increases 33 00:02:41,100 --> 00:02:45,260 the other one also increase if one variable 34 00:02:45,260 --> 00:02:47,980 increases the other will also decrease so they 35 00:02:47,980 --> 00:02:52,980 have the same direction either up or down so we 36 00:02:52,980 --> 00:02:58,140 have to know number one the form of the 37 00:02:58,140 --> 00:03:02,140 relationship this one could be linear here we 38 00:03:02,140 --> 00:03:06,890 focus just on linear relationship between X and Y. 39 00:03:08,050 --> 00:03:13,730 The second, we have to know the direction of the 40 00:03:13,730 --> 00:03:21,270 relationship. This direction might be positive or 41 00:03:21,270 --> 00:03:22,350 negative relationship. 42 00:03:25,150 --> 00:03:27,990 In addition to that, we have to know the strength 43 00:03:27,990 --> 00:03:33,760 of the relationship between the two variables of 44 00:03:33,760 --> 00:03:37,320 interest the strength can be classified into three 45 00:03:37,320 --> 00:03:46,480 categories either strong, moderate or there exists 46 00:03:46,480 --> 00:03:50,580 a weak relationship so it could be positive 47 00:03:50,580 --> 00:03:53,320 -strong, positive-moderate or positive-weak, the 48 00:03:53,320 --> 00:03:58,360 same for negative so by using scatter plot we can 49 00:03:58,360 --> 00:04:02,530 determine the form either linear or non-linear, 50 00:04:02,690 --> 00:04:06,130 but here we are focusing on just linear 51 00:04:06,130 --> 00:04:10,310 relationship. Also, we can determine the direction 52 00:04:10,310 --> 00:04:12,870 of the relationship. We can say there exists 53 00:04:12,870 --> 00:04:15,910 positive or negative based on the scatter plot. 54 00:04:16,710 --> 00:04:19,530 Also, we can know the strength of the 55 00:04:19,530 --> 00:04:23,130 relationship, either strong, moderate or weak. For 56 00:04:23,130 --> 00:04:29,810 example, suppose we have again weights and ages. 57 00:04:30,390 --> 00:04:33,590 And we know that there are two types of variables 58 00:04:33,590 --> 00:04:36,710 in this case. One is called dependent and the 59 00:04:36,710 --> 00:04:41,330 other is independent. So if we, as we explained 60 00:04:41,330 --> 00:04:47,890 before, is the dependent variable and A is 61 00:04:47,890 --> 00:04:48,710 independent variable. 62 00:04:52,690 --> 00:04:57,270 Always dependent 63 00:04:57,270 --> 00:04:57,750 variable 64 00:05:00,400 --> 00:05:05,560 is denoted by Y and always on the vertical axis so 65 00:05:05,560 --> 00:05:11,300 here we have weight and independent variable is 66 00:05:11,300 --> 00:05:17,760 denoted by X and X is in the X axis or horizontal 67 00:05:17,760 --> 00:05:26,300 axis now scatter plot for example here child with 68 00:05:26,300 --> 00:05:30,820 age 2 years his weight is 8 So two years, for 69 00:05:30,820 --> 00:05:36,760 example, this is eight. So this star represents 70 00:05:36,760 --> 00:05:42,320 the first pair of observation, age of two and 71 00:05:42,320 --> 00:05:46,820 weight of eight. The other child, his weight is 72 00:05:46,820 --> 00:05:52,860 four years, and the corresponding weight is 15. 73 00:05:53,700 --> 00:05:58,970 For example, this value is 15. The same for the 74 00:05:58,970 --> 00:06:02,430 other points. Here we can know the direction. 75 00:06:04,910 --> 00:06:10,060 In this case they exist. Positive. Form is linear. 76 00:06:12,100 --> 00:06:16,860 Strong or weak or moderate depends on how these 77 00:06:16,860 --> 00:06:20,260 values are close to the straight line. Closer 78 00:06:20,260 --> 00:06:24,380 means stronger. So if the points are closer to the 79 00:06:24,380 --> 00:06:26,620 straight line, it means there exists stronger 80 00:06:26,620 --> 00:06:30,800 relationship between the two variables. So closer 81 00:06:30,800 --> 00:06:34,480 means stronger, either positive or negative. In 82 00:06:34,480 --> 00:06:37,580 this case, there exists positive. Now for the 83 00:06:37,580 --> 00:06:42,360 negative association or relationship, we have the 84 00:06:42,360 --> 00:06:46,060 other direction, it could be this one. So in this 85 00:06:46,060 --> 00:06:49,460 case there exists linear but negative 86 00:06:49,460 --> 00:06:51,900 relationship, and this negative could be positive 87 00:06:51,900 --> 00:06:56,100 or negative, it depends on the points. So it's 88 00:06:56,100 --> 00:07:02,660 positive relationship. The other direction is 89 00:07:02,660 --> 00:07:06,460 negative. So the points, if the points are closed, 90 00:07:06,820 --> 00:07:10,160 then we can say there exists strong negative 91 00:07:10,160 --> 00:07:14,440 relationship. So by using scatter plot, we can 92 00:07:14,440 --> 00:07:17,280 determine all of these. 