1 00:00:05,600 --> 00:00:09,760 Last time, we talked about the types of samples 2 00:00:09,760 --> 00:00:15,880 and introduced two 3 00:00:15,880 --> 00:00:20,120 types of samples. One is called non-probability 4 00:00:20,120 --> 00:00:23,900 samples, and the other one is probability samples. 5 00:00:25,120 --> 00:00:30,560 And also, we have discussed two types of non 6 00:00:30,560 --> 00:00:33,620 -probability, which are judgment and convenience. 7 00:00:35,100 --> 00:00:39,500 For the product samples, we also produced four 8 00:00:39,500 --> 00:00:46,560 types, random sample, systematic, stratified, and 9 00:00:46,560 --> 00:00:54,400 clustered sampling. That was last Sunday. Let's 10 00:00:54,400 --> 00:01:02,550 see the comparison between these sampling data. A 11 00:01:02,550 --> 00:01:05,370 simple, random sample, systematic random sample, 12 00:01:05,510 --> 00:01:09,930 first, for these two techniques. First of all, 13 00:01:09,970 --> 00:01:13,590 they are simple to use because we just use the 14 00:01:13,590 --> 00:01:18,750 random tables, random number tables, or by using 15 00:01:18,750 --> 00:01:27,690 any statistical software. But the disadvantage of 16 00:01:27,690 --> 00:01:28,490 this technique 17 00:01:37,590 --> 00:01:40,830 So it might be this sample is not representative 18 00:01:40,830 --> 00:01:44,530 of the entire population. So this is the mainly 19 00:01:44,530 --> 00:01:50,230 disadvantage of this sampling technique. So it can 20 00:01:50,230 --> 00:01:56,250 be used unless the population is not symmetric or 21 00:01:56,250 --> 00:02:00,090 the population is not heterogeneous. I mean if the 22 00:02:00,090 --> 00:02:04,510 population has the same characteristics, then we 23 00:02:04,510 --> 00:02:08,870 can use simple or systematic sample. But if there 24 00:02:08,870 --> 00:02:12,110 are big differences or big disturbances between 25 00:02:12,990 --> 00:02:15,510 the items of the population, I mean between or 26 00:02:15,510 --> 00:02:21,550 among the individuals. In this case, stratified 27 00:02:21,550 --> 00:02:26,190 sampling is better than using a simple random 28 00:02:26,190 --> 00:02:30,170 sample. Stratified samples ensure representation 29 00:02:30,170 --> 00:02:33,010 of individuals across the entire population. If 30 00:02:33,010 --> 00:02:36,800 you remember last time we said a IUG population 31 00:02:36,800 --> 00:02:40,340 can be splitted according to gender, either males 32 00:02:40,340 --> 00:02:44,440 or females, or can be splitted according to 33 00:02:44,440 --> 00:02:48,840 students' levels. First level, second level, and 34 00:02:48,840 --> 00:02:51,960 fourth level, and so on. The last type of sampling 35 00:02:51,960 --> 00:02:55,340 was clusters. Cluster sampling is more cost 36 00:02:55,340 --> 00:02:59,940 effective. Because in this case, you have to split 37 00:02:59,940 --> 00:03:03,140 the population into many clusters, then you can 38 00:03:03,140 --> 00:03:08,320 choose a random of these clusters. Also, it's less 39 00:03:08,320 --> 00:03:12,720 efficient unless you use a large sample. For this 40 00:03:12,720 --> 00:03:16,460 reason, it's more cost effective than using the 41 00:03:16,460 --> 00:03:20,640 other sampling techniques. So, these techniques 42 00:03:20,640 --> 00:03:23,700 are used based on the study you have. Sometimes 43 00:03:23,700 --> 00:03:26,100 simple random sampling is fine, and you can go 44 00:03:26,100 --> 00:03:29,360 ahead and use it. Most of the time, stratified 45 00:03:29,360 --> 00:03:33,340 random sampling is much better. So, it depends on 46 00:03:33,340 --> 00:03:36,940 the population you have underlying your study. 47 00:03:37,680 --> 00:03:40,240 That was what we talked about last Sunday. 48 00:03:43,860 --> 00:03:47,780 Now, suppose we design a questionnaire or survey. 49 00:03:48,640 --> 00:03:52,980 You have to know, number one, what's the purpose 50 00:03:52,980 --> 00:03:59,600 of the survey. In this case, you can determine the 51 00:03:59,600 --> 00:04:02,040 frame of the population. Next, 52 00:04:05,480 --> 00:04:07,660 survey 53 00:04:13,010 --> 00:04:18,350 Is the survey based on a probability sample? If 54 00:04:18,350 --> 00:04:20,830 the answer is yes, then go ahead and use one of 55 00:04:20,830 --> 00:04:22,910 the non-probability sampling techniques, either 56 00:04:22,910 --> 00:04:26,610 similar than some certified cluster or systematic. 57 00:04:28,830 --> 00:04:33,330 Next, we have to distinguish between four types of 58 00:04:33,330 --> 00:04:38,770 errors, at least now. One is called coverage 59 00:04:38,770 --> 00:04:44,560 error. You have to ask yourself, is the frame 60 00:04:44,560 --> 00:04:48,880 appropriate? I mean, frame appropriate means that 61 00:04:48,880 --> 00:04:52,540 you have all the individual list, then you can 62 00:04:52,540 --> 00:04:56,040 choose one of these. For example, suppose we 63 00:04:56,040 --> 00:05:01,500 divide Gaza Strip into four governorates. North 64 00:05:01,500 --> 00:05:06,800 Gaza, Gaza Middle Area, Khanon, and Rafah. So we 65 00:05:06,800 --> 00:05:12,950 have five sections of five governments. In this 66 00:05:12,950 --> 00:05:17,430 case, if you, so that's your frame. Now, if you 67 00:05:17,430 --> 00:05:19,890 exclude one, for example, and that one is 68 00:05:19,890 --> 00:05:22,350 important for you, but you exclude it for some 69 00:05:22,350 --> 00:05:26,270 reasons, in this case, you will have coverage as 70 00:05:26,270 --> 00:05:32,600 well, because you excluded one group out of five 71 00:05:32,600 --> 00:05:36,140 and that group may be important for your study. 72 00:05:36,840 --> 00:05:41,740 Next is called non-response error. Suppose I 73 00:05:41,740 --> 00:05:48,040 attributed my questionnaire for 100 students and I 74 00:05:48,040 --> 00:05:52,840 gave each one 30 minutes to answer the 75 00:05:52,840 --> 00:05:56,380 questionnaire or to fill up the questionnaire, but 76 00:05:56,380 --> 00:06:01,230 I didn't follow up. The response in this case, it 77 00:06:01,230 --> 00:06:05,890 might be you will get something error, and that 78 00:06:05,890 --> 00:06:09,010 error refers to non-responsive, so you have to 79 00:06:09,010 --> 00:06:12,190 follow up, follow up. It means maybe sometimes you 80 00:06:12,190 --> 00:06:14,670 need to clarify the question you have in your 81 00:06:14,670 --> 00:06:19,510 questionnaire so that the respondent understand 82 00:06:19,510 --> 00:06:21,850 what do you mean exactly by that question 83 00:06:21,850 --> 00:06:25,550 otherwise if you don't follow up it means it may 84 00:06:25,550 --> 00:06:30,550 be there is an error, and that error is called non 85 00:06:30,550 --> 00:06:35,170 -response. The other type of error is called 86 00:06:35,170 --> 00:06:37,150 measurement error, which is one of the most 87 00:06:37,150 --> 00:06:39,910 important errors, and we have to avoid. 