XYZ Mat 120 Module 1 Lesson 1 8 — Transcript

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1. 0:00this is section 1.8
2. 0:04and module 1. we're looking at
3. 0:07linear regression we're going to talk
4. 0:09about correlation
5. 0:10and regression a correlation exists
6. 0:13between two variables when the values of
7. 0:15these variables are somehow associated
8. 0:17with the values of
9. 0:18another variable and so we're going to
10. 0:20be looking at an x and a y
11. 0:21variable are they related is there some
12. 0:23kind of relationship
13. 0:25so a linear correlation exists between
14. 0:28two variables when
15. 0:29the plotted lines result in a pattern
16. 0:32that can be approximated
17. 0:34by a straight line so what we want to
18. 0:36look and see
19. 0:37is can we have a straight line
20. 0:40so there are types of correlation the
21. 0:42first is the positive correlation
22. 0:44the positive correlation is if you have
23. 0:47data
24. 0:48when we plot it you can have a
25. 0:52line a line going right through there
26. 0:54and we can see that this
27. 0:55data pretty much lines up in
28. 0:58a positive slope on this line
29. 1:02all the dots are not exactly on the line
30. 1:04but they're all very close
31. 1:06so there is a positive correlation so
32. 1:09as x increases y increases and that
33. 1:12makes that a positive correlation
34. 1:15let's look at a negative correlation
35. 1:16that's just like a line with a negative
36. 1:18slope
37. 1:19so we'll have a line we've got a
38. 1:21negative slope coming in this direction
39. 1:23again all the dots are not on the line
40. 1:25but they're very very close
41. 1:27and so we can say that it's a negative
42. 1:28correlation as
43. 1:30x increases the y's
44. 1:33decrease and what we're going to look at
45. 1:36is something that's a correlation
46. 1:39coefficient in just a minute we're going
47. 1:41to call that r so we're going to come
48. 1:42back to this
49. 1:44if there's no correlation then there is
50. 1:46no distinct pattern
51. 1:47i can't draw a straight line in this way
52. 1:50or that way because of those dots
53. 1:52they're all over the place and so there
54. 1:54might be no correlation now we're going
55. 1:57to look at the strength of the
56. 1:58correlation
57. 2:00the things that we looked at just a
58. 2:01minute ago the first one we saw was a
59. 2:03positive correlation
60. 2:04we're going to say that r is
61. 2:09.859 r is .859
62. 2:12if r equals one then you're going to
63. 2:16have a perfect straight line
64. 2:18but it doesn't equal one but .859
65. 2:21is pretty close to 100 percent so pretty
66. 2:24close to 1.
67. 2:26now look at this negative this is a
68. 2:27better correlation because
69. 2:29our line almost lines up on the dots a
70. 2:34little bit better than on the one for
71. 2:35the positive
72. 2:36but it is negative so our r for this one
73. 2:38is negative 0.97
74. 2:40because it's almost a perfect negative
75. 2:42correlation
76. 2:44so if it were perfect then it would be r
77. 2:47equals negative one and you'd have a
78. 2:48perfect straight line
79. 2:50with a negative slope then we come down
80. 2:53to this one that was a
81. 2:54scattered bunch of data no correlation
82. 2:59r is .074
83. 3:020 7 4 that's way away from 1
84. 3:05in either the positive or the negative
85. 3:07direction
86. 3:09so therefore 0.074 makes sense
87. 3:13sometimes you have a relationship but it
88. 3:15might be non-linear
89. 3:17so for this particular one you can see
90. 3:20you've got a relationship on these dots
91. 3:21but it's a non-linear relationship
92. 3:25now a linear correlation
93. 3:28uses the coefficient r it measures
94. 3:32the strength of that linear relationship
95. 3:36between the x and the y values
96. 3:39so as i said just a minute ago
97. 3:46well let's see what we need to look at
98. 3:50before we look at this example
99. 3:52is that if r
100. 3:55equals 1 that's a perfect
101. 3:59positive correlation now it might not be
102. 4:03exactly one it could be 0.95 it could be
103. 4:060.8
104. 4:06and it's still going to be really close
105. 4:10but maybe not perfect if r equals
106. 4:13negative 1
107. 4:15then you have a perfect negative
108. 4:17correlation
109. 4:19and again it might not be exactly
110. 4:21negative 1 it might be negative
111. 4:230.97 it might be negative 0.85 but it
112. 4:26means that our dots
113. 4:27are pretty close in that direction if r
114. 4:30equals 0
115. 4:31then there is no correlation no
116. 4:34relationship the dots are
117. 4:35all over the place and so they don't
118. 4:38have a relationship
119. 4:40so let's look at a couple of homework
120. 4:42examples this one
121. 4:44says we have some data they've already
122. 4:46put the data in the x and y
123. 4:48table and that's one of the things
124. 4:50you're going to have to do in one of
125. 4:51your homework problems
126. 4:53is you're going to have to plot x and y
127. 4:56and this one comes up with a model that
128. 4:59says
129. 4:59our line for this data is y
130. 5:02equals 1.94 x plus 6.8
131. 5:07i don't know if you've recently taken