93 00:07:20,840 --> 00:07:24,460 and direction and strength now here the two 94 00:07:24,460 --> 00:07:27,060 variables we are talking about are numerical 95 00:07:27,060 --> 00:07:30,480 variables so the two variables here are numerical 96 00:07:30,480 --> 00:07:35,220 variables so we are talking about quantitative 97 00:07:35,220 --> 00:07:39,850 variables but remember in chapter 11 We talked 98 00:07:39,850 --> 00:07:43,150 about the relationship between two qualitative 99 00:07:43,150 --> 00:07:47,450 variables. So we use chi-square test. Here we are 100 00:07:47,450 --> 00:07:49,630 talking about something different. We are talking 101 00:07:49,630 --> 00:07:52,890 about numerical variables. So we can use scatter 102 00:07:52,890 --> 00:07:58,510 plot, number one. Next correlation analysis is 103 00:07:58,510 --> 00:08:02,090 used to measure the strength of the association 104 00:08:02,090 --> 00:08:05,190 between two variables. And here again, we are just 105 00:08:05,190 --> 00:08:09,560 talking about linear relationship. So this chapter 106 00:08:09,560 --> 00:08:13,340 just covers the linear relationship between the 107 00:08:13,340 --> 00:08:17,040 two variables. Because sometimes there exists non 108 00:08:17,040 --> 00:08:23,180 -linear relationship between the two variables. So 109 00:08:23,180 --> 00:08:26,120 correlation is only concerned with the strength of 110 00:08:26,120 --> 00:08:30,500 the relationship. No causal effect is implied with 111 00:08:30,500 --> 00:08:35,220 correlation. We just say that X affects Y, or X 112 00:08:35,220 --> 00:08:39,580 explains the variation in Y. Scatter plots were 113 00:08:39,580 --> 00:08:43,720 first presented in Chapter 2, and we skipped, if 114 00:08:43,720 --> 00:08:48,480 you remember, Chapter 2. And it's easy to make 115 00:08:48,480 --> 00:08:52,620 scatter plots for Y versus X. In Chapter 3, we 116 00:08:52,620 --> 00:08:56,440 talked about correlation, so correlation was first 117 00:08:56,440 --> 00:09:00,060 presented in Chapter 3. But here I will give just 118 00:09:00,060 --> 00:09:07,240 a review for computation about correlation 119 00:09:07,240 --> 00:09:11,460 coefficient or coefficient of correlation. First, 120 00:09:12,800 --> 00:09:15,680 coefficient of correlation measures the relative 121 00:09:15,680 --> 00:09:19,920 strength of the linear relationship between two 122 00:09:19,920 --> 00:09:23,740 numerical variables. So here, we are talking about 123 00:09:23,740 --> 00:09:28,080 numerical variables. Sample correlation 124 00:09:28,080 --> 00:09:31,500 coefficient is given by this equation. which is 125 00:09:31,500 --> 00:09:36,180 sum of the product of xi minus x bar, yi minus y 126 00:09:36,180 --> 00:09:41,100 bar, divided by n minus 1 times standard deviation 127 00:09:41,100 --> 00:09:44,960 of x times standard deviation of y. We know that x 128 00:09:44,960 --> 00:09:47,240 bar and y bar are the means of x and y 129 00:09:47,240 --> 00:09:51,360 respectively. And Sx, Sy are the standard 130 00:09:51,360 --> 00:09:55,540 deviations of x and y values. And we know this 131 00:09:55,540 --> 00:09:58,460 equation before. But there is another equation 132 00:09:58,460 --> 00:10:05,330 that one can be used For computation, which is 133 00:10:05,330 --> 00:10:09,290 called shortcut formula, which is just sum of xy 134 00:10:09,290 --> 00:10:15,310 minus n times x bar y bar divided by square root 135 00:10:15,310 --> 00:10:18,690 of this quantity. And we know this equation from 136 00:10:18,690 --> 00:10:23,650 chapter three. Now again, x bar and y bar are the 137 00:10:23,650 --> 00:10:30,060 means. Now the question is, Do outliers affect the 138 00:10:30,060 --> 00:10:36,440 correlation? For sure, yes. Because this formula 139 00:10:36,440 --> 00:10:39,940 actually based on the means and the standard 140 00:10:39,940 --> 00:10:44,300 deviations, and these two measures are affected by 141 00:10:44,300 --> 00:10:47,880 outliers. So since R is a function of these two 142 00:10:47,880 --> 00:10:51,340 statistics, the means and standard deviations, 143 00:10:51,940 --> 00:10:54,280 then outliers will affect the value of the 144 00:10:54,280 --> 00:10:55,940 correlation coefficient. 145 00:10:57,890 --> 00:11:01,170 Some features about the coefficient of 146 00:11:01,170 --> 00:11:09,570 correlation. Here rho is the population 147 00:11:09,570 --> 00:11:13,210 coefficient of correlation, and R is the sample 148 00:11:13,210 --> 00:11:17,730 coefficient of correlation. Either rho or R have 149 00:11:17,730 --> 00:11:21,390 the following features. Number one, unity free. It 150 00:11:21,390 --> 00:11:24,890 means R has no units. For example, here we are 151 00:11:24,890 --> 00:11:28,820 talking about whales. And weight in kilograms, 152 00:11:29,300 --> 00:11:33,700 ages in years. And for example, suppose the 153 00:11:33,700 --> 00:11:37,080 correlation between these two variables is 0.8. 154 00:11:38,620 --> 00:11:41,760 It's unity free, so it's just 0.8. So there is no 155 00:11:41,760 --> 00:11:45,640 unit. You cannot say 0.8 kilogram per year or 156 00:11:45,640 --> 00:11:51,040 whatever it is. So just 0.8. So the first feature 157 00:11:51,040 --> 00:11:53,360 of the correlation coefficient is unity-free. 158 00:11:54,180 --> 00:11:56,340 Number two ranges between negative one and plus 159 00:11:56,340 --> 00:12:00,380 one. So R is always, or rho, is always between 160 00:12:00,380 --> 00:12:04,560 minus one and plus one. So minus one smaller than 161 00:12:04,560 --> 00:12:07,340 or equal to R smaller than or equal to plus one. 162 00:12:07,420 --> 00:12:11,420 So R is always in this range. So R cannot be 163 00:12:11,420 --> 00:12:15,260 smaller than negative one or greater than plus 164 00:12:15,260 --> 00:12:20,310 one. The closer to minus one or negative one, the 165 00:12:20,310 --> 00:12:23,130 stronger negative relationship between or linear 166 00:12:23,130 --> 00:12:26,770 relationship between x and y. So, for example, if 167 00:12:26,770 --> 00:12:33,370 R is negative 0.85 or R is negative 0.8. Now, this 168 00:12:33,370 --> 00:12:39,690 value is closer to minus one than negative 0.8. So 169 00:12:39,690 --> 00:12:43,230 negative 0.85 is stronger than negative 0.8. 