1 00:00:06,760 --> 00:00:10,320 Eventually I will give some practice problem for 2 00:00:10,320 --> 00:00:15,040 chapter eight. Generally speaking, there are three 3 00:00:15,040 --> 00:00:19,800 types of questions. The first type, multiple 4 00:00:19,800 --> 00:00:22,940 choice, so MCQ questions. 5 00:00:36,250 --> 00:00:41,490 The other type of problems will be true or false. 6 00:00:42,890 --> 00:00:54,870 Part B, Part C, three response problems. 7 00:00:56,510 --> 00:01:00,210 So three types of questions. Multiple choice, we 8 00:01:00,210 --> 00:01:03,130 have four answers. You have to select correct one. 9 00:01:06,060 --> 00:01:11,660 True or false problems. And the last part, free 10 00:01:11,660 --> 00:01:14,800 response problems. Here we'll talk about one of 11 00:01:14,800 --> 00:01:20,740 these. I will cover multiple choice questions as 12 00:01:20,740 --> 00:01:24,200 well as true and false. Let's start with number 13 00:01:24,200 --> 00:01:30,200 one for multiple choice. The width of a confidence 14 00:01:30,200 --> 00:01:36,050 interval estimate for a proportion will be Here we 15 00:01:36,050 --> 00:01:39,690 are talking about the width of a confidence 16 00:01:39,690 --> 00:01:40,230 interval. 17 00:01:43,070 --> 00:01:48,770 Estimates for a proportion will be narrower for 99 18 00:01:48,770 --> 00:01:56,180 % confidence than for a 9%. For 95 confidence? No, 19 00:01:56,280 --> 00:01:59,120 because as we know that as the confidence level 20 00:01:59,120 --> 00:02:03,120 increases, the width becomes wider. So A is 21 00:02:03,120 --> 00:02:10,440 incorrect. Is this true? B. Wider for sample size 22 00:02:10,440 --> 00:02:14,900 of 100 than for a sample size of 50? False, 23 00:02:15,020 --> 00:02:20,400 because as sample size increases, The sampling 24 00:02:20,400 --> 00:02:23,600 error goes down. That means the width of the 25 00:02:23,600 --> 00:02:28,700 interval becomes smaller and smaller. Yes, for N. 26 00:02:30,300 --> 00:02:37,100 Part C. Normal for 90% confidence, then for 95% 27 00:02:37,100 --> 00:02:40,620 confidence. That's correct. So C is the correct 28 00:02:40,620 --> 00:02:43,640 answer. Part D. Normal when the sampling 29 00:02:43,640 --> 00:02:49,100 proportion is 50%. is incorrect because if we have 30 00:02:49,100 --> 00:02:52,920 smaller than 50%, we'll get smaller confidence, 31 00:02:53,320 --> 00:02:56,620 smaller weight of the confidence. So C is the 32 00:02:56,620 --> 00:03:02,720 correct answer. Any question? So C is the correct 33 00:03:02,720 --> 00:03:06,180 answer because as C level increases, the 34 00:03:06,180 --> 00:03:08,760 confidence interval becomes narrower. 35 00:03:11,040 --> 00:03:12,920 Let's move to the second one. 36 00:03:16,540 --> 00:03:19,900 A 99% confidence interval estimate can be 37 00:03:19,900 --> 00:03:23,940 interpreted to mean that. Let's look at the 38 00:03:23,940 --> 00:03:28,040 interpretation of the 99% confidence interval. 39 00:03:28,940 --> 00:03:29,660 Part eight. 40 00:03:33,360 --> 00:03:38,820 If all possible samples are taken and confidence 41 00:03:38,820 --> 00:03:43,070 interval estimates are developed, 99% of them 42 00:03:43,070 --> 00:03:45,750 would include the true population mean somewhere 43 00:03:45,750 --> 00:03:46,790 within their interval. 