88 00:06:42,950 --> 00:06:45,830 It's called measurement error. Good questions 89 00:06:45,830 --> 00:06:50,710 elicit good responses. It means, suppose, for 90 00:06:50,710 --> 00:06:57,700 example, my question is, I feel this candidate is 91 00:06:57,700 --> 00:07:01,740 good for us. What do you think? It's my question. 92 00:07:02,820 --> 00:07:08,000 I feel this candidate, candidate A, whatever he 93 00:07:08,000 --> 00:07:15,000 is, is good for us. What do you think? For sure 94 00:07:15,000 --> 00:07:18,080 there's abundant answer will be yes. I agree with 95 00:07:18,080 --> 00:07:22,660 you. So that means you design the question in the 96 00:07:22,660 --> 00:07:28,100 way that you will know they respond directly, that 97 00:07:28,100 --> 00:07:31,980 he will answer yes or no depends on your design of 98 00:07:31,980 --> 00:07:36,580 the question. So it means leading question. So 99 00:07:36,580 --> 00:07:40,160 measurement error. So but if we have good 100 00:07:40,160 --> 00:07:43,260 questions, just ask any question for the 101 00:07:43,260 --> 00:07:48,060 respondent, and let him or let his answer based on 102 00:07:48,890 --> 00:07:52,850 what exactly he thinks about it. So don't force 103 00:07:52,850 --> 00:07:56,490 the respondent to answer the question in the 104 00:07:56,490 --> 00:07:59,250 direction you want to be. Otherwise you will get 105 00:07:59,250 --> 00:08:03,570 something called Measurement Error. Do you think? 106 00:08:04,910 --> 00:08:06,770 Give me an example of Measurement Error. 107 00:08:09,770 --> 00:08:12,450 Give me an example of Measurement Error. Just ask 108 00:08:12,450 --> 00:08:16,370 a question in a way that the respondent will 109 00:08:16,370 --> 00:08:20,750 answer, I mean, his answer will be the same as you 110 00:08:20,750 --> 00:08:24,770 think about it. 111 00:08:30,130 --> 00:08:35,130 Maybe I like coffee, do you like coffee or tea? So 112 00:08:35,130 --> 00:08:37,390 maybe he will go with your answer. In this case 113 00:08:37,390 --> 00:08:41,630 it's measurement. Another example. 114 00:09:00,260 --> 00:09:00,860 Exactly. 115 00:09:07,960 --> 00:09:12,420 So it means that if you design a question in the 116 00:09:12,420 --> 00:09:15,420 way that you will get the same answer you think 117 00:09:15,420 --> 00:09:18,910 about it, it means that you will have something 118 00:09:18,910 --> 00:09:21,310 called measurement error. The last type is 119 00:09:21,310 --> 00:09:25,490 sampling error. Sampling error always happens, 120 00:09:25,710 --> 00:09:29,990 always exists. For example, suppose you are around 121 00:09:29,990 --> 00:09:33,150 50 students in this class. Suppose I select 122 00:09:33,150 --> 00:09:40,130 randomly 20 of you, and I am interested suppose in 123 00:09:40,130 --> 00:09:44,090 your age. Maybe for this sample. 124 00:09:46,590 --> 00:09:53,610 I will get an average of your age of 19 years 125 00:09:53,610 --> 00:09:57,370 someone 126 00:09:57,370 --> 00:10:02,050 select another sample from the same population 127 00:10:02,050 --> 00:10:08,670 with the same size maybe 128 00:10:08,670 --> 00:10:13,530 the average of your age is not equal to 19 years 129 00:10:13,530 --> 00:10:16,370 maybe 19 years, 3 months 130 00:10:19,330 --> 00:10:24,790 Someone else maybe also select the same number of 131 00:10:24,790 --> 00:10:28,910 students, but the average of the class might be 20 132 00:10:28,910 --> 00:10:33,690 years. So, the first one, second tier, each of them 133 00:10:33,690 --> 00:10:37,830 has different sample statistics. I mean different 134 00:10:37,830 --> 00:10:42,710 sample means. This difference or this error 135 00:10:42,710 --> 00:10:46,470 actually is called sampling error and always 136 00:10:46,470 --> 00:10:52,040 happens. So, now we have five types of errors. One 137 00:10:52,040 --> 00:10:54,580 is called coverage error. In this case, you have 138 00:10:54,580 --> 00:10:59,360 a problem with the frame. The other type is called 139 00:10:59,360 --> 00:11:03,260 non-response error. It means you have a problem with 140 00:11:03,260 --> 00:11:06,620 following up. Measurement error. It means you have 141 00:11:07,810 --> 00:11:11,370 bad questionnaire design. The last type is called 142 00:11:11,370 --> 00:11:14,130 sampling error, and this one always happens and 143 00:11:14,130 --> 00:11:18,390 actually we would like to have this error. I mean 144 00:11:18,390 --> 00:11:22,250 this sampling error as small as possible. So, these are 145 00:11:22,250 --> 00:11:25,890 the steps you have to follow up when you design 146 00:11:25,890 --> 00:11:26,430 the questionnaire. 147 00:11:29,490 --> 00:11:34,710 So again, for these types of errors, the first one 148 00:11:34,710 --> 00:11:40,170 coverage error, or selection bias. This type of 149 00:11:40,170 --> 00:11:43,830 error exists if some groups are excluded from the 150 00:11:43,830 --> 00:11:48,950 frame, and have no chance of being selected. That's 151 00:11:48,950 --> 00:11:51,970 the first type of error, coverage error. So, it 152 00:11:51,970 --> 00:11:55,690 means there is a problem on the population frame. 153 00:11:56,510 --> 00:12:00,450 Non-response error bias. It means people who don't 154 00:12:00,450 --> 00:12:03,230 respond may be different from those who do 155 00:12:03,230 --> 00:12:09,730 respond. For example, suppose I have a sample of 156 00:12:09,730 --> 00:12:11,410 tennis students. 