170 00:12:44,590 --> 00:12:48,470 Because we are looking for closer to minus 1. 171 00:12:49,570 --> 00:12:55,310 Minus 0.8, the value itself is greater than minus 172 00:12:55,310 --> 00:12:59,610 0.85. But this value is closer to minus 1 than 173 00:12:59,610 --> 00:13:03,790 minus 0.8. So we can say that this relationship is 174 00:13:03,790 --> 00:13:05,070 stronger than the other one. 175 00:13:07,870 --> 00:13:11,730 Also, the closer to plus 1, the stronger the 176 00:13:11,730 --> 00:13:16,040 positive linear relationship. Here, suppose R is 0 177 00:13:16,040 --> 00:13:22,740 .7 and another R is 0.8. 0.8 is closer to plus one 178 00:13:22,740 --> 00:13:26,740 than 0.7, so 0.8 is stronger. This one makes 179 00:13:26,740 --> 00:13:31,800 sense. The closer to zero, the weaker relationship 180 00:13:31,800 --> 00:13:35,420 between the two variables. For example, suppose R 181 00:13:35,420 --> 00:13:40,720 is plus or minus 0.05. This value is very close to 182 00:13:40,720 --> 00:13:44,420 zero. It means there exists weak. relationship. 183 00:13:44,980 --> 00:13:47,960 Sometimes we can say that there exists moderate 184 00:13:47,960 --> 00:13:57,080 relationship if R is close to 0.5. So it could be 185 00:13:57,080 --> 00:14:01,360 classified into these groups closer to minus 1, 186 00:14:01,500 --> 00:14:06,220 closer to 1, 0.5 or 0. So we can know the 187 00:14:06,220 --> 00:14:11,680 direction by the sign of R negative it means 188 00:14:11,680 --> 00:14:14,320 because here our ranges as we mentioned between 189 00:14:14,320 --> 00:14:19,520 minus one and plus one here zero so this these 190 00:14:19,520 --> 00:14:24,560 values it means there exists negative above zero 191 00:14:24,560 --> 00:14:26,760 all the way up to one it means there exists 192 00:14:26,760 --> 00:14:31,020 positive relationship between the two variables so 193 00:14:31,020 --> 00:14:35,520 the sign gives the direction of the relationship 194 00:14:36,720 --> 00:14:40,840 The absolute value gives the strength of the 195 00:14:40,840 --> 00:14:43,500 relationship between the two variables. So the 196 00:14:43,500 --> 00:14:49,260 same as we had discussed before. Now, some types 197 00:14:49,260 --> 00:14:51,880 of scatter plots for different types of 198 00:14:51,880 --> 00:14:54,740 relationship between the two variables is 199 00:14:54,740 --> 00:14:59,100 presented in this 223 00:16:53,260 --> 00:16:58,860 individual and also weights are increased by three 224 00:16:58,860 --> 00:17:03,080 units for each person. In this case there exists 225 00:17:03,080 --> 00:17:06,820 a perfect relationship but that never happened in 226 00:17:06,820 --> 00:17:13,300 real life. So perfect means all points are lie on 227 00:17:13,300 --> 00:17:16,260 the straight line otherwise if the points are 228 00:17:16,260 --> 00:17:21,230 close Then we can say there exists strong. Here if 229 00:17:21,230 --> 00:17:24,750 you look carefully at these points corresponding 230 00:17:24,750 --> 00:17:30,150 to this regression line, it looks like not strong 231 00:17:30,150 --> 00:17:32,630 because some of the points are not close, so you 232 00:17:32,630 --> 00:17:35,450 can say there exists maybe moderate negative 233 00:17:35,450 --> 00:17:39,530 relationship. This one, most of the points are 234 00:17:39,530 --> 00:17:42,390 scattered away from the straight line, so there 235 00:17:42,390 --> 00:17:46,930 exists weak relationship. So by just looking at 236 00:17:46,930 --> 00:17:50,290 the scatter path, sometimes you can, sometimes 237 00:17:50,290 --> 00:17:53,290 it's hard to tell, but most of the time you can 238 00:17:53,290 --> 00:17:58,250 tell at least the direction, positive or negative, 239 00:17:59,410 --> 00:18:04,150 the form, linear or non-linear, or the strength of 240 00:18:04,150 --> 00:18:09,100 the relationship. The last one here, now x 241 00:18:09,100 --> 00:18:13,800 increases, y remains the same. For example, 242 00:18:13,880 --> 00:18:18,580 suppose x is 1, y is 10. x increases to 2, y still 243 00:18:18,580 --> 00:18:22,220 is 10. So as x increases, y stays the same 244 00:18:22,220 --> 00:18:26,140 position, it means there is no linear relationship 245 00:18:26,140 --> 00:18:28,900 between the two variables. So based on the scatter 246 00:18:28,900 --> 00:18:33,240 plot you can have an idea about the relationship 247 00:18:33,240 --> 00:18:37,800 between the two variables. Here I will give a 248 00:18:37,800 --> 00:18:41,120 simple example in order to determine the 249 00:18:41,120 --> 00:18:45,160 correlation coefficient. A real estate agent 250 00:18:45,160 --> 00:18:50,380 wishes to examine the relationship between selling 251 00:18:50,380 --> 00:18:54,580 the price of a home and its size measured in 252 00:18:54,580 --> 00:18:57,140 square feet. So in this case, there are two 253 00:18:57,140 --> 00:19:02,400 variables of interest. One is called selling price 254 00:19:02,400 --> 00:19:13,720 of a home. So here, selling price of a home and 255 00:19:13,720 --> 00:19:18,020 its size. Now, selling price in $1,000. 256 00:19:25,360 --> 00:19:29,380 And size in feet squared. Here we have to 257 00:19:29,380 --> 00:19:35,640 distinguish between dependent and independent. So 258 00:19:35,640 --> 00:19:39,740 your dependent variable is house price, sometimes 259 00:19:39,740 --> 00:19:41,620 called response variable. 