44 00:03:49,750 --> 00:03:53,210 Here we are talking about the population mean. It 45 00:03:53,210 --> 00:03:57,890 says that 99% of them of these intervals would 46 00:03:57,890 --> 00:04:00,790 include the true population mean somewhere within 47 00:04:00,790 --> 00:04:05,490 their interval. It's correct. Why false? Why is it 48 00:04:05,490 --> 00:04:10,620 false? This is correct answer, because it's 49 00:04:10,620 --> 00:04:15,240 mentioned that 99% of these confidence intervals 50 00:04:15,240 --> 00:04:19,600 will contain the true population mean somewhere 51 00:04:19,600 --> 00:04:22,900 within their interval. So A is correct. Let's look 52 00:04:22,900 --> 00:04:30,880 at B. B says we have 99% confidence that we have 53 00:04:30,880 --> 00:04:34,160 selected a sample whose interval does include the 54 00:04:34,160 --> 00:04:39,160 population mean. Also, this one is correct. Again, 55 00:04:39,300 --> 00:04:42,540 it's mentioned that 99% confidence that we have 56 00:04:42,540 --> 00:04:47,080 selected sample whose interval does include. So 57 00:04:47,080 --> 00:04:52,600 it's correct. So C is both of the above and D none 58 00:04:52,600 --> 00:04:54,840 of the above. So C is the correct answer. So 59 00:04:54,840 --> 00:04:59,080 sometimes maybe there is only one answer. Maybe in 60 00:04:59,080 --> 00:05:03,360 other problems, it might be two answers are 61 00:05:03,360 --> 00:05:07,150 correct. So for this one, B and C. I'm sorry, A 62 00:05:07,150 --> 00:05:11,790 and B are correct, so C is the correct answer. 63 00:05:14,270 --> 00:05:17,530 Number three, which of the following is not true 64 00:05:17,530 --> 00:05:20,610 about the student's T distribution? Here, we are 65 00:05:20,610 --> 00:05:25,550 talking about the not true statement about the 66 00:05:25,550 --> 00:05:31,110 student T distribution, A. It has more data in the 67 00:05:31,110 --> 00:05:35,780 tails. and less in the center than does the normal 68 00:05:35,780 --> 00:05:38,580 distribution. That's correct because we mentioned 69 00:05:38,580 --> 00:05:44,160 last time that T distribution, the tail is fatter 70 00:05:44,160 --> 00:05:48,000 than the Z normal. So that means it has more data 71 00:05:48,000 --> 00:05:52,100 in the tails and less data in the center. So 72 00:05:52,100 --> 00:05:52,720 that's correct. 73 00:05:58,000 --> 00:06:01,020 It is used to construct confidence intervals for 74 00:06:01,020 --> 00:06:03,220 the population mean when the population standard 75 00:06:03,220 --> 00:06:07,400 deviation is known. No, we use z instead of t, so 76 00:06:07,400 --> 00:06:11,680 this one is incorrect about t. It is well-shaped 77 00:06:11,680 --> 00:06:17,320 and symmetrical, so that's true, so we are looking 78 00:06:17,320 --> 00:06:21,900 for the incorrect statement. D, as the number of 79 00:06:21,900 --> 00:06:23,260 degrees of freedom increases, 80 00:06:25,850 --> 00:06:31,510 The T distribution approaches the normal. That's 81 00:06:31,510 --> 00:06:36,830 true. So which one? P. So P is incorrect. So 82 00:06:36,830 --> 00:06:39,670 number four. Extra. 83 00:06:42,010 --> 00:06:47,090 Can you explain the average total compensation of 84 00:06:47,090 --> 00:06:51,830 CEOs in the service industry? Data were randomly 85 00:06:51,830 --> 00:06:57,480 collected from 18 CEOs and 19 employees. 97% 86 00:06:57,480 --> 00:07:06,040 confidence interval was calculated to be $281, 87 00:07:07,040 --> 00:07:09,020 $260, 88 00:07:10,060 --> 00:07:13,780 $5836, 89 00:07:14,800 --> 00:07:19,300 and $180. Which of the following interpretations 90 00:07:19,300 --> 00:07:27,310 is correct? Part number A. It says 97% of the 91 00:07:27,310 --> 00:07:32,450 sample data compensation value between these two 92 00:07:32,450 --> 00:07:37,310 values, correct or incorrect statement. Because it 93 00:07:37,310 --> 00:07:44,310 says 97% of the sample data. For the confidence 94 00:07:44,310 --> 00:07:48,310 center value, we are looking for the average, not 95 00:07:48,310 --> 00:07:51,690 for the population, not for the sample. So A is 96 00:07:51,690 --> 00:07:55,890 incorrect. Because A, it says here 97% of the 97 00:07:55,890 --> 00:07:58,690 sampling total. Sample total, we are looking for 98 