157 00:12:15,910 --> 00:12:22,950 And I got responses from number two, number five, 158 00:12:24,120 --> 00:12:29,420 and number 10. So, I have these points of view for 159 00:12:29,420 --> 00:12:34,860 these three students. Now, the other seven students 160 00:12:34,860 --> 00:12:41,100 might be they have different opinions. So, the only 161 00:12:41,100 --> 00:12:45,220 thing you have, the opinions of just the three, 162 00:12:45,520 --> 00:12:47,800 and maybe the rest have different opinions, it 163 00:12:47,800 --> 00:12:50,680 means in this case you will have something called 164 00:12:50,680 --> 00:12:54,270 non-responsiveness. Or the same as we said before, 165 00:12:54,810 --> 00:12:58,870 if your question is designed in a correct way. The 166 00:12:58,870 --> 00:13:02,130 other type, sample error, variations from sample 167 00:13:02,130 --> 00:13:06,230 to sample will always exist. As I mentioned here, 168 00:13:06,230 --> 00:13:10,470 we select six samples, each one has different 169 00:13:10,470 --> 00:13:14,730 sample mean. The other type, Measurement Error, 170 00:13:15,310 --> 00:13:19,200 due to weakness in question design. So that's the 171 00:13:19,200 --> 00:13:25,600 type of survey errors. So, one more time, average 172 00:13:25,600 --> 00:13:32,120 error, it means you exclude a group or groups from 173 00:13:32,120 --> 00:13:35,080 the frame. So, in this case, suppose I excluded 174 00:13:35,080 --> 00:13:40,380 these from my frame. So I just select the sample 175 00:13:40,380 --> 00:13:45,940 from all of these, except this portion, or these 176 00:13:45,940 --> 00:13:49,300 two groups. Non-response error means you don't 177 00:13:49,300 --> 00:13:52,920 have a follow-up on non-responses. Sampling error, 178 00:13:54,060 --> 00:13:58,720 random sample gives different sample statistics. 179 00:13:59,040 --> 00:14:01,400 So it means random differences from sample to 180 00:14:01,400 --> 00:14:05,760 sample. Finally, measurement error, bad or leading 181 00:14:05,760 --> 00:14:09,260 questions. This is one of the most important ones 182 00:14:09,260 --> 00:14:15,310 that you have to avoid. So, that's the first part 183 00:14:15,310 --> 00:14:20,910 of this chapter, assembling techniques. Do you 184 00:14:20,910 --> 00:14:24,990 have any questions? Next, we'll talk about 185 00:14:24,990 --> 00:14:30,050 assembling distributions. So far, up to this 186 00:14:30,050 --> 00:14:35,690 point. I mean, at the end of chapter 6, we 187 00:14:35,690 --> 00:14:40,840 discussed the probability, for example, of 188 00:14:40,840 --> 00:14:46,580 computing X greater than, for example, 7. For 189 00:14:46,580 --> 00:14:53,260 example, suppose X represents your score in 190 00:14:53,260 --> 00:14:55,020 business statistics course. 191 00:14:57,680 --> 00:15:02,680 And suppose we know that X is normally distributed 192 00:15:02,680 --> 00:15:10,860 with a mean of 80, standard deviation of 10. My 193 00:15:10,860 --> 00:15:15,360 question was, in chapter 6, what's the probability 194 00:15:15,360 --> 00:15:23,380 that the student scores more than 70? Suppose we 195 00:15:23,380 --> 00:15:26,720 select randomly one student, and the question is, 196 00:15:26,840 --> 00:15:29,980 what's the probability that his score, so just for 197 00 223 00:17:30,370 --> 00:17:33,890 score of the student. Now we have to use something 224 00:17:33,890 --> 00:17:38,470 other called x bar. I'm interested in the average 225 00:17:38,470 --> 00:17:46,770 of this. So x bar minus the mean of not x, x bar, 226 00:17:47,550 --> 00:17:52,550 then divided by sigma x bar. So this is my new, 227 00:17:53,270 --> 00:17:54,770 the score. 228 00:17:57,820 --> 00:18:00,680 Here, there are three questions. Number one, 229 00:18:03,680 --> 00:18:11,000 what's the shape of the distribution of X bar? So, 230 00:18:11,040 --> 00:18:13,340 we are asking about the shape of the distribution. 231 00:18:14,560 --> 00:18:19,290 It might be normal. If the entire population that 232 00:18:19,290 --> 00:18:22,390 we select a sample from is normal, I mean if the 233 00:18:22,390 --> 00:18:24,450 population is normally distributed, then you 234 00:18:24,450 --> 00:18:27,110 select a random sample of that population, it 235 00:18:27,110 --> 00:18:30,590 makes sense that the sample is also normal, so any 236 00:18:30,590 --> 00:18:33,030 statistic is computed from that sample is also 237 00:18:33,030 --> 00:18:35,810 normally distributed, so it makes sense. If the 238 00:18:35,810 --> 00:18:38,450 population is normal, then the shape is also 239 00:18:38,450 --> 00:18:43,650 normal. But if the population is unknown, you 240 00:18:43,650 --> 00:18:46,550 don't have any information about the underlying 241 00:18:46,550 --> 00:18:50,530 population, then you cannot say it's normal unless 242 00:18:50,530 --> 00:18:53,790 you have certain condition that we'll talk about 243 00:18:53,790 --> 00:18:57,510 maybe after 30 minutes. So, exactly, if the 244 00:18:57,510 --> 00:18:59,610 population is normal, then the shape is also 245 00:18:59,610 --> 00:19:01,910 normal, but otherwise, we have to think about it. 246 00:19:02,710 --> 00:19:06,890 This is the first question. Now, there are two 247 00:19:06,890 --> 00:19:11,910 unknowns in this equation. We have to know the 248 00:19:11,910 --> 00:19:17,980 mean, Or x bar, so the mean of x bar is not given, 249 00:19:18,520 --> 00:19:23,200 the mean means the center. So the second question, 250 00:19:23,440 --> 00:19:26,920 what's the center of the distribution? In this 251 00:19:26,920 --> 00:19:29,980 case, the mean of x bar. So we are looking at 252 00:19:29,980 --> 00:19:32,920 what's the mean of x bar. The third question is 253 00:19:32,920 --> 00:19:39,350 sigma x bar is also unknown, spread. Now shape, 254 00:19:39,850 --> 00:19:44,590 center, spread, these are characteristics, these 255 00:19:44,590 --> 00:19:47,770 characteristics in this case sampling 256 00:19:47,770 --> 00:19:50,490 distribution, exactly which is called sampling 257 00:19:50,490 --> 00:19:56,130 distribution. So by sampling distribution we mean 258 00:19:56,130 --> 00:20:00,110 that, by sampling distribution, we mean that you 259 00:20:00,110 --> 00:20:05,840 have to know the center of distribution, I mean 260 00:20:05,840 --> 00:20:08,760 the mean of the statistic you are interested in. 261 00:20:09,540 --> 00:20:14,620 Second, the spread or the variability of the 262 00:20:14,620 --> 00:20:17,600 sample statistic also you are interested in. In 263 00:20:17,600 --> 00:20:21,240 addition to that, you have to know the shape of 264 00:20:21,240 --> 00:20:25,080 the statistic. So three things we have to know, 265 00:20:25,820 --> 00:20:31,980 center, spread and shape. So that's what we'll 266 00:20:31,980 --> 00:20:37,340 talk about now. So now sampling distribution is a 267 00:20:37,340 --> 00:20:41,640 distribution of all of the possible values of a 268 00:20:41,640 --> 00:20:46,040 sample statistic. This sample statistic could be 269 00:20:46,040 --> 00:20:50,240 sample mean, could be sample variance, could be 270 00:20:50,240 --> 00:20:53,500 sample proportion, because any population has 271 00:20:53,500 --> 00:20:57,080 mainly three characteristics, mean, standard 272 00:20:57,080 --> 00:20:59,040 deviation, and proportion. 