260 00:19:45,750 --> 00:19:49,490 The independent variable is the size, which is in 261 00:19:49,490 --> 00:19:54,570 square feet, sometimes called sub-planetary 262 00:19:54,570 --> 00:19:54,850 variable. 263 00:19:59,570 --> 00:20:06,370 So my Y is ceiling rise, and size is square feet, 264 00:20:07,530 --> 00:20:12,910 or size of the house. In this case, there are 10. 265 00:20:14,290 --> 00:20:17,890 It's sample size is 10. So the first house with 266 00:20:17,890 --> 00:20:26,850 size 1,400 square feet, it's selling price is 245 267 00:20:26,850 --> 00:20:31,670 multiplied by 1,000. Because these values are in 268 00:20:31,670 --> 00:20:37,950 $1,000. Now based on this data, you can first plot 269 00:20:37,950 --> 00:20:46,590 the scatterplot of house price In Y direction, the 270 00:20:46,590 --> 00:20:51,870 vertical direction. So here is house. And rise. 271 00:20:54,230 --> 00:21:01,470 And size in the X axis. You will get this scatter 272 00:21:01,470 --> 00:21:07,370 plot. Now, the data here is just 10 points, so 273 00:21:07,370 --> 00:21:12,590 sometimes it's hard to tell. the relationship 274 00:21:12,590 --> 00:21:15,510 between the two variables if your data is small. 275 00:21:16,510 --> 00:21:21,170 But just this example for illustration. But at 276 00:21:21,170 --> 00:21:25,370 least you can determine that there exists linear 277 00:21:25,370 --> 00:21:28,810 relationship between the two variables. It is 278 00:21:28,810 --> 00:21:35,490 positive. So the form is linear. Direction is 279 00:21:35,490 --> 00:21:41,880 positive. Weak or strong or moderate. Sometimes 280 00:21:41,880 --> 00:21:45,620 it's not easy to tell if it is strong or moderate. 281 00:21:47,720 --> 00:21:50,120 Now if you look at these points, some of them are 282 00:21:50,120 --> 00:21:53,700 close to the straight line and others are away 283 00:21:53,700 --> 00:21:56,700 from the straight line. So maybe there exists 284 00:21:56,700 --> 00:22:02,720 moderate for example, but you cannot say strong. 285 00:22:03,930 --> 00:22:08,210 Here, strong it means the points are close to the 286 00:22:08,210 --> 00:22:11,890 straight line. Sometimes it's hard to tell the 287 00:22:11,890 --> 00:22:15,230 strength of the relationship, but you can know the 288 00:22:15,230 --> 00:22:20,990 form or the direction. But to measure the exact 289 00:22:20,990 --> 00:22:24,130 strength, you have to measure the correlation 290 00:22:24,130 --> 00:22:29,810 coefficient, R. Now, by looking at the data, you 291 00:22:29,810 --> 00:22:31,430 can compute 292 00:22:33,850 --> 00:22:42,470 The sum of x values, y values, sum of x squared, 293 00:22:43,290 --> 00:22:48,170 sum of y squared, also sum of xy. Now plug these 294 00:22:48,170 --> 00:22:50,610 values into the formula we have for the shortcut 295 00:22:50,610 --> 00:22:58,210 formula. You will get R to be 0.76 around 76. 296 00:23:04,050 --> 00:23:10,170 So there exists positive, moderate relationship 297 00:23:10,170 --> 00:23:13,770 between selling 298 00:23:13,770 --> 00:23:19,850 price of a home and its size. So that means if the 299 00:23:19,850 --> 00:23:24,670 size increases, the selling price also increases. 300 00:23:25,310 --> 00:23:29,550 So there exists positive relationship between the 301 00:23:29,550 --> 00:23:30,310 two variables. 302 00:23:35,800 --> 00:23:40,300 Strong it means close to 1, 0.8, 0.85, 0.9, you 303 00:23:40,300 --> 00:23:44,400 can say there exists strong. But fields is not 304 00:23:44,400 --> 00:23:47,960 strong relationship, you can say it's moderate 305 00:23:47,960 --> 00:23:53,440 relationship. Because it's close if now if you 306 00:23:53,440 --> 00:23:57,080 just compare this value and other data gives 9%. 307 00:23:58,830 --> 00:24:03,790 Other one gives 85%. So these values are much 308 00:24:03,790 --> 00:24:08,550 closer to 1 than 0.7, but still this value is 309 00:24:08,550 --> 00:24:09,570 considered to be high. 310 00:24:15,710 --> 00:24:16,810 Any question? 311 00:24:19,850 --> 00:24:22,810 Next, I will give some introduction to regression 312 00:24:22,810 --> 00:24:23,390 analysis. 313 00:24:26,970 --> 00:24:32,210 Regression analysis used to number one, predict 314 00:24:32,210 --> 00:24:35,050 the value of a dependent variable based on the 315 00:24:35,050 --> 00:24:39,250 value of at least one independent variable. So by 316 00:24:39,250 --> 00:24:42,490 using the data we have for selling price of a home 317 00:24:42,490 --> 00:24:48,370 and size, you can predict the selling price by 318 00:24:48,370 --> 00:24:51,510 knowing the value of its size. So suppose for 319 00:24:51,510 --> 00:24:54,870 example, You know that the size of a house is 320 00:24:54,870 --> 00:25:03,510 1450, 1450 square feet. What do you predict its 321 00:25:03,510 --> 00:25:10,190 size, its sale or price? So by using this value, 322 00:25:10,310 --> 00:25:16,510 we can predict the selling price. Next, explain 323 00:25:16,510 --> 00:25:19,890 the impact of changes in independent variable on 324 00:25:19,890 --> 00:25:23,270 the dependent variable. You can say, for example, 325 00:25:23,510 --> 00:25:30,650 90% of the variability in the dependent variable 326 00:25:30,650 --> 00:25:36,790 in selling price is explained by its size. So we 327 00:25:36,790 --> 00:25:39,410 can predict the value of dependent variable based 328 00:25:39,410 --> 00:25:42,890 on a value of one independent variable at least. 329 00:25:43,870 --> 00:25:47,090 Or also explain the impact of changes in 330 00:25:47,090 --> 00:25:49,550 independent variable on the dependent variable. 