00:07:58,690 --> 00:08:02,390 the average of the population. So A is incorrect 99 00:08:02,390 --> 00:08:09,150 statement. B, we are 97% confident that the mean 100 00:08:09,150 --> 00:08:15,890 of the sample. So it's false. Because the 101 00:08:15,890 --> 00:08:18,470 confidence about the entire population is about 102 00:08:18,470 --> 00:08:24,160 the population mean. So B is incorrect. C. In the 103 00:08:24,160 --> 00:08:27,760 population of the surface industry, here we have 104 00:08:27,760 --> 00:08:33,240 97% of them will have a total death toll. Also, 105 00:08:33,360 --> 00:08:37,480 this one is incorrect because it mentions in the 106 00:08:37,480 --> 00:08:39,900 population. Here we are talking about total, but 107 00:08:39,900 --> 00:08:44,000 we are looking for the average. Now, part D. We 108 00:08:44,000 --> 00:08:47,360 are 97% confident that the average total 109 00:08:50,460 --> 00:08:53,440 So this one is correct statement. So D is the 110 00:08:53,440 --> 00:08:55,840 correct statement. So for the confidence interval, 111 00:08:55,840 --> 00:08:59,040 we are looking for population, number one. Number 112 00:08:59,040 --> 00:09:03,520 two, the average of that population. So D is the 113 00:09:03,520 --> 00:09:07,260 correct answer. Let's go back to part A. In part 114 00:09:07,260 --> 00:09:10,420 A, it says sample total. So this is incorrect. 115 00:09:11,380 --> 00:09:15,140 Next one. The mean of the sample. We are looking 116 00:09:15,140 --> 00:09:17,440 for the mean of the population. So B is incorrect. 117 00:09:18,040 --> 00:09:25,240 Part C. It mentions here population, but total. So 118 00:09:25,240 --> 00:09:30,300 this one is incorrect. Finally here, we are 97% 119 00:09:30,300 --> 00:09:34,680 confident that the average total. So this one is 120 00:09:34,680 --> 00:09:39,360 true of all. So here we have population and the 121 00:09:39,360 --> 00:09:42,100 average of that population. So it makes sense that 122 00:09:42,100 --> 00:09:43,260 this is the correct answer. 123 00:09:46,520 --> 00:09:47,660 Number five. 124 00:09:59,690 --> 00:10:03,290 Number five, confidence interval. Confidence 125 00:10:03,290 --> 00:10:06,610 interval was used to estimate the proportion of 126 00:10:06,610 --> 00:10:10,170 statistics students that are females. A random 127 00:10:10,170 --> 00:10:16,170 sample of 72 statistics students generated the 128 00:10:16,170 --> 00:10:22,970 following 90% confidence interval, 0.438 129 00:10:22,970 --> 00:10:28,150 and 0.640. 130 00:10:28,510 --> 00:10:32,890 42, based on the interval above the population 131 00:10:32,890 --> 00:10:38,230 proportion of females equals to 0.6. So here we 132 00:10:38,230 --> 00:10:44,310 have confidence interval for the female proportion 133 00:10:44,310 --> 00:10:52,990 ranges between 0.438 up to 0.642. Based on this 134 00:10:52,990 --> 00:10:57,050 interval. Is the population proportion of females 135 00:10:57,050 --> 00:10:58,770 equal 60%? 136 00:11:03,410 --> 00:11:06,690 So here we have from this point all the way up to 137 00:11:06,690 --> 00:11:10,610 0.6. Is the population proportion of females equal 138 00:11:10,610 --> 00:11:16,250 to 0.6? No. The answer is no, but know what? 139 00:11:16,850 --> 00:11:24,320 Number A. No, and we are 90% sure of it. No, the 140 00:11:24,320 --> 00:11:31,200 proportion is 54.17. See, maybe 60% is a 141 00:11:31,200 --> 00:11:33,760 believable value of population proportion based on 142 00:11:33,760 --> 00:11:38,080 information about it. He said yes, and we are 90% 143 00:11:38,080 --> 00:11:44,300 sure of it. So which one is correct? Farah. Which 144 00:11:44,300 --> 00:11:44,900 one is correct? 145 00:11:50,000 --> 00:11:56,760 B says the proportion is 54. 