273 00:21:01,520 --> 00:21:04,260 So again, a sampling distribution is a 274 00:21:04,260 --> 00:21:07,400 distribution of all of the possible values of a 275 00:21:07,400 --> 00:21:11,960 sample statistic or a given size sample selected 276 00:21:11,960 --> 00:21:20,020 from a population. For example, suppose you sample 277 00:21:20,020 --> 00:21:23,420 50 students from your college regarding their mean 278 00:21:23,420 --> 00:21:30,640 GPA. GPA means Graduate Point Average. Now, if you 279 00:21:30,640 --> 00:21:35,280 obtain many different samples of size 50, you will 280 00:21:35,280 --> 00:21:38,760 compute a different mean for each sample. As I 281 00:21:38,760 --> 00:21:42,680 mentioned here, I select a sample the same sizes, 282 00:21:43,540 --> 00:21:47,580 but we obtain different sample statistics, I mean 283 00:21:47,580 --> 00:21:54,260 different sample means. We are interested in the 284 00:21:54,260 --> 00:21:59,760 distribution of all potential mean GBA We might 285 00:21:59,760 --> 00:22:04,040 calculate for any given sample of 50 students. So 286 00:22:04,040 --> 00:22:09,440 let's focus into these values. So we have again a 287 00:22:09,440 --> 00:22:14,580 random sample of 50 sample means. So we have 1, 2, 288 00:22:14,700 --> 00:22:18,480 3, 4, 5, maybe 50, 6, whatever we have. So select 289 00:22:18,480 --> 00:22:22,660 a random sample of size 20. Maybe we repeat this 290 00:22:22,660 --> 00:22:28,590 sample 10 times. So we end with 10. different 291 00:22:28,590 --> 00:22:31,650 values of the simple means. Now we have new ten 292 00:22:31,650 --> 00:22:38,130 means. Now the question is, what's the center of 293 00:22:38,130 --> 00:22:42,590 these values, I mean for the means? What's the 294 00:22:42,590 --> 00:22:46,250 spread of the means? And what's the shape of the 295 00:22:46,250 --> 00:22:50,310 means? So these are the mainly three questions. 296 00:22:53,510 --> 00:22:56,810 For example, let's get just simple example and 297 00:22:56,810 --> 00:23:04,370 that we have only population of size 4. In the 298 00:23:04,370 --> 00:23:09,950 real life, the population size is much bigger than 299 00:23:09,950 --> 00:23:14,970 4, but just for illustration. 300 00:23:17,290 --> 00:23:20,190 Because size 4, I mean if the population is 4, 301 00:23:21,490 --> 00:23:24,950 it's a small population. So we can take all the 302 00:23:24,950 --> 00:23:27,610 values and find the mean and standard deviation. 303 00:23:28,290 --> 00:23:31,430 But in reality, we have more than that. So this 304 00:23:31,430 --> 00:23:37,390 one just for as example. So let's suppose that we 305 00:23:37,390 --> 00:23:42,790 have a population of size 4. So n equals 4. 306 00:23:46,530 --> 00:23:54,030 And we are interested in the ages. And suppose the 307 00:23:54,030 --> 00:23:58,930 values of X, X again represents H, 308 00:24:00,690 --> 00:24:01,810 and the values we have. 309 00:24:06,090 --> 00:24:08,930 So these are the four values we have. 310 00:24:12,050 --> 00:24:16,910 Now simple calculation will 311 00:24:16,910 --> 00:24:19,910 give you the mean, the population mean. 312 00:24:25,930 --> 00:24:30,410 Just add these values and divide by the operation 313 00:24:30,410 --> 00:24:35,410 size, we'll get 21 years. And sigma, as we 314 00:24:35,410 --> 00:24:39,450 mentioned in chapter three, square root of this 315 00:24:39,450 --> 00:24:44,550 quantity will give 2.236 316 00:24:44,550 --> 00:24:49,450 years. So simple calculation will give these 317 00:24:49,450 --> 00:24:54,470 results. Now if you look at distribution of these 318 00:24:54,470 --> 00:24:58,430 values, Because as I mentioned, we are looking for 319 00:24:58,430 --> 00:25:03,810 center, spread, and shape. The center is 21 of the 320 00:25:03,810 --> 00:25:09,430 exact population. The variation is around 2.2. 321 00:25:10,050 --> 00:25:14,770 Now, the shape of distribution. Now, 18 represents 322 00:25:14,770 --> 00:25:15,250 once. 323 00:25:17,930 --> 00:25:22,350 I mean, we have only one 18, so 18 divided one 324 00:25:22,350 --> 00:25:29,830 time over 425. 20% represent also 25%, the same as 325 00:25:29,830 --> 00:25:33,030 for 22 or 24. In this case, we have something 326 00:25:33,030 --> 00:25:37,530 called uniform distribution. In this case, the 327 00:25:37,530 --> 00:25:43,330 proportions are the same. So, the mean, not 328 00:25:43,330 --> 00:25:48,030 normal, it's uniform distribution. The mean is 21, 329 00:25:48,690 --> 00:25:52,490 standard deviation is 2.236, and the distribution 330 00:25:52,490 --> 00:25:58,840 is uniform. Okay, so that's center, spread and 331 00:25:58,840 --> 00:26:02,920 shape of the true population we have. Now suppose 332 00:26:02,920 --> 00:26:03,520 for example, 333 00:26:06,600 --> 00:26:12,100 we select a random sample of size 2 from this 334 00:26:12,100 --> 00:26:12,620 population. 335 00:26:15,740 --> 00:26:21,500 So we select a sample of size 2. We have 18, 20, 336 00:26:21,600 --> 00:26:25,860 22, 24 years. We have four students, for example. 337 00:26:27,760 --> 00:26:31,140 And we select a sample of size two. So the first 338 00:26:31,140 --> 00:26:40,820 one could be 18 and 18, 18 and 20, 18 and 22. So 339 00:26:40,820 --> 00:26:47,400 we have 16 different samples. So number of samples 340 00:26:47,400 --> 00:26:54,500 in this case is 16. Imagine that we have five. I 341 00:26:54,500 --> 00:27:00,220 mean the population size is 5 and so on. So the 342 00:27:00,220 --> 00:27:06,000 rule is number 343 00:27:06,000 --> 00:27:13,020 of samples in this case and the volume is million. 344 00:27:14,700 --> 00:27:19,140 Because we have four, four squared is sixteen, 345 00:27:19,440 --> 00:27:26,940 that's all. 5 squared, 25, and so on. Now, we have 346 00:27:26,940 --> 00:27:31,740 16 different samples. For sure, we will have 347 00:27:31,740 --> 00:27:37,940 different sample means. Now, for the first sample, 348 00:27:39,560 --> 00:27:47,200 18, 18, the average is also 18. The next one, 18, 349 00:27:47,280 --> 00:27:50,040 20, the average is 19. 