331 00:25:51,420 --> 00:25:53,920 Sometimes there exists more than one independent 332 00:25:53,920 --> 00:25:59,680 variable. For example, maybe there are more than 333 00:25:59,680 --> 00:26:04,500 one variable that affects a price, a selling 334 00:26:04,500 --> 00:26:10,300 price. For example, beside selling 335 00:26:10,300 --> 00:26:16,280 price, beside size, maybe location. 336 00:26:19,480 --> 00:26:23,580 Maybe location is also another factor that affects 337 00:26:23,580 --> 00:26:27,360 the selling price. So in this case there are two 338 00:26:27,360 --> 00:26:32,240 variables. If there exists more than one variable, 339 00:26:32,640 --> 00:26:36,080 in this case we have something called multiple 340 00:26:36,080 --> 00:26:38,680 linear regression. 341 00:26:42,030 --> 00:26:46,710 Here, we just talk about one independent variable. 342 00:26:47,030 --> 00:26:51,610 There is only, in this chapter, there is only one 343 00:26:51,610 --> 00:26:58,330 x. So it's called simple linear 344 00:26:58,330 --> 00:26:59,330 regression. 345 00:27:02,190 --> 00:27:07,930 The calculations for multiple takes time. So we 346 00:27:07,930 --> 00:27:11,430 are going just to cover one independent variable. 347 00:27:11,930 --> 00:27:14,290 But if there exists more than one, in this case 348 00:27:14,290 --> 00:27:18,250 you have to use some statistical software as SPSS. 349 00:27:18,470 --> 00:27:23,390 Because in that case you can just select a 350 00:27:23,390 --> 00:27:25,970 regression analysis from SPSS, then you can run 351 00:27:25,970 --> 00:27:28,590 the multiple regression without doing any 352 00:27:28,590 --> 00:27:34,190 computations. But here we just covered one 353 00:27:34,190 --> 00:27:36,820 independent variable. In this case, it's called 354 00:27:36,820 --> 00:27:41,980 simple linear regression. Again, the dependent 355 00:27:41,980 --> 00:27:44,600 variable is the variable we wish to predict or 356 00:27:44,600 --> 00:27:50,020 explain, the same as weight. Independent variable, 357 00:27:50,180 --> 00:27:52,440 the variable used to predict or explain the 358 00:27:52,440 --> 00:27:54,000 dependent variable. 359 00:27:57,400 --> 00:28:00,540 For simple linear regression model, there is only 360 00:28:00,540 --> 00:28:01,800 one independent variable. 361 00:28:04,830 --> 00:28:08,450 Another example for simple linear regression. 362 00:28:08,770 --> 00:28:11,590 Suppose we are talking about your scores. 363 00:28:14,210 --> 00:28:17,770 Scores is the dependent variable can be affected 364 00:28:17,770 --> 00:28:21,050 by number of hours. 365 00:28:25,130 --> 00:28:31,030 Hour of study. Number of studying hours. 366 00:28:36,910 --> 00:28:39,810 Maybe as number of studying hour increases, your 367 00:28:39,810 --> 00:28:43,390 scores also increase. In this case, if there is 368 00:28:43,390 --> 00:28:46,330 only one X, one independent variable, it's called 369 00:28:46,330 --> 00:28:51,110 simple linear regression. Maybe another variable, 370 00:28:52,270 --> 00:28:59,730 number of missing classes or 371 00:28:59,730 --> 00:29:03,160 attendance. As number of missing classes 372 00:29:03,160 --> 00:29:06,380 increases, your score goes down. That means there 373 00:29:06,380 --> 00:29:09,400 exists negative relationship between missing 374 00:29:09,400 --> 00:29:13,540 classes and your score. So sometimes, maybe there 375 00:29:13,540 --> 00:29:16,580 exists positive or negative. It depends on the 376 00:29:16,580 --> 00:29:20,040 variable itself. In this case, if there are more 377 00:29:20,040 --> 00:29:23,180 than one variable, then we are talking about 378 00:29:23,180 --> 00:29:28,300 multiple linear regression model. But here, we 379 00:29:28,300 --> 00:29:33,630 have only one independent variable. In addition to 380 00:29:33,630 --> 00:29:37,230 that, a relationship between x and y is described 381 00:29:37,230 --> 00:29:40,850 by a linear function. So there exists a straight 382 00:29:40,850 --> 00:29:46,270 line between the two variables. The changes in y 383 00:29:46,270 --> 00:29:50,210 are assumed to be related to changes in x only. So 384 00:29:50,210 --> 00:29:54,270 any change in y is related only to changes in x. 385 00:29:54,730 --> 00:29:57,810 So that's the simple case we have for regression, 386 00:29:58,890 --> 00:30:01,170 that we have only one independent 387 00:30:03,890 --> 00:30:07,070 Variable. Types of relationships, as we mentioned, 388 00:30:07,210 --> 00:30:12,190 maybe there exist linear, it means there exist 389 00:30:12,190 --> 00:30:16,490 straight line between X and Y, either linear 390 00:30:16,490 --> 00:30:22,050 positive or negative, or sometimes there exist non 391 00:30:22,050 --> 00:30:25,830 -linear relationship, it's called curved linear 392 00:30:25,830 --> 00:30:29,290 relationship. The same as this one, it's parabola. 393 00:30:32,570 --> 00:30:35,150 Now in this case there is no linear relationship 394 00:30:35,150 --> 00:30:39,690 but there exists curved linear or something like 395 00:30:39,690 --> 00:30:45,910 this one. So these types of non-linear 396 00:30:45,910 --> 00:30:49,530 relationship between the two variables. Here we 397 00:30:49,530 --> 00:30:54,070 are covering just the linear relationship between 398 00:30:54,070 --> 00:30:56,570 the two variables. So based on the scatter plot 399 00:30:56,570 --> 00:31:00,620 you can determine the direction. The form, the 400 00:31:00,620 --> 00:31:03,860 strength. Here, the form we are talking about is 401 00:31:03,860 --> 00:31:04,720 just linear. 