54 if we take the 146 00:11:56,760 --> 00:12:01,120 average of these two values, the answer is 54. But 147 00:12:01,120 --> 00:12:04,960 the true proportion is not the average of the two 148 00:12:04,960 --> 00:12:07,640 endpoints. 149 00:12:08,440 --> 00:12:09,500 So B is incorrect. 150 00:12:12,780 --> 00:12:16,320 If you look at A, the answer is no. And we 151 00:12:16,320 --> 00:12:20,440 mentioned before that this interval may Or may not 152 00:12:20,440 --> 00:12:25,380 contains the true proportion, so A is incorrect. 153 00:12:26,700 --> 00:12:32,640 Now C, maybe. So C is the correct statement, maybe 154 00:12:32,640 --> 00:12:35,820 60% is believable value of the population 155 00:12:35,820 --> 00:12:39,020 proportion based on the information about. So C is 156 00:12:39,020 --> 00:12:44,440 the correct answer. A6, number six. 157 00:12:48,590 --> 00:12:49,550 Number six. 158 00:13:21,280 --> 00:13:23,800 So up to this point, we have the same information 159 00:13:23,800 --> 00:13:27,440 for the previous problem. Using the information 160 00:13:27,440 --> 00:13:31,440 about what total size sample would be necessary if 161 00:13:31,440 --> 00:13:35,460 we wanted to estimate the true proportion within 162 00:13:35,460 --> 00:13:43,620 minus positive or minus 0.108 using 95% 163 00:13:43,620 --> 00:13:46,320 confidence. Now here we are looking for the sample 164 00:13:46,320 --> 00:13:49,160 size that is required to estimate the true 165 00:13:49,160 --> 00:13:53,720 proportion to be within 8% plus or minus 8% using 166 00:13:53,720 --> 00:13:54,720 95% confidence. 167 00:13:58,640 --> 00:14:05,360 The formula first, n equals z squared c plus one. 168 00:14:08,740 --> 00:14:14,240 We have pi, one minus pi divided by e squared. 169 00:14:15,740 --> 00:14:21,120 Now, pi is not given. So in this case either we 170 00:14:21,120 --> 00:14:25,880 use a sinus sample in order to estimate the sample 171 00:14:25,880 --> 00:14:30,400 proportion, Or use Pi to be 0.5. So in this case 172 00:14:30,400 --> 00:14:35,900 we have to use Pi 1 half. If you remember last 173 00:14:35,900 --> 00:14:39,720 time I gave you this equation. Z alpha over 2 174 00:14:39,720 --> 00:14:44,820 divided by 2 squared. So we have this equation. 175 00:14:45,900 --> 00:14:49,280 Because Pi is not given, just use Pi to be 1 half. 176 00:14:50,060 --> 00:14:54,880 Or you may use this equation. shortcut formula. In 177 00:14:54,880 --> 00:15:02,120 this case, here we are talking about 95%. So 178 00:15:02,120 --> 00:15:07,240 what's the value of Z? 196. 2 times E. 179 00:15:10,100 --> 00:15:17,140 E is 8%. So 196 divided by 2 times E, the quantity 180 00:15:17,140 --> 00:15:25,540 squared. Now the answer of this problem 150. So 181 00:15:25,540 --> 00:15:28,720 approximately 150. 182 00:15:29,160 --> 00:15:33,520 So 150 is the correct answer. So again, here we 183 00:15:33,520 --> 00:15:41,460 used pi to be 1 half because P is not given. And 184 00:15:41,460 --> 00:15:46,580 simple calculation results in 150 for the sample 185 00:15:46,580 --> 00:15:49,820 size. So P is the correct answer, 7. 186 00:15:55,220 --> 00:15:56,000 Number seven. 187 00:16:00,480 --> 00:16:03,820 Number seven. When determining the sample size 188 00:16:03,820 --> 00:16:05,820 necessarily for estimating the true population 189 00:16:05,820 --> 00:16:10,560 mean, which factor is not considered when sampling 190 00:16:10,560 --> 00:16:14,960 with replacement? Now here, if you remember the 191 00:16:14,960 --> 00:16:17,960 formula for the sample size. 192 00:16:38,820 --> 00:16:43,460 Now, which factor is not considered when sampling 193 00:16:43,460 --> 00:16:47,120 without weight replacement? Now, the population 194 00:16:47,120 --> 00:16:51,460 size, the population size is not in this quantity, 195 00:16:51,600 --> 00:16:54,420 so A is the correct answer. B, the population 196 00:16:54,420 --> 00:16:58,820 standard deviation, sigma is here. C, the level of 197 00:16:58,820 --> 00:17:03,090 confidence desired in the estimate, we have Z. The 198 00:17:03,090 --> 00:17:06,470 allowable or tolerable seminar, we have it here. 199 00:17:07,370 --> 00:17:10,770 So eight is the correct answer. 