350 00:27:54,790 --> 00:27:59,770 20, 18, 24, the average is 21, and so on. So now 351 00:27:59,770 --> 00:28:05,450 we have 16 sample means. Now this is my new 352 00:28:05,450 --> 00:28:10,510 values. It's my sample. This sample has different 353 00:28:10,510 --> 00:28:16,050 sample means. Now let's take these values and 354 00:28:16,050 --> 00:28:23,270 compute average, sigma, and the shape of the 355 00:28:23,270 --> 00:28:29,200 distribution. So again, we have a population of 356 00:28:29,200 --> 00:28:35,240 size 4, we select a random cell bone. of size 2 357 00:28:35,240 --> 00:28:39,060 from that population, we end with 16 random 358 00:28:39,060 --> 00:28:43,620 samples, and they have different sample means. 359 00:28:43,860 --> 00:28:46,700 Might be two of them are the same. I mean, we have 360 00:28:46,700 --> 00:28:52,220 18 just repeated once, but 19 repeated twice, 23 361 00:28:52,220 --> 00:28:59,220 times, 24 times, and so on. 22 three times, 23 362 00:28:59,220 --> 00:29:04,270 twice, 24 once. So it depends on The sample means 363 00:29:04,270 --> 00:29:07,210 you have. So we have actually different samples. 364 00:29:14,790 --> 00:29:18,970 For example, let's look at 24 and 22. What's the 365 00:29:18,970 --> 00:29:22,790 average of these two values? N divided by 2 will 366 00:29:22,790 --> 00:29:24,290 give 22. 367 00:29:33,390 --> 00:29:35,730 So again, we have 16 sample means. 368 00:29:38,610 --> 00:29:41,550 Now look first at the shape of the distribution. 369 00:29:43,110 --> 00:29:47,490 18, as I mentioned, repeated once. So 1 over 16. 370 00:29:48,430 --> 00:29:57,950 19 twice. 23 times. 1 four times. 22 three times. 371 00:29:58,940 --> 00:30:03,340 then twice then once now the distribution was 372 00:30:03,340 --> 00:30:07,960 uniform remember now it becomes normal 373 00:30:07,960 --> 00:30:10,780 distribution so the first one x1 is normal 374 00:30:10,780 --> 00:30:16,340 distribution so it has normal distribution so 375 00:30:16,340 --> 00:30:20,040 again the shape of x1 looks like normal 376 00:30:20,040 --> 00:30:26,800 distribution we need to compute the center of X 377 00:30:26,800 --> 00:30:32,800 bar, the mean of X bar. We have to add the values 378 00:30:32,800 --> 00:30:36,380 of X bar, the sample mean, then divide by the 379 00:30:36,380 --> 00:30:42,800 total number of size, which is 16. So in this 380 00:30:42,800 --> 00:30:51,720 case, we got 21, which is similar to the one for 381 00:30:51,720 --> 00:30:55,950 the entire population. So this is the first 382 00:30:55,950 --> 00:30:59,930 unknown parameter. The mu of x bar is the same as 383 00:30:59,930 --> 00:31:05,490 the population mean mu. The second one, the split 384 00:31:05,490 --> 00:31:13,450 sigma of x bar by using the same equation 385 00:31:13,450 --> 00:31:17,170 we have, sum of x bar in this case minus the mean 386 00:31:17,170 --> 00:31:21,430 of x bar squared, then divide this quantity by the 387 00:31:21,430 --> 00:31:26,270 capital I which is 16 in this case. So we will end 388 00:31:26,270 --> 00:31:28,510 with 1.58. 389 00:31:31,270 --> 00:31:36,170 Now let's compare population standard deviation 390 00:31:36,170 --> 00:31:42,210 and the sample standard deviation. First of all, 391 00:31:42,250 --> 00:31:45,050 you see that these two values are not the same. 392 00:31:47,530 --> 00:31:50,370 The population standard deviation was 2.2, around 393 00:31:50,370 --> 00:31:57,310 2.2. But for the sample, for the sample mean, it's 394 00:31:57,310 --> 00:32:02,690 1.58, so that means sigma of X bar is smaller than 395 00:32:02,690 --> 00:32:03,710 sigma of X. 396 00:32:07,270 --> 00:32:12,010 It means exactly, the variation of X bar is always 397 00:32:12,010 --> 00:32:15,770 smaller than the variation of X, always. 398 00:32:20,420 --> 00:32:26,480 So here is the comparison. The distribution was 399 00:32:26,480 --> 00:32:32,000 uniform. It's no longer uniform. It looks like a 400 00:32:32,000 --> 00:32:36,440 bell shape. The mean of X is 21, which is the same 401 00:32:36,440 --> 00:32:40,440 as the mean of X bar. But the standard deviation 402 00:32:40,440 --> 00:32:44,200 of the population is larger than the standard 403 00:32:44,200 --> 00:32:48,060 deviation of the sample mean or the average. 404 00:32:53,830 --> 00:32:58,090 Different samples of the same sample size from the 405 00:32:58,090 --> 00:33:00,790 same population will yield different sample means. 406 00:33:01,450 --> 00:33:06,050 We know that. If we have a population and from 407 00:33:06,050 --> 00:33:08,570 that population, so we have this big population, 408 00:33:10,250 --> 00:33:15,010 from this population suppose we selected 10 409 00:33:15,010 --> 00:33:19,850 samples, sample 1 with size 50. 410 00:33:21,540 --> 00:33:26,400 Another sample, sample 2 with the same size. All 411 00:33:26,400 --> 00:33:29,980 the way, suppose we select 10 samples, sample 10, 412 00:33: 445 00:36:23,500 --> 00:36:28,660 smaller than sigma of the standard deviation of 446 00:36:28,660 --> 00:36:33,180 normalization. Now if you look at the relationship 447 00:36:33,180 --> 00:36:36,380 between the standard error of X bar and the sample 448 00:36:36,380 --> 00:36:41,760 size, we'll see that as the sample size increases, 449 00:36:42,500 --> 00:36:46,440 sigma of X bar decreases. So if we have large 450 00:36:46,440 --> 00:36:51,200 sample size, I mean instead of selecting a random 451 00:36:51,200 --> 00:36:53,520 sample of size 2, if you select a random sample of 452 00:36:53,520 --> 00:36:56,900 size 3 for example, you will get sigma of X bar 453 00:36:56,900 --> 00:37:03,140 less than 1.58. So note that standard error of the 454 00:37:03,140 --> 00:37:09,260 mean decreases as the sample size goes up. So as n 455 00:37:09,260 --> 00:37:13,000 increases, sigma of x bar goes down. So there is 456 00:37:13,000 --> 00:37:17,440 an inverse relationship between the standard error of 457 00:37:17,440 --> 00:37:21,900 the mean and the sample size. So now we answered 458 00:37:21,900 --> 00:37:24,660 the three questions. The shape looks like a bell 459 00:37:24,660 --> 00:37:31,290 shape. If we select our sample from a normal 460 00:37:31,290 --> 00:37:37,850 population with a mean equal to the population mean 461 00:37:37,850 --> 00:37:40,530 and standard deviation of the standard error equals 462 00:37:40,530 --> 00:37:48,170 sigma over the square root of n. So now, let's talk 463 00:37:48,170 --> 00:37:53,730 about the sampling distribution of the sample mean if 464 00:37:53,730 --> 00:37:59,170 the population is normal. So now, my population is 465 00:37:59,170 --> 00:38:03,830 normally distributed, and we are interested in the 466 00:38:03,830 --> 00:38:06,430 sampling distribution of the sample mean of X