402 00:31:08,700 --> 00:31:13,260 Now, another type of relationship, the strength of 403 00:31:13,260 --> 00:31:16,940 the relationship. Here, the points, either for 404 00:31:16,940 --> 00:31:20,570 this graph or the other one, These points are 405 00:31:20,570 --> 00:31:24,570 close to the straight line, it means there exists 406 00:31:24,570 --> 00:31:28,210 strong positive relationship or strong negative 407 00:31:28,210 --> 00:31:31,230 relationship. So it depends on the direction. So 408 00:31:31,230 --> 00:31:35,710 strong either positive or strong negative. Here 409 00:31:35,710 --> 00:31:38,850 the points are scattered away from the regression 410 00:31:38,850 --> 00:31:41,790 line, so you can say there exists weak 411 00:31:41,790 --> 00:31:45,090 relationship, either weak positive or weak 412 00:31:45,090 --> 00:31:49,650 negative. It depends on the direction of the 413 00:31:49,650 --> 00:31:54,270 relationship between the two variables. Sometimes 414 00:31:54,270 --> 00:31:59,680 there is no relationship or actually there is no 415 00:31:59,680 --> 00:32:02,340 linear relationship between the two variables. If 416 00:32:02,340 --> 00:32:05,660 the points are scattered away from the regression 417 00:32:05,660 --> 00:32:09,800 line, I mean you cannot determine if it is 418 00:32:09,800 --> 00:32:13,160 positive or negative, then there is no 419 00:32:13,160 --> 00:32:16,220 relationship between the two variables, the same 420 00 445 00:34:40,270 --> 00:34:43,850 could be negative or 446 00:34:46,490 --> 00:34:49,350 Maybe the straight line passes through the origin 447 00:34:49,350 --> 00:34:56,990 point. So in this case, beta zero equals zero. So 448 00:34:56,990 --> 00:34:59,890 it could be positive and negative or equal zero, 449 00:35:00,430 --> 00:35:05,510 but still we have positive relationship. That 450 00:35:05,510 --> 00:35:09,970 means the value of beta zero, the sign of beta 451 00:35:09,970 --> 00:35:13,310 zero does not affect the relationship between Y 452 00:35:13,310 --> 00:35:17,850 and X. Because here in the three cases, there 453 00:35:17,850 --> 00:35:22,390 exists positive relationship, but beta zero could 454 00:35:22,390 --> 00:35:25,370 be positive or negative or equal zero, but still 455 00:35:25,370 --> 00:35:31,720 we have positive relationship. I mean, you cannot 456 00:35:31,720 --> 00:35:35,060 determine by looking at beta 0, you cannot 457 00:35:35,060 --> 00:35:37,940 determine if there is a positive or negative 458 00:35:37,940 --> 00:35:41,720 relationship. The other term is beta 1. Beta 1 is 459 00:35:41,720 --> 00:35:46,900 the population slope coefficient. Now, the sign of 460 00:35:46,900 --> 00:35:50,010 the slope determines the direction of the 461 00:35:50,010 --> 00:35:54,090 relationship. That means if the slope has positive 462 00:35:54,090 --> 00:35:56,570 sign, it means there exists positive relationship. 463 00:35:57,330 --> 00:35:59,370 Otherwise if it is negative, then there is 464 00:35:59,370 --> 00:36:01,390 negative relationship between the two variables. 465 00:36:02,130 --> 00:36:05,310 So the sign of the slope determines the direction. 466 00:36:06,090 --> 00:36:11,290 But the sign of beta zero has no meaning about the 467 00:36:11,290 --> 00:36:15,470 relationship between Y and X. X is your 468 00:36:15,470 --> 00:36:19,630 independent variable, Y is your independent 469 00:36:19,630 --> 00:36:19,650 your independent variable, Y is your independent 470 00:36:19,650 --> 00:36:21,250 variable, Y is your independent variable, Y is 471 00:36:21,250 --> 00:36:24,370 variable, Y is your independent variable, Y is 472 00:36:24,370 --> 00:36:24,430 variable, Y is your independent variable, Y is 473 00:36:24,430 --> 00:36:24,770 your independent variable, Y is your independent 474 00:36:24,770 --> 00:36:27,490 variable, Y is your independent variable, Y is 475 00:36:27,490 --> 00:36:30,110 your independent variable, Y is your It means 476 00:36:30,110 --> 00:36:32,450 there are some errors you don't know about it 477 00:36:32,450 --> 00:36:36,130 because you ignore some other variables that may 478 00:36:36,130 --> 00:36:39,410 affect the selling price. Maybe you select a 479 00:36:39,410 --> 00:36:42,490 random sample, that sample is small. Maybe there 480 00:36:42,490 --> 00:36:46,270 is a random, I'm sorry, there is sampling error. 481 00:36:47,070 --> 00:36:52,980 So all of these are called random error term. So 482 00:36:52,980 --> 00:36:57,420 all of them are in this term. So epsilon I means 483 00:36:57,420 --> 00:37:00,340 something you don't include in your regression 484 00:37:00,340 --> 00:37:03,280 modeling. For example, you don't include all the 485 00:37:03,280 --> 00:37:06,180 independent variables that affect Y, or your 486 00:37:06,180 --> 00:37:09,700 sample size is not large enough. So all of these 487 00:37:09,700 --> 00:37:14,260 measured in random error term. So epsilon I is 488 00:37:14,260 --> 00:37:18,840 random error component, beta 0 plus beta 1X is 489 00:37:18,840 --> 00:37:25,070 called linear component. So that's the simple 490 00:37:25,070 --> 00:37:31,430 linear regression model. Now, the data you have, 491 00:37:32,850 --> 00:37:38,210 the blue circles represent the observed value. So 492 00:37:38,210 --> 00:37:47,410 these blue circles are the observed values. So we 493 00:37:47,410 --> 00:37:49,370 have observed. 