The Most Simple Introduction to Hypothesis Testing! - Statistics Help — Transcript

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1. 0:00welcome to Quant Concepts education this
2. 0:03is a quick 10-minute lecture on the
3. 0:05fundamentals of hypothesis
4. 0:07testing this is a crucial Topic in any
5. 0:09study of statistics and quantitative
6. 0:11methods I assume the audience has no
7. 0:14background in
8. 0:16statistics so let's say you hang out
9. 0:18with your friend Sam one day and you
10. 0:20both decide to go bowling and while
11. 0:23you're driving to the bowling alley with
12. 0:24Sam he keeps on saying mate my bowling
13. 0:27average is so good I've got a long-term
14. 0:29aage a of 150
15. 0:32R now in case you don't know 150 is an
16. 0:36excellent average for bowling it means
17. 0:38Sam is a very very good bowler so let's
18. 0:41say you play three games of bowling and
19. 0:43over the three games Sam's average score
20. 0:46is a dismal
21. 0:4740 now in such a scenario do you believe
22. 0:50him do you believe Sam's claim that his
23. 0:53long-term bowling average is 150 and
24. 0:56more importantly is Sam a dodgy friend
25. 1:02you would probably not believe him why
26. 1:05is this because if Sam's long-term
27. 1:07average really is 150 then this means he
28. 1:10is a very good bowler and if he is a
29. 1:13very good bowler it is highly unlikely
30. 1:15that he would have scored a miserable
31. 1:17average of 40 over your three games with
32. 1:19him now let's rewind the clock a little
33. 1:21bit and say you play the three games
34. 1:23with Sam but this time his average over
35. 1:26the three games is 140 now in this this
36. 1:30case do you believe
37. 1:31him you'd be much more likely to believe
38. 1:34Sam in the second scenario yeah because
39. 1:36scoring 140 is very close to his claimed
40. 1:39long-term average score of
41. 1:42150 on top of that if 150 is his
42. 1:46long-term average it doesn't necessarily
43. 1:48mean that he will score 150 every game
44. 1:52what it does mean is that Sam will score
45. 1:54an average of 150 over many games maybe
46. 1:58Sam underperformed today because he had
47. 2:00a bad day maybe he didn't like the
48. 2:03bowling ball maybe he met a pretty girl
49. 2:05in the morning and his love struck who
50. 2:07knows there are a million reasons why he
51. 2:10may have
52. 2:11underperformed but the main point we're
53. 2:13trying to get at is that you are more
54. 2:15likely to believe Sam's claim as 140 is
55. 2:18very close to his claimed average of
56. 2:21150 so we can see that there are two
57. 2:24extreme cases here one where you'll be
58. 2:26unlikely to believe Sam and will call
59. 2:28him a liar and another where you'll be
60. 2:31likely to believe him and still be
61. 2:33friends now can you tell me at what
62. 2:36point between 40 and 140 do you make the
63. 2:40decision to believe Sam or
64. 2:43not well this is actually quite
65. 2:45subjective but in your mind you will
66. 2:48have a cut off score which you will use
67. 2:49to determine whether Sam's claim is
68. 2:51correct or not for example you may tell
69. 2:54yourself that if Sam scores an average
70. 2:56over our three games that is below 120 I
71. 2:59won't believe his claim however if he
72. 3:02does score an average over our three
73. 3:04games that is above 120 I will believe
74. 3:06him this is quite intuitive so Sam makes
75. 3:10a claim that his long-term bowling
76. 3:12average is
77. 3:13150 and if his average over the three
78. 3:16games with you Falls below a particular
79. 3:18value such as 120 you will reject his
80. 3:22claim easy enough
81. 3:25yeah now let's have a look at a possible
82. 3:28probability distribution of Sam's
83. 3:30bowling scores assuming that his claim
84. 3:32is correct because at the end of the day
85. 3:34Sam's your friend you will give him the
86. 3:36benefit of the doubt until proven
87. 3:39otherwise so this is known as a
88. 3:42probability density function which we
89. 3:44will discuss in more detail later in the
90. 3:46lecture all you need to know for now is
91. 3:48that the x-axis contains all possible
92. 3:51values for Sam's average bowling score
93. 3:52for your three games with him and the y-
94. 3:55AIS contains
95. 3:56probabilities therefore the higher the
96. 3:59graph at a particular bowling score the
97. 4:01higher the probability of that bowling
98. 4:02score
99. 4:04occurring okay so given that Sam's claim
100. 4:07is correct and that his long-term
101. 4:09average is
102. 4:10150 then we would expect his bowling
103. 4:13scores to be very close to 150 hence the
104. 4:16graph Peaks at
105. 4:18150 this means that we would expect with
106. 4:20a high probability that Sam's bowling
107. 4:22scores will be close to 150 assuming his
108. 4:25claim is
109. 4:26correct moreover you can also see that
110. 4:29that values that are far from 150 have a
111. 4:32lower probability of occurring as the
112. 4:34graph is lower at bowling scores further
113. 4:37away from
114. 4:38150 this makes sense because if Sam's
115. 4:41average really is 150 and he really is a
116. 4:44good bowler then there should be a low
117. 4:46probability that he scores very poorly
118. 4:48in
119. 4:50Bowling now remember our cutoff value
120. 4:53is0 that is you will believe Sam if he
121. 4:57scores over 120 in your three games with
122. 4:59him
123. 5:00and you won't believe him
124. 5:02otherwise the Shaded region is known as
125. 5:05a rejection region if Sam's average
126. 5:07score over his three games with you
127. 5:09Falls below
128. 5:11120 you will reject his claim that his
129. 5:13long-term average bowling score is
130. 5:18150 let's picture another scenario it's