132. 5:08any algebra classes but if you have
133. 5:11this should remind you of the slope
134. 5:13formula
135. 5:14y equals mx plus b and so that's
136. 5:18the thing that we're doing we're looking
137. 5:20at lines is it a line that goes this way
138. 5:22is it a line that that
139. 5:24goes that way or is there no perfect
140. 5:27line
141. 5:27so they're telling us that that is going
142. 5:29to be our line
143. 5:30and we're asked to find some things it
144. 5:32says now use this model
145. 5:34to estimate when x equals 5
146. 5:39find y so all i'm going to do is i'm
147. 5:42going to solve for y
148. 5:43so i'm going to say y is equal to
149. 5:471.94 they told me they want x to be 5
150. 5:51plus 6.48 we're going to do this
151. 5:54calculation
152. 5:56so in your calculator 1.94
153. 6:00times 5 plus
154. 6:046.48 and what you say we get is 16.18 so
155. 6:09what we've done
156. 6:10is we said when x equals 5
157. 6:14then y equals 16.18
158. 6:18so we've created an ordered pair and
159. 6:20then it asks us to do the same thing but
160. 6:22this time let x
161. 6:23equal to 10 so we'll take our same
162. 6:26formula
163. 6:27that they gave us 1.94 times
164. 6:3110 plus 6.48
165. 6:36and that's how i get my 25.88
166. 6:39so i've said okay so now when x equals
167. 6:4210
168. 6:44y equals 25.88
169. 6:48and that is using just an equation
170. 6:52now let's see how do we find all this
171. 6:53stuff so our last example is very
172. 6:55comprehensive it's got a lot of things
173. 6:57going on
174. 6:58it's looking at lengths and weights of
175. 7:00alligators
176. 7:01and so we have notice an x
177. 7:05and a y and what we're going to do is in
178. 7:08our calculator we're going to go to stat
179. 7:11we're going to go to enter and what i
180. 7:13did is i already
181. 7:15put an l1 all of the x values
182. 7:18116 80 128 101
183. 7:2277 93 70.
184. 7:27um 84 106 and 97
185. 7:31and then i went over to y and i put all
186. 7:33of the y values
187. 7:34in l2 so in l2 i put 651
188. 7:38563 782 515
189. 7:42360 506 434
190. 7:46486 712 and 580. so i've entered those
191. 7:49in l1
192. 7:50and l2 now we're asked to find r
193. 7:54so let's see how we do that we'll go to
194. 7:57stat
195. 7:58we'll go to calc this time we're going
196. 8:00to go down to
197. 8:01linear regression notice on your
198. 8:04calculator that says
199. 8:05ax plus b that's like the y equals mx
200. 8:08plus b it's linear regression
201. 8:10we'll hit enter x list should be l1 the
202. 8:13y list should be l2
203. 8:15and don't put anything in any of the
204. 8:17others so the frequency list just clear
205. 8:19that out
206. 8:20the store reg equation just leave that
207. 8:23out and then calculate it
208. 8:25and you should get something that looks
209. 8:26like this you're going to have y equals
210. 8:28ax plus b
211. 8:30you have the a value you have the b
212. 8:31value you have an r
213. 8:33squared and you have an r the question
214. 8:35is what is
215. 8:36r r is the correlation coefficient
216. 8:39that's 0.872
217. 8:41what does that mean before we do
218. 8:43anything with that
219. 8:450.872 means there
220. 8:47is a pretty good relationship when we
221. 8:50dot
222. 8:51when we plot these points that you'll
223. 8:54get
224. 8:55nearly a straight line so it's not
225. 8:57exactly straight
226. 8:58it's not perfect but that .87 is a
227. 9:02pretty good positive relationship and
228. 9:04that's how that's going to look
229. 9:05so that means the longer the alligator
230. 9:07is the more it's going to weigh
231. 9:10then we want to find the equation of the
232. 9:11regression line well that's
233. 9:13ax plus b notice we have up here a
234. 9:17is 6.12 so we'll just go ahead and write
235. 9:19that down a equals 6.12
236. 9:22you see that in your calculator b is
237. 9:24negative 24.10
238. 9:27so b is negative 24.10
239. 9:30but we want a x we want to put the x
240. 9:33there
241. 9:34so when i write this equation i'm going
242. 9:36to write y
243. 9:37equals 6.12 x minus
244. 9:4124.10 or you can just put 24.1
245. 9:45then it says okay now you've got the
246. 9:47equation predict the weight
247. 9:50of an alligator now remember weight is
248. 9:51your y value predict the weight
249. 9:54that is 87 inches long that means let x
250. 9:57equal 87.
251. 9:59so i'm going to take my equation that i
252. 10:01just found
253. 10:02and do 6.12
254. 10:05times 87 because that's now x minus
255. 10:1024.10 and then we'll get 508.34
256. 10:15so if an alligator is 87
257. 10:18inches long then the weight is predicted
258. 10:20to be about 508.34 based on this
259. 10:23knowing it's not exact that's just a
260. 10:26really close
261. 10:27estimate would it be meaningful to use
262. 10:30this model to predict
263. 10:32the weight of an alligator that is 235
264. 10:34inches long
265. 10:36so let's look at that i said no for that
266. 10:38if i'm looking for 235 look at all of
267. 10:41these values
268. 10:43235 is nowhere near these
269. 10:46the largest link the longest length you
270. 10:48see is 128
271. 10:50so it would not be meaningful because of
272. 10:53the data values that we're given