494 00:37:52,980 --> 00:37:57,940 Y observed value of Y for each value X. The 495 00:37:57,940 --> 00:38:03,360 regression line is the blue, the red one. It's 496 00:38:03,360 --> 00:38:07,560 called the predicted values. Predicted Y. 497 00:38:08,180 --> 00:38:14,760 Predicted Y is denoted always by Y hat. Now the 498 00:38:14,760 --> 00:38:19,740 difference between Y and Y hat. It's called the 499 00:38:19,740 --> 00:38:20,200 error term. 500 00:38:24,680 --> 00:38:28,000 It's actually the difference between the observed 501 00:38:28,000 --> 00:38:31,600 value and its predicted value. Now, the predicted 502 00:38:31,600 --> 00:38:34,720 value can be determined by using the regression 503 00:38:34,720 --> 00:38:39,180 line. So this line is the predicted value of Y for 504 00:38:39,180 --> 00:38:44,480 XR. Again, beta zero is the intercept. As we 505 00:38:44,480 --> 00:38:46,260 mentioned before, it could be positive or negative 506 00:38:46,260 --> 00:38:52,600 or even equal zero. The slope is changing Y. 507 00:38:55,140 --> 00:38:57,580 Divide by change of x. 508 00:39:01,840 --> 00:39:07,140 So these are the components for the simple linear 509 00:39:07,140 --> 00:39:10,840 regression model. Y again represents the 510 00:39:10,840 --> 00:39:14,960 independent variable. Beta 0 y intercept. Beta 1 511 00:39:14,960 --> 00:39:17,960 is your slope. And the slope determines the 512 00:39:17,960 --> 00:39:20,900 direction of the relationship. X independent 513 00:39:20,900 --> 00:39:25,270 variable epsilon i is the random error term. Any 514 00:39:25,270 --> 00:39:25,650 question? 515 00:39:31,750 --> 00:39:36,610 The relationship may be positive or negative. It 516 00:39:36,610 --> 00:39:37,190 could be negative. 517 00:39:40,950 --> 00:39:42,710 Now, for negative relationship, 518 00:39:57,000 --> 00:40:04,460 Or negative, where beta zero is negative. 519 00:40:04,520 --> 00:40:08,700 Or beta 520 00:40:08,700 --> 00:40:09,740 zero equals zero. 521 00:40:16,680 --> 00:40:20,620 So here there exists negative relationship, but 522 00:40:20,620 --> 00:40:22,060 beta zero may be positive. 523 00:40:25,870 --> 00:40:30,210 So again, the sign of beta 0 also does not affect 524 00:40:30,210 --> 00:40:31,990 the relationship between the two variables. 525 00:40:36,230 --> 00:40:40,590 Now, we don't actually know the values of beta 0 526 00:40:40,590 --> 00:40:44,510 and beta 1. We are going to estimate these values 527 00:40:44,510 --> 00:40:48,110 from the sample we have. So the simple linear 528 00:40:48,110 --> 00:40:50,970 regression equation provides an estimate of the 529 00:40:50,970 --> 00:40:55,270 population regression line. So here we have Yi hat 530 00:40:55,270 --> 00:41:00,010 is the estimated or predicted Y value for 531 00:41:00,010 --> 00:41:00,850 observation I. 532 00:41:03,530 --> 00:41:08,220 The estimate of the regression intercept P0. The 533 00:41:08,220 --> 00:41:11,360 estimate of the regression slope is b1, and this 534 00:41:11,360 --> 00:41:16,680 is your x, all independent variable. So here is 535 00:41:16,680 --> 00:41:20,340 the regression equation. Simple linear regression 536 00:41:20,340 --> 00:41:24,400 equation is given by y hat, the predicted value of 537 00:41:24,400 --> 00:41:29,380 y equals b0 plus b1 times x1. 538 00:41:31,240 --> 00:41:35,960 Now these coefficients, b0 and b1 can be computed 539 00:41:37,900 --> 00:41:43,040 by the following equations. So the regression 540 00:41:43,040 --> 00:41:52,920 equation is 541 00:41:52,920 --> 00:41:57,260 given by y hat equals b0 plus b1x. 542 00:41:59,940 --> 00:42:06,140 Now the slope, b1, is r times standard deviation 543 00:42:06,140 --> 00:42:10,540 of y Times standard deviation of x. This is the 544 00:42:10,540 --> 00:42:13,820 simplest equation to determine the value of the 545 00:42:13,820 --> 00:42:18,980 star. B1r, r is the correlation coefficient. Sy is 546 00:42:18,980 --> 00:42:25,080 xr, the standard deviations of y and x. Where b0, 547 00:42:25,520 --> 00:42:30,880 which is y intercept, is y bar minus b x bar, or 548 00:42:30,880 --> 00:42:38,100 b1 x bar. Sx, as we know, is the sum of x minus y 549 00:42:38,100 --> 00:42:40,460 squared divided by n minus 1 under square root, 550 00:42:40,900 --> 00:42:47,060 similarly for y values. So this, how can we, these 551 00:42:47,060 --> 00:42:52,380 formulas compute the values of b0 and b1. So we 552 00:42:52,380 --> 00:42:54,600 are going to use these equations in order to 553 00:42:54,600 --> 00:42:58,960 determine the values of b0 and b1. 554 00:43:04,670 --> 00:43:07,710 Now, what's your interpretation about the slope 555 00:43:07,710 --> 00:43:13,130 and the intercept? For example, suppose we are 556 00:43:13,130 --> 00:43:18,610 talking about your score Y and 557 00:43:18,610 --> 00:43:22,110 X number of missing classes. 558 00:43:29,210 --> 00:43:35,460 And suppose, for example, Y hat Equal 95 minus 5x. 559 00:43:37,780 --> 00:43:41,420 Now let's see what's the interpretation of B0. 560 00:43:42,300 --> 00:43:45,060 This is B0. So B0 is 95. 