200 00:17:13,290 --> 00:17:13,630 Eight. 201 00:17:20,020 --> 00:17:22,600 Supposedly, I'm supposed to focus on the companies 202 00:17:22,600 --> 00:17:25,640 that you're working on now. It turns out to be one 203 00:17:25,640 --> 00:17:28,360 of them. I'm not sure if I'm on the right track. 204 00:17:28,420 --> 00:17:31,040 To make more use of it as a reference for the 205 00:17:31,040 --> 00:17:31,780 update. 206 00:17:43,820 --> 00:17:47,240 Now, which of the following will result in a 207 00:17:47,240 --> 00:17:50,100 reduced interval width? So here we are talking 208 00:17:50,100 --> 00:17:55,580 about reducing the width of the interval. Number 209 00:17:55,580 --> 00:17:58,120 one. Here, if you look carefully at this equation, 210 00:17:59,040 --> 00:18:08,140 increase the sample size, the error Z up over 2 211 00:18:08,140 --> 00:18:17,180 sigma over n. So this is the error state. Now, 212 00:18:17,240 --> 00:18:20,360 based on this equation, if we increase the sample 213 00:18:20,360 --> 00:18:25,180 size, the error will decrease. That means we 214 00:18:25,180 --> 00:18:28,360 reduce the interval with it. So A is the correct 215 00:18:28,360 --> 00:18:31,560 answer. Look at B. Increase the confidence level. 216 00:18:34,030 --> 00:18:36,770 Increasing the confidence level, it means increase 217 00:18:36,770 --> 00:18:41,030 Z, increase E, that means we have wider confidence 218 00:18:41,030 --> 00:18:43,910 interval, so B is incorrect. Increase the 219 00:18:43,910 --> 00:18:46,790 population mean, it doesn't matter actually, so 220 00:18:46,790 --> 00:18:50,250 it's not correct. Increase the sample mean also. 221 00:18:50,770 --> 00:18:54,990 So C and D are incorrect totally, so B is 222 00:18:54,990 --> 00:18:57,670 incorrect, so E is the correct answer. So the 223 00:18:57,670 --> 00:19:00,630 correct answer is A, nine. 224 00:19:07,140 --> 00:19:10,500 A major department store chain is interested in 225 00:19:10,500 --> 00:19:13,560 estimating the average amount each credit and 226 00:19:13,560 --> 00:19:16,560 customers spent on their first visit to the 227 00:19:16,560 --> 00:19:21,380 chain's new store in the mall. 15 credit cards 228 00:19:21,380 --> 00:19:26,540 accounts were randomly sampled and analyzed with 229 00:19:26,540 --> 00:19:29,320 the following results. So here we have this 230 00:19:29,320 --> 00:19:34,880 information about the 15 data points. We have x 231 00:19:34,880 --> 00:19:35,220 bar. 232 00:19:38,550 --> 00:19:42,150 of $50.5. 233 00:19:43,470 --> 00:19:47,390 And S squared, the sample variance is 400. 234 00:19:49,890 --> 00:19:52,750 Construct 95 confidence interval for the average 235 00:19:52,750 --> 00:19:55,570 amount it's credit card customer spent on their 236 00:19:55,570 --> 00:20:01,230 first visit to the chain. It's a new store. It's 237 00:20:01,230 --> 00:20:04,310 in the mall, assuming the amount spent follows a 238 00:20:04,310 --> 00:20:05,010 normal distribution. 239 00:20:08,090 --> 00:20:13,150 In this case, we should use T instead of Z. So the 240 00:20:13,150 --> 00:20:16,310 formula should be X bar plus or minus T, alpha 241 00:20:16,310 --> 00:20:17,610 over 2S over root N. 242 00:20:23,110 --> 00:20:29,350 So X bar is 50.5. T, we should use the T table. In 243 00:20:29,350 --> 00:20:34,010 this case, here we are talking about 95%. 