bar. 467 00:38:07,630 --> 00:38:11,330 If the population is normally distributed with 468 00:38:11,330 --> 00:38:14,870 mean mu and standard deviation sigma, in this 469 00:38:14,870 --> 00:38:18,590 case, the sampling distribution of X bar is also 470 00:38:18,590 --> 00:38:22,870 normally distributed, so this is the shape. With 471 00:38:22,870 --> 00:38:27,070 the mean of X bar equals mu and sigma of X bar equals 472 00:38:27,070 --> 00:38:35,470 sigma over the square root of n. So again, if we sample from a normal 473 00:38:35,470 --> 00:38:39,650 population, I mean my sampling technique, I select 474 00:38:39,650 --> 00:38:44,420 a random sample from a normal population. Then if 475 00:38:44,420 --> 00:38:47,640 we are interested in the standard distribution of 476 00:38:47,640 --> 00:38:51,960 X bar, then that distribution is normally 477 00:38:51,960 --> 00:38:56,000 distributed with a mean equal to mu and standard 478 00:38:56,000 --> 00:39:02,540 deviation sigma over mu. So that's the shape. It's 479 00:39:02,540 --> 00:39:05,670 normal. The mean is the same as the population 480 00:39:05,670 --> 00:39:09,030 mean, and the standard deviation of x bar equals 481 00:39:09,030 --> 00:39:16,130 sigma over the square root of n. So now let's go back to the z 482 00:39:16,130 --> 00:39:21,130 -score we discussed before. If you remember, I 483 00:39:21,130 --> 00:39:25,150 mentioned before 484 00:39:25,150 --> 00:39:32,720 that the z-score, generally speaking, is X minus the mean 485 00:39:32,720 --> 00:39:34,740 of X divided by sigma X. 486 00:39:37,640 --> 00:39:41,620 And we know that Z has a standard normal 487 00:39:41,620 --> 00:39:48,020 distribution with a mean of zero and a variance of one. In 488 00:39:48,020 --> 00:39:52,860 this case, we are looking for the sampling 489 00:39:52,860 --> 00:39:59,350 -distribution of X bar. So Z equals X bar. minus 490 00:39:59,350 --> 00:40:06,050 the mean of x bar divided by sigma of x bar. So 491 00:40:06,050 --> 00:40:10,770 the same equation, but different statistics. In the 492 00:40:10,770 --> 00:40:15,770 first one, we have x, for example, which represents the 493 00:40:15,770 --> 00:40:20,370 score. Here, my sample statistic is the sample 494 00:40:20,370 --> 00:40:22,890 mean, which represents the average of the scores. 495 00:40:23,470 --> 00:40:29,460 So x bar, minus its mean, I mean the mean of x 496 00:40:29,460 --> 00:40:37,280 bar, divided by its standard error. So x bar minus 497 00:40:37,280 --> 00:40:41,000 the mean of x bar divided by sigma of x bar. By 498 00:40:41,000 --> 00:40:48,020 using that mu of x bar equals mu, and sigma of x 499 00:40:48,020 --> 00:40:51,240 bar equals sigma over the square root of n, we will end with 500 00:40:51,240 --> 00:40:52,600 this equation z square. 501 00:40:56,310 --> 00:41:00,790 So this equation will be used instead of using the 502 00:41:00,790 --> 00:41:04,650 previous one. So z square equals sigma, I'm sorry, 503 00:41:04,770 --> 00:41:08,470 z equals x bar minus the mean divided by sigma 504 00:41:08,470 --> 00:41:13,310 bar, where x bar is the sample mean, mu is the 505 00:41:13,310 --> 00:41:15,990 population mean, sigma is the population standard 506 00:41:15,990 --> 00:41:19,810 deviation, and n is the sample size. So that's the 507 00:41:19,810 --> 00:41:22,490 difference between chapter six, 508 00:41:25,110 --> 00:41:32,750 and that one we have only x minus y by sigma. Here 509 00:41:32,750 --> 00:41:36,450 we are interested in x bar minus the mean of x bar 510 00:41:36,450 --> 00:41:40,290 which is mu. And sigma of x bar equals sigma over the square root of n. 511 00:41:47,970 --> 00:41:52,010 Now when we are saying that mu of x bar equals mu, 512 00:41:54,530 --> 00:42:01,690 That means the expected value of 513 00:42:01,690 --> 00:42:05,590 the sample mean equals the population mean. When 514 00:42:05,590 --> 00:42:08,610 we are saying mean of X bar equals mu, it means 515 00:42:08,610 --> 00:42:13,270 the expected value of X bar equals mu. In other 516 00:42:13,270 --> 00:42:20,670 words, the expectation of X bar equals mu. If this 517 00:42:20,670 --> 00:42:27,900 happens, we say that X bar is an unbiased 518 00:42:27,900 --> 00:42:31,420 estimator 519 00:42:31,420 --> 00:42:35,580 of 520 00:42:35,580 --> 00:42:40,620 mu. So this is a new definition, an unbiased 521 00:42:40,620 --> 00:42:45,490 estimator X bar is called an unbiased estimator if 522 00:42:45,490 --> 00:42:49,410 this condition is satisfied. I mean, if the mean 523 00:42:49,410 --> 00:42:54,450 of X bar or if the expected value of X bar equals 524 00:42:54,450 --> 00:42:57,790 the population mean, in this case, we say that X 525 00:42:57,790 --> 00:43:02,450 bar is a good estimator of Mu. Because on average, 526 00:43:05,430 --> 00:43:08,230 The expected value of X bar equals the population 527 00:43:08,230 --> 00:43:14,970 mean, so in this case, X bar is a good estimator of 528 00:43:14,970 --> 00:43:20,410 Mu. Now if you compare the two distributions, 529 00:43:22,030 --> 00:43:27,510 a normal distribution here with the population mean Mu 530 00:43:27,510 --> 00:43:30,550 and a standard deviation for example sigma. 531 00:43:33,190 --> 00:43:40,590 That's for the scores, the scores. Now instead of 532 00:43:40,590 --> 00:43:43,690 the scores above, we have x bar, the sample mean. 533 00:43:44,670 --> 00:43:48,590 Again, the mean of x bar is the same as the 534 00:43:48,590 --> 00:43:52,990 population mean. Both means are the same, mu of x 535 00:43:52,990 --> 00:43:57,130 bar equals mu. But if you look at the spread of 536 00:43:57,130 --> 00:44:00,190 the second distribution, it is more than the 537 00:44:00,190 --> 00:44:03,350 other one. So that's the comparison between the 538 00:44:03,350 --> 00:44:05,530 two populations. 539 00:44:07,050 --> 00:44:13,390 So again, to compare or to figure out the 540 00:44:13,390 --> 00:44:17,910 relationship between sigma of x bar and the sample 541 00:44:17,910 --> 00:44:22,110 size. Suppose we have this blue normal 542 00:44:22,110 --> 00:44:28,590 distribution with a sample size of say 10 or 30, for 543 00:44:28,590 --> 00:44:28,870 example. 