561 00:43:47,660 --> 00:43:51,960 And B1 is 5. Now what's your interpretation about 562 00:43:51,960 --> 00:43:57,740 B0 and B1? B0 is the estimated mean value of Y 563 00:43:57,740 --> 00:44:02,560 when the value of X is 0. that means if the 564 00:44:02,560 --> 00:44:08,500 student does not miss any class that means x 565 00:44:08,500 --> 00:44:13,260 equals zero in this case we predict or we estimate 566 00:44:13,260 --> 00:44:19,880 the mean value of his score or her score is 95 so 567 00:44:19,880 --> 00:44:27,500 95 it means when x is zero if x is zero then we 568 00:44:27,500 --> 00:44:35,350 expect his or Here, the score is 95. So that means 569 00:44:35,350 --> 00:44:39,830 B0 is the estimated mean value of Y when the value 570 00:44:39,830 --> 00:44:40,630 of X is 0. 571 00:44:43,370 --> 00:44:46,590 Now, what's the meaning of the slope? The slope in 572 00:44:46,590 --> 00:44:51,290 this case is negative Y. B1, which is the slope, 573 00:44:51,590 --> 00:44:57,610 is the estimated change in the mean of Y. as a 574 00:44:57,610 --> 00:45:03,050 result of a one unit change in x for example let's 575 00:45:03,050 --> 00:45:07,070 compute y for different values of x suppose x is 576 00:45:07,070 --> 00:45:15,510 one now we predict his score to be 95 minus 5 577 00:45:15,510 --> 00:45:25,470 times 1 which is 90 when x is 2 for example Y hat 578 00:45:25,470 --> 00:45:28,570 is 95 minus 5 times 2, so that's 85. 579 00:45:31,950 --> 00:45:39,970 So for each one unit, there is a drop by five 580 00:45:39,970 --> 00:45:43,750 units in his score. That means if number of 581 00:45:43,750 --> 00:45:47,550 missing classes increases by one unit, then his or 582 00:45:47,550 --> 00:45:51,790 her weight is expected to be reduced by five units 583 00:45:51,790 --> 00:45:56,150 because the sign is negative. another example 584 00:45:56,150 --> 00:46:05,910 suppose again we are interested in whales and 585 00:46:05,910 --> 00:46:16,170 angels and imagine that just 586 00:46:16,170 --> 00:46:21,670 for example y equal y hat equals three plus four x 587 00:46:21,670 --> 00:46:29,830 now y hat equals 3 if x equals zero. That has no 588 00:46:29,830 --> 00:46:34,510 meaning because you cannot say age of zero. So 589 00:46:34,510 --> 00:46:40,450 sometimes the meaning of y intercept does not make 590 00:46:40,450 --> 00:46:46,150 sense because you cannot say x equals zero. Now 591 00:46:46,150 --> 00:46:50,690 for the stock of four, that means as his or her 592 00:46:50,690 --> 00:46:55,550 weight increases by one year, Then we expect his 593 00:46:55,550 --> 00:47:00,470 weight to increase by four kilograms. So as one 594 00:47:00,470 --> 00:47:05,130 unit increase in x, y is our, his weight is 595 00:47:05,130 --> 00:47:10,150 expected to increase by four units. So again, 596 00:47:10,370 --> 00:47:16,950 sometimes we can interpret the y intercept, but in 597 00:47:16,950 --> 00:47:18,670 some cases it has no meaning. 598 00:47:24,970 --> 00:47:27,190 Now for the previous example, for the selling 599 00:47:27,190 --> 00:47:32,930 price of a home and its size, B1rSy divided by Sx, 600 00:47:33,790 --> 00:47:43,550 r is computed, r is found to be 76%, 76%Sy divided 601 00:47:43,550 --> 00:47:49,990 by Sx, that will give 0.109. B0y bar minus B1x 602 00:47:49,990 --> 00:47:50,670 bar, 603 00:47:53,610 --> 00:48:00,150 Y bar for this data is 286 minus D1. So we have to 604 00:48:00,150 --> 00:48:03,490 compute first D1 because we use it in order to 605 00:48:03,490 --> 00:48:08,590 determine D0. And calculation gives 98. So that 606 00:48:08,590 --> 00:48:16,450 means based on these equations, Y hat equals 0 607 00:48:16,450 --> 00:48:22,990 .10977 plus 98.248. 608 00:48:24,790 --> 00:48:29,370 times X. X is the size. 609 00:48:32,890 --> 00:48:39,830 0.1 B1 610 00:48:39,830 --> 00:48:45,310 is 611 00:48:45,310 --> 00:48:56,650 0.1, B0 is 98, so 98.248 plus B1. So this is your 612 00:48:56,650 --> 00:49:03,730 regression equation. So again, the intercept is 613 00:49:03,730 --> 00:49:09,750 98. So this amount, the segment is 98. Now the 614 00:49:09,750 --> 00:49:14,790 slope is 0.109. So house price, the expected value 615 00:49:14,790 --> 00:49:21,270 of house price equals B098 plus 0.109 square feet. 616 00:49:23,150 --> 00:49:27,630 So that's the prediction line for the house price. 617 00:49:28,510 --> 00:49:34,370 So again, house price equal B0 98 plus 0.10977 618 00:49:34,370 --> 00:49:36,930 times square root. Now, what's your interpretation 619 00:49:36,930 --> 00:49:41,950 about B0 and B1? B0 is the estimated mean value of 620 00:49:41,950 --> 00:49:46,430 Y when the value of X is 0. So if X is 0, this 621 00:49:46,430 --> 00:49:52,980 range of X observed X values and you have a home 622 00:49:52,980 --> 00:49:57,860 or a house of size zero. So that means this value 623 00:49:57,860 --> 00:50:02,680 has no meaning. Because a house cannot have a 624 00:50:02,680 --> 00:50:06,400 square footage of zero. So B0 has no practical 625 00:50:06,400 --> 00:50:10,040 application in this case. So sometimes it makes 626 00:50:10,040 --> 00:50:17,620 sense, in other cases it doesn't have that. So for 627 00:50:17,620 --> 00:50:21,790 this specific example, B0 has no practical 628 00:50:21,790 --> 00:50:28,210 application in this case. But B1 which is 0.1097, 629 00:50:28,930 --> 00:50:33,050 B1 estimates the change in the mean value of Y as 630 00:50:33,050 --> 00:50:36,730 a result of one unit increasing X. So for this 631 00:50:36,730 --> 00:50:41,640 value which is 0.109, it means This fellow tells 632 00:50:41,640 --> 00:50:46,420 us that the mean value of a house can increase by 633 00:50:46,420 --> 00:50:52,280 this amount, increase by 0.1097, but we have to 634 00:50:52,280 --> 0 667 00:53:54,190 --> 00:53:57,770 that's 668 00:53:57,770 --> 00:53:57,990 all