244 00:20:36,830 --> 00:20:44,130 So that means alpha is 5%, alpha over 2, 0, 2, 5. 245 00:20:44,770 --> 00:20:48,930 So now we are looking for 2, 0, 2, 5, and degrees 246 00:20:48,930 --> 00:20:55,170 of freedom. N is 15. It says that 15 credit cards. 247 00:20:55,770 --> 00:20:59,110 So 15 credit cards accounts for random samples. So 248 00:20:59,110 --> 00:21:05,470 N equals 15. So since N is 15, Degrees of freedom 249 00:21:05,470 --> 00:21:09,850 is 14. Now we may use the normal, the T table in 250 00:21:09,850 --> 00:21:16,250 order to find the value of T in 251 00:21:16,250 --> 00:21:19,270 the upper tier actually. So what's the value if 252 00:21:19,270 --> 00:21:26,350 you have the table? So look at degrees of freedom 253 00:21:26,350 --> 00:21:33,090 14 under the probability of 0 to 5. 254 00:21:40,190 --> 00:21:45,050 So again, we are looking for degrees of freedom 255 00:21:45,050 --> 00:21:49,170 equal 14 under 2.5%. 256 00:22:04,850 --> 00:22:11,390 0.5 plus or minus 2 257 00:22:11,390 --> 00:22:18,390 .1448. S squared is given 400. Take square root of 258 00:22:18,390 --> 00:22:25,570 this quantity 20 over root n over root 15. And the 259 00:22:25,570 --> 00:22:30,350 answer, just simple calculation will give 260 00:22:34,250 --> 00:22:38,410 This result, so D is the correct answer. So the 261 00:22:38,410 --> 00:22:45,870 answer should be 50.5 plus or minus 11.08. So D is 262 00:22:45,870 --> 00:22:49,170 the correct answer. So this one is straightforward 263 00:22:49,170 --> 00:22:52,990 calculation, gives part D to be the correct 264 00:22:52,990 --> 00:22:55,750 answer. Any question? 265 00:22:58,510 --> 00:23:00,110 11, 10? 266 00:23:03,110 --> 00:23:07,250 Private colleges and universities rely on money 267 00:23:07,250 --> 00:23:12,730 contributed by individuals and corporations for 268 00:23:12,730 --> 00:23:17,950 their operating expenses. Much of this money is 269 00:23:17,950 --> 00:23:24,090 put into a fund called an endowment, and the 270 00:23:24,090 --> 00:23:27,530 college spends only the interest earned by the 271 00:23:27,530 --> 00:23:33,130 fund. Now, here we have a recent It says that a 272 00:23:33,130 --> 00:23:35,310 recent survey of eight private colleges in the 273 00:23:35,310 --> 00:23:39,450 United States revealed the following endowment in 274 00:23:39,450 --> 00:23:44,350 millions of dollars. So we have this data. So it's 275 00:23:44,350 --> 00:23:50,650 raw data. Summary statistics yield export to be 276 00:23:50,650 --> 00:23:53,010 180. 277 00:23:57,010 --> 00:23:57,850 So export. 278 00:24:07,070 --> 00:24:12,130 Now if this information is not given, you have to 279 00:24:12,130 --> 00:24:15,170 compute the average and standard deviation by the 280 00:24:15,170 --> 00:24:19,860 equations we know. But here, the mean and standard 281 00:24:19,860 --> 00:24:23,200 deviation are given. So just use this information 282 00:24:23,200 --> 00:24:27,480 anyway. Calculate 95 confidence interval for the 283 00:24:27,480 --> 00:24:30,140 mean endowment of all private colleges in the 284 00:24:30,140 --> 00:24:34,520 United States, assuming a normal distribution for 285 00:24:34,520 --> 00:24:39,300 the endowment. Here we have 95%. 286 00:24:39,300 --> 00:24:42,600 Now 287 00:24:42,600 --> 00:24:48,480 what's the sample size? It says that eight. So N 288 00:24:48,480 --> 00:24:53,900 is eight. So again, simple calculation. So 289 00:24:53,900 --> 00:24:59,680 explore, plus or minus T, S over root N. So use 290 00:24:59,680 --> 00:25:04,200 the same idea for the previous one. And the answer 291 00:25:04,200 --> 00:25:10,420 for number 10 is part D. So D is the correct 292 00:25:10,420 --> 00:25:17,380 answer. So again, For eleven, D is the correct 293 00:25:17,380 --> 00:25:22,680 answer. For ten, D is the correct answer. Next. So 294 00:25:22,680 --> 00:25:26,280 this one is similar to the one we just did. 295 00:25:30,660 --> 00:25:31,260 Eleven. 