544 00:44:32,220 --> 00:44:37,880 As n gets bigger and bigger, sigma of x bar 545 00:44:37,880 --> 00:44:41,800 becomes smaller and smaller. If you look at the 546 00:44:41,800 --> 00:44:44,760 red one, maybe if the red one has n equal to 100, 547 00:44:45,700 --> 00:44:48,780 we'll get this spread. But for the other one, we 548 00:44:48,780 --> 00:44:55,240 have a larger spread. So as n increases, sigma of x 549 00:44:55,240 --> 00:44:59,860 bar decreases. So this, the blue one for a smaller 550 00:44:59,860 --> 00:45:06,240 sample size. The red one for a larger sample size. 551 00:45:06,840 --> 00:45:11,120 So again, as n increases, sigma of x bar goes down 552 00:45:11,120 --> 00:45:12,040 four degrees. 553 00:45:21,720 --> 00:45:29,480 Next, let's use this fact to 554 00:45:29,480 --> 00:45:37,440 figure out an interval for the sample mean with 90 555 00:45:37,440 --> 00:45:42,140 % confidence and suppose the population we have is 556 00:45:42,140 --> 00:45:49,500 normal with a mean of 368 and sigma of 15 and suppose 557 00:45:49,500 --> 00:45:52,900 we select a random sample of a size of 25 and the question 558 00:45:52,900 --> 00:45:57,600 is find symmetrically distributed interval around 559 00:45:57,600 --> 00:46:03,190 the mean that will include 95% of the sample means 560 00:46:03,190 --> 00:46:08,610 when mu equals 368, sigma is 15, and your sample 561 00:46:08,610 --> 00:46:13,830 size is 25. So in this case, we are looking for 562 00:46:13,830 --> 00:46:17,150 the 563 00:46:17,150 --> 00:46:19,110 estimation of the sample mean. 564 00:46:23,130 --> 00:46:24,970 And we have this information, 565 00:46:28,910 --> 00:46:31,750 Sigma is 15 and N is 25. 566 00:46:35,650 --> 00:46:38,890 The problem mentioned there, we have a symmetric 567 00:46:38,890 --> 00:46:48,490 distribution and this area is 95% bisymmetric and 568 00:46:48,490 --> 00:46:52,890 we have only 5% out. So that means half to the 569 00:46:52,890 --> 00:46:56,490 right and half to the left. 570 00:46:59,740 --> 00:47:02,640 And let's see how we can compute these two values. 571 00:47:03,820 --> 00:47:11,440 The problem says that the average is 368 572 00:47:11,440 --> 00:47:18,660 for this data and the standard deviation sigma of 573 00:47:18,660 --> 00:47:28,510 15. He asked about what are the values of x bar. I 574 00:47:28,510 --> 00:47:32,430 mean, we have to find the interval of x bar. Let's 575 00:47:32,430 --> 00:47:36,130 see. If you remember last time, z score was x 576 00:47:36,130 --> 00:47:41,130 minus mu divided by sigma. But now we have x bar. 577 00:47:41,890 --> 00:47:45,850 So your z score should be x bar minus mu divided by 578 00:47:45,850 --> 00:47:50,850 sigma over the square root of n. Now cross multiplication, you 579 00:47:50,850 --> 00:47:55,970 will get x bar minus mu equals z sigma over the square root 580 00:47:55,970 --> 00:48:01,500 of n. That means x bar equals mu plus z sigma over 581 00:48:01,500 --> 00:48:04,440 the square root of n. Exactly the same equation we got in 582 00:48:04,440 --> 00:48:09,840 chapter six, but there, in that one, we have x 583 00:48:09,840 --> 00:48:13,700 equals mu plus z sigma. Now we have x bar equals 584 00:48:13,700 --> 00:48:18,200 mu plus z sigma over the square root of n, because we have 585 00:48:18,200 --> 00:48:23,000 different statistics. It's x bar instead of x. Now 586 00:48:23,000 --> 00:48:28,510 we are looking for these two values. Now let's 587 00:48:28,510 --> 00:48:29,410 compute z-score. 588 00:48:32,450 --> 00:48:36,830 The z-score for this point, which has an area of 2.5% 589 00:48:36,830 --> 00:48:41,930 below it, is the same as the z-score, but in the 590 00:48:41,930 --> 00:48:48,670 opposite direction. If you remember, we got this 591 00:48:48,670 --> 00:48:49,630 value, 1.96. 592 00:48:52,790 --> 00:48:58,080 So my z-score is negative 1.96 to the left. and 1 593 00:48:58,080 --> 00:49:08,480 .9621 so now my x bar in the lower limit in this 594 00:49:08,480 --> 00:49:17,980 side on the left side equals mu which is 368 minus 595 00:49:17,980 --> 00:49:29,720 1.96 times sigma which is 15 divide by the square root of 25. 596 00:49:30,340 --> 00:49:34,980 So that's the value of the sample mean in the 597 00:49:34,980 --> 00:49:39,740 lower limit, or lower bound. On the other hand, 598 00:49:42,320 --> 00:49:49,720 expand our limit to the other hand equals 316 plus 1.96 599 00:49:49,720 --> 00:49:56,100 sigma over the square root of n. Simple calculation will give this 600 00:49:56,100 --> 00:49:56,440 result. 601 00:49:59,770 --> 00:50:06,870 The first X bar for the lower limit is 362.12, the 602 00:50:06,870 --> 00:50:10,050 other is 373.1. 603 00:50:11,450 --> 00:50:17,170 So again for this data, for this example, the mean 604 00:50:17,170 --> 00:50:23,030 was, the population mean was 368, the population 605 00:50:23,030 --> 00:50:26,310 has a standard deviation of 15, we select a random 606 00:50:26,310 --> 00:50:31,070 sample of size 25, Then we end with this result 607 00:50:31,070 --> 00:50:41,110 that 95% of all sample means of sample size 25 are 608 00:50:41,110 --> 00:50:44,810 between these two values. It means that we have 609 00:50:44,810 --> 00:50:49,530 this big population and this population is 610 00:50:49,530 --> 00:50:55,240 symmetric, it's normal. And we know that The mean of 611 00:50:55,240 --> 00:51:00,680 this population is 368 with a sigma of 15. 612 00:51:02,280 --> 00:51:08,320 We select from this population many samples. Each 613 00:51:08,320 --> 00:51:11,600 one has a size of 25. 614 00:51:15,880 --> 00:51:20,940 Suppose, for example, we select 100 samples, 100 615 00:51:20,940 --> 00:51:27,260 random samples. So we end with different sample 616 00:51:27,260 --> 00:51:27,620 means. 617 00:51:33,720 --> 00:51:39,820 So we have 100 new sample means. In this case, you 618 00:51:39,820 --> 00:51:46,320 can say that 95 out of these, 95 out of 100, it 619 00:51:46,320 --> 00:51:52,560 means 95, one of these sample means. have values 620 00:51:52,560 --> 00:52:01,720 between 362.12 and 373.5. And what's remaining? 621 00:52:03,000 --> 00:52:07,940 Just five of these sample means would be out of 622 00:52:07,940 --> 00:52:13,220 this interval either below 362 or above the upper 623 00:52:13,220 --> 00:52:17,720 limit. So you are 95% sure that 624 00:52:21,230 --> 00:52:24,350 the sample mean lies between these two points. 625 00:52:25,410 --> 00:52:29,470 So, 5% of the sample means will be out. Make 626 00:52:29,470 --> 00:52:37,510 sense? Imagine that I have selected 200 samples. 627 00:52:40,270 --> 00:52:46,330 Now, how many X bar will be between these two 628 00:52:46,330 --> 00:52:54,140 values? 95% of these 200. So how many 95%? How 629 00:52:54,140 --> 00:52:56,060 many means in this case? 630 00:52:58,900 --> 00:53:04,600 95% out of 200 is 190. 