131. 5:21your mom's birthday and you decide to
132. 5:23take her out bowling and in the car
133. 5:26whilst driving to the bowling alley your
134. 5:28mom tells you honey did you know that
135. 5:31I'm an excellent bowler in fact dear my
136. 5:34long-term average for bowling is
137. 5:37150
138. 5:39interesting so you play three games with
139. 5:41your beloved mother and her average
140. 5:43score over the three games is a
141. 5:45depressing 40 now what do you do do you
142. 5:49believe your mom's claim that her
143. 5:51long-term average is
144. 5:52150 or do you call your own mother a
145. 5:56liar well it's a tough situation
146. 6:00firstly it's highly unlikely that her
147. 6:02claim is correct however it is possible
148. 6:06that she may just be having a really bad
149. 6:09day now you're more likely to give your
150. 6:11mom the benefit of the doubt than you
151. 6:13are to Sam right because the last thing
152. 6:16you want is to call your mom a liar when
153. 6:18in fact she really was just having a bad
154. 6:21day now that's just plain
155. 6:24cruel so when rejecting your mom's claim
156. 6:27of having a long-term average of 50 you
157. 6:30want to be really really sure that she's
158. 6:33lying before you do
159. 6:35so so what do you do if you want to be
160. 6:38more sure that a claim is false before
161. 6:41rejecting it
162. 6:43simple we simply use a lower cut off
163. 6:47value for example you may have told
164. 6:50yourself that you'll believe Sam if his
165. 6:52average score over the three games with
166. 6:54you is above
167. 6:55120 as you love your mom and are willing
168. 6:58to give her the benefit of the dad out
169. 7:00you may tell yourself that you will
170. 7:01believe your mom's claim if her average
171. 7:04score over the three games with you is
172. 7:06above
173. 7:0750 so your cut off value for Sam is 120
174. 7:11and your cut off value for your mom is
175. 7:13only
176. 7:1450 because calling Sam a lie is no big
177. 7:17deal he lies all the time so you have a
178. 7:20higher cut off value for him but calling
179. 7:22your M A Lie is a massive deal so you
180. 7:25have to be very sure she is lying before
181. 7:27you do so so you'll have a lower cut off
182. 7:29value for her besides you never really
183. 7:32like Sam anyway and your mom is cooking
184. 7:34you a nice dinner tonight even though it
185. 7:36is her birthday you lazy
186. 7:39bugger so the probability distribution
187. 7:42of your mom's bowling scores will look
188. 7:44something like
189. 7:46this in this case however notice that
190. 7:49the cut off value is only 50 whereas for
191. 7:51Sam it was
192. 7:53120 therefore the Shaded region or the
193. 7:55rejection region is much smaller in your
194. 7:58mom's case
195. 8:00what the smaller rejection region
196. 8:01signifies is that you are less likely to
197. 8:04call your mom a liar when she is in fact
198. 8:06telling the truth because doing so will
199. 8:08send you straight to Hell In the case
200. 8:11for Sam the repercussions of calling him
201. 8:14a liar when he is in fact telling you
202. 8:16the truth is not so harsh so his
203. 8:18rejection region is of a larger
204. 8:21size hence the size of the rejection
205. 8:24region and how far the cut off value is
206. 8:26from the claimed average is determined
207. 8:28by you the researcher it depends on how
208. 8:32sure you want to be when rejecting the
209. 8:35claim now time for some statistical
210. 8:38jargon if you've understood the lecture
211. 8:40so far then you're already Miles Ahead
212. 8:42in learning hypothesis testing all we
213. 8:45need to do now is to add the statistical
214. 8:47names to the ideas that we've just
215. 8:50discussed the null hypothesis is the
216. 8:53claim we are trying to test it is
217. 8:55denoted by
218. 8:58h0 in in our particular case we are
219. 9:01trying to test Sam's claim that his
220. 9:03long-term average bowling score is
221. 9:05150 so the null hypothesis is that Sam's
222. 9:08long-term bowling average is equal to
223. 9:12150 the alternate hypothesis usually
224. 9:16denoted by H1 is the counter claim that
225. 9:19is what must be true if the null
226. 9:22hypothesis is
227. 9:24false so in our little example the
228. 9:27alternate hypothesis is that Sam's
229. 9:30long-term bowling average is below
230. 9:32150 and he is a liar and a dodgy
231. 9:37friend the sample statistic usually
232. 9:40denoted by X is the observed sample
233. 9:42estimate that we use to determine
234. 9:44whether the null hypothesis is false or
235. 9:47not in this case the sample statistic is
236. 9:51Sam's average score for the three games
237. 9:53you played with
238. 9:54him the critical value is simply the cut
239. 9:57off value you assign to the sample
240. 9:59statistic to determine whether the null
241. 10:02hypothesis or claim is rejected or not
242. 10:05in our example the critical value is
243. 10:08120 as you tell yourself that if Sam's
244. 10:11average score for the three games you
245. 10:13play with him is below 120 you will
246. 10:16reject his
247. 10:18claim finally the significance level
248. 10:21measures how sure you want to be when
249. 10:23rejecting the null
250. 10:26hypothesis the smaller the significance
251. 10:28level the the more sure you are when
252. 10:30rejecting the null hypothesis and the
253. 10:33smaller the rejection
254. 10:34region so in your mom's case you will
255. 10:37use a smaller significance level to
256. 10:40determine if her claim is false or not
257. 10:43for Sam's case you are willing to use a
258. 10:45larger significance
259. 10:48level thank you for listening to Quant
260. 10:50Concepts education come and visit us at
261. 10:54www. quantcon ups.com