296 00:25:47,140 --> 00:25:51,140 Here it says that rather than examine the records 297 00:25:51,140 --> 00:25:56,220 of all students, the dean took a random sample of 298 00:25:56,220 --> 00:26:01,340 size 200. So we have large university. Here we 299 00:26:01,340 --> 00:26:04,860 took representative sample of size 200. 300 00:26:26,980 --> 00:26:31,900 How many students would be to be assembled? It 301 00:26:31,900 --> 00:26:34,540 says that if the dean wanted to estimate the 302 00:26:34,540 --> 00:26:38,040 proportion of all students, The saving financial 303 00:26:38,040 --> 00:26:46,100 aid to within 3% with 99% probability. How many 304 00:26:46,100 --> 00:26:51,620 students would need to be sampled? So we have the 305 00:26:51,620 --> 00:26:56,920 formula, if you remember, n equals z y 1 minus y 306 00:26:56,920 --> 00:27:00,860 divided by e. So we have z squared. 307 00:27:03,640 --> 00:27:09,200 Now, y is not given. If Pi is not given, we have 308 00:27:09,200 --> 00:27:14,180 to look at either B or 0.5. Now in this problem, 309 00:27:15,000 --> 00:27:18,900 it says that Dean selected 200 students, and he 310 00:27:18,900 --> 00:27:23,800 finds that out of this number, 118 of them are 311 00:27:23,800 --> 00:27:26,480 receiving financial aid. So based on this 312 00:27:26,480 --> 00:27:30,480 information, we can compute B. So B is x over n. 313 00:27:30,700 --> 00:27:34,840 It's 118 divided by 200. And this one gives? 314 00:27:41,090 --> 00:27:46,310 So in this case, out of 200 students, 118 of them 315 00:27:46,310 --> 00:27:49,630 are receiving financial aid. That means the 316 00:27:49,630 --> 00:27:53,730 proportion, the sample proportion, is 118 divided 317 00:27:53,730 --> 00:27:57,690 by 200, which is 0.59. So we have to use this 318 00:27:57,690 --> 00:28:03,830 information instead of pi. So n equals, 319 00:28:08,050 --> 00:28:15,120 now it's about 99%. 2.85. Exactly, it's one of 320 00:28:15,120 --> 00:28:21,380 these. We have 2.57 and 321 00:28:21,380 --> 00:28:30,220 8. It says 99%. So 322 00:28:30,220 --> 00:28:32,720 here we have 99%. So what's left? 323 00:28:42,180 --> 00:28:47,320 0.5 percent, this area. 0.5 to the right and 0.5 324 00:28:47,320 --> 00:28:52,500 to the left, so 005. Now if you look at 2.5 under 325 00:28:52,500 --> 00:28:57,280 7, the answer is 0051, the other one 0049. 326 00:28:59,840 --> 00:29:04,600 So either this one or the other value, so 2.57. or 327 00:29:04,600 --> 00:29:07,600 2.58, it's better to take the average of these 328 00:29:07,600 --> 00:29:13,320 two. Because 005 lies exactly between these two 329 00:29:13,320 --> 00:29:20,780 values. So the score in this case, either 2.75 or 330 00:29:20,780 --> 00:29:30,880 2.78, or the average. And the exact one, 2.7, I'm 331 00:29:30,880 --> 00:29:34,680 sorry, 2.576. The exact answer. 332 00:29:38,000 --> 00:29:40,700 It's better to use the average if you don't 333 00:29:40,700 --> 00:29:46,100 remember the exact answer. So it's the exact one. 334 00:29:47,480 --> 00:29:53,440 But 2.575 is okay. Now just use this equation, 2 335 00:29:53,440 --> 00:30:02,020 .575 times square, times 59. 336 00:30:03,900 --> 00:30:09,440 1 minus 59 divided by the error. It's three 337 00:30:09,440 --> 00:30:17,800 percent. So it's 0.0312 squared. So the answer in 338 00:30:17,800 --> 00:30:23,420 this case is part 2 339 00:30:23,420 --> 00:30:30,300 .57 times 59 times 41 divided by 03 squared. The 340 00:30:30,300 --> 00:30:31,140 answer is part. 341 00:30:41,650 --> 00:30:46,530 You will get the exact answer if you use 2.576. 