631 00:53:05,480 --> 00:53:12,200 190. Just multiply. 95 multiplies by 200. It will 632 00:53:12,200 --> 00:53:13,160 give you 190. 633 00:53:22,740 --> 00:53:29,860 values between 362 667 00:56:00,160 --> 00:56:03,640 larger and larger, or gets larger and larger, then 668 00:56:03,640 --> 00:56:06,860 the standard distribution of X bar is 669 00:56:06,860 --> 00:56:14,090 approximately normal in this. Again, look at the 670 00:56:14,090 --> 00:56:19,630 blue curve. Now, this one looks like skewed 671 00:56:19,630 --> 00:56:20,850 distribution to the right. 672 00:56:24,530 --> 00:56:28,730 Now, as the sample gets large enough, then it 673 00:56:28,730 --> 00:56:33,470 becomes normal. So, the sample distribution 674 00:56:33,470 --> 00:56:37,350 becomes almost normal regardless of the shape of 675 00:56:37,350 --> 00:56:41,570 the population. I mean if you sample from unknown 676 00:56:41,570 --> 00:56:46,590 population, and that one has either right skewed 677 00:56:46,590 --> 00:56:52,130 or left skewed, if the sample size is large, then 678 00:56:52,130 --> 00:56:55,810 the sampling distribution of X bar becomes almost 679 00:56:55,810 --> 00:57:01,530 normal distribution regardless of the… so that’s 680 00:57:01,530 --> 00:57:06,830 the central limit theorem. So again, if the 681 00:57:06,830 --> 00:57:10,980 population is not normal, The condition is only 682 00:57:10,980 --> 00:57:15,360 you have to select a large sample. In this case, 683 00:57:15,960 --> 00:57:19,340 the central tendency mu of X bar is same as mu. 684 00:57:20,000 --> 00:57:24,640 The variation is also sigma over root N. 685 00:57:28,740 --> 00:57:32,120 So again, standard distribution of X bar becomes 686 00:57:32,120 --> 00:57:38,620 normal as N. The theorem again says If we select a 687 00:57:38,620 --> 00:57:42,500 random sample from unknown population, then the 688 00:57:42,500 --> 00:57:44,560 standard distribution of X part is approximately 689 00:57:44,560 --> 00:57:53,580 normal as long as N gets large enough. Now the 690 00:57:53,580 --> 00:57:57,100 question is how large is large enough? 691 00:58:00,120 --> 00:58:06,530 There are two cases, or actually three cases. For 692 00:58:06,530 --> 00:58:11,310 most distributions, if you don’t know the exact 693 00:58:11,310 --> 00:58:18,670 shape, n above 30 is enough to use or to apply 694 00:58:18,670 --> 00:58:22,290 that theorem. So if n is greater than 30, it will 695 00:58:22,290 --> 00:58:24,650 give a standard distribution that is nearly 696 00:58:24,650 --> 00:58:29,070 normal. So if my n is large, it means above 30, or 697 00:58:29,070 --> 00:58:33,450 30 and above this. For fairly symmetric 698 00:58:33,450 --> 00:58:35,790 distribution, I mean for nearly symmetric 699 00:58:35,790 --> 00:58:38,630 distribution, the distribution is not exactly 700 00:58:38,630 --> 00:58:42,910 normal, but approximately normal. In this case, N 701 00:58:42,910 --> 00:58:46,490 to be large enough if it is above 15. So, N 702 00:58:46,490 --> 00:58:48,770 greater than 15 will usually have same 703 00:58:48,770 --> 00:58:50,610 distribution as almost normal. 704 00:58:55,480 --> 00:58:57,840 For normal population, as we mentioned, of 705 00:58:57,840 --> 00:59:00,740 distributions, the semantic distribution of the 706 00:59:00,740 --> 00:59:02,960 mean is always. 707 00:59:06,680 --> 00:59:12,380 Okay, so again, there are three cases. For most 708 00:59:12,380 --> 00:59:16,280 distributions, N to be large, above 30. In this 709 00:59:16,280 --> 00:59:20,460 case, the distribution is nearly normal. For 710 00:59:20,460 --> 00:59:24,300 fairly symmetric distributions, N above 15 gives 711 00:59:24,660 --> 00:59:28,960 almost normal distribution. But if the population 712 00:59:28,960 --> 00:59:32,400 by itself is normally distributed, always the 713 00:59:32,400 --> 00:59:35,800 sample mean is normally distributed. So that’s the 714 00:59:35,800 --> 00:59:37,300 three cases. 715 00:59:40,040 --> 00:59:47,480 Now for this example, suppose we have a 716 00:59:47,480 --> 00:59:49,680 population. It means we don’t know the 717 00:59:49,680 --> 00:59:52,900 distribution of that population. And that 718 00:59:52,900 --> 00:59:57,340 population has mean of 8. Standard deviation of 3. 719 00:59:58,200 --> 01:00:01,200 And suppose a random sample of size 36 is 720 01:00:01,200 --> 01:00:04,780 selected. In this case, the population is not 721 01:00:04,780 --> 01:00:07,600 normal. It says A population, so you don’t know 722 01:00:07,600 --> 01:00:12,340 the exact distribution. But N is large. It’s above 723 01:00:12,340 --> 01:00:15,060 30, so you can apply the central limit theorem. 724 01:00:15,920 --> 01:00:20,380 Now we ask about what’s the probability that a 725 01:00:20,380 --> 01:00:25,920 sample means. is between what’s the probability 726 01:00:25,920 --> 01:00:29,240 that the same element is between these two values. 727 01:00:32,180 --> 01:00:36,220 Now, the difference between this lecture and the 728 01:00:36,220 --> 01:00:39,800 previous ones was, here we are interested in the 729 01:00:39,800 --> 01:00:44,440 exponent of X. Now, even if the population is not 730 01:00:44,440 --> 01:00:47,080 normally distributed, the central limit theorem 731 01:00:47,080 --> 01:00:51,290 can be abused because N is large enough. So now, 732 01:00:51,530 --> 01:00:57,310 the mean of X bar equals mu, which is eight, and 733 01:00:57,310 --> 01:01:02,170 sigma of X bar equals sigma over root N, which is 734 01:01:02,170 --> 01:01:07,150 three over square root of 36, which is one-half. 735 01:01:11,150 --> 01:01:17,210 So now, the probability of X bar greater than 7.8, 736 01:01:17,410 --> 01:01:21,890 smaller than 8.2, Subtracting U, then divide by 737 01:01:21,890 --> 01:01:26,210 sigma over root N from both sides, so 7.8 minus 8 738 01:01:26,210 --> 01:01:30,130 divided by sigma over root N. Here we have 8.2 739 01:01:30,130 --> 01:01:33,230 minus 8 divided by sigma over root N. I will end 740 01:01:33,230 --> 01:01:38,150 with Z between minus 0.4 and 0.4. Now, up to this 741 01:01:38,150 --> 01:01:43,170 step, it’s in U, for chapter 7. Now, Z between 742 01:01:43,170 --> 01:01:47,630 minus 0.4 up to 0.4, you have to go back. And use 743 01:01:47,630 --> 01:01:51,030 the table in chapter 6, you will end with this 744 01:01:51,030 --> 01:01:54,530 result. So the only difference here, you have to 745 01:01:54,530 --> 01:01:55,790 use sigma over root N.