342 00:30:48,190 --> 00:30:51,230 You will get the exact answer. But anyway, if you 343 00:30:51,230 --> 00:30:53,310 use one of these, you will get approximate answer 344 00:30:53,310 --> 00:30:56,430 to be 1784. 345 00:30:58,590 --> 00:31:04,430 Any question? So in this case, we used the sample 346 00:31:04,430 --> 00:31:11,240 proportion instead of 0.5, because the dean 347 00:31:11,240 --> 00:31:14,120 selected a random sample of size 200, and he finds 348 00:31:14,120 --> 00:31:19,200 that 118 of them are receiving financial aid. That 349 00:31:19,200 --> 00:31:24,980 means the sample proportion is 118 divided by 200, 350 00:31:25,360 --> 00:31:30,420 which gives 0.59. So we have to use 59% as the 351 00:31:30,420 --> 00:31:38,360 sample proportion. Is it clear? Next, number 352 00:31:38,360 --> 00:31:38,760 three. 353 00:31:41,700 --> 00:31:45,860 An economist is interested in studying the incomes 354 00:31:45,860 --> 00:31:51,620 of consumers in a particular region. The 355 00:31:51,620 --> 00:31:56,400 population standard deviation is known to be 1 356 00:31:56,400 --> 00:32:00,560 ,000. A random sample of 50 individuals resulted 357 00:32:00,560 --> 00:32:06,460 in an average income of $15,000. What is the 358 00:32:06,460 --> 00:32:11,520 weight of the 90% confidence interval? So here in 359 00:32:11,520 --> 00:32:16,560 this example, the population standard deviation 360 00:32:16,560 --> 00:32:21,480 sigma is known. So sigma is $1000. 361 00:32:24,600 --> 00:32:32,280 Random sample of size 50 is selected. This sample 362 00:32:32,280 --> 00:32:41,430 gives an average of $15,000 ask 363 00:32:41,430 --> 00:32:48,150 about what is the width of the 90% confidence 364 00:32:48,150 --> 00:32:55,630 interval. So again, here 365 00:32:55,630 --> 00:32:58,710 we are asking about the width of the confidence 366 00:32:58,710 --> 00:33:02,570 interval. If we have a random sample of size 50, 367 00:33:03,320 --> 00:33:07,560 And that sample gives an average of $15,000. And 368 00:33:07,560 --> 00:33:10,940 we know that the population standard deviation is 369 00:33:10,940 --> 00:33:17,580 1,000. Now, what's the width of the 90% confidence 370 00:33:17,580 --> 00:33:21,800 interval? Any idea of this? 371 00:33:33,760 --> 00:33:40,020 So idea number one is fine. You may calculate the 372 00:33:40,020 --> 00:33:43,400 lower limit and upper limit. And the difference 373 00:33:43,400 --> 00:33:46,640 between these two gives the width. So idea number 374 00:33:46,640 --> 00:33:51,360 one, the width equals the distance between upper 375 00:33:51,360 --> 00:33:59,070 limit our limit minus lower limit. Now this 376 00:33:59,070 --> 00:34:03,270 distance gives a width, that's correct. Let's see. 377 00:34:04,710 --> 00:34:07,910 So in other words, you have to find the confidence 378 00:34:07,910 --> 00:34:12,070 interval by using this equation x bar plus or 379 00:34:12,070 --> 00:34:17,070 minus z sigma over root n, x bar is given. 380 00:34:20,190 --> 00:34:28,690 Now for 90% we know that z equals 1.645 sigma is 381 00:34:28,690 --> 00:34:32,670 1000 divided 382 00:34:32,670 --> 00:34:40,850 by root 50 plus or minus. By calculator, 1000 383 00:34:40,850 --> 00:34:45,010 times this number divided by root 50 will give 384 00:34:45,010 --> 00:34:49,190 around 385 00:34:49,190 --> 00:34:50,730 232.6. 386 00:34:58,290 --> 00:35:05,790 So the upper limit is this value and lower limit 387 00:35:05,790 --> 00:35:09,650 147671. 388 00:35:11,350 --> 00:35:14,250 So now the upper limit and lower limit are 389 00:35:14,250 --> 00:35:18,590 computed. Now the difference between these two 390 00:35:18,590 --> 00:35:24,010 values will give the weight. If you subtract these 391 00:35:24,010 --> 00:35:26,030 two values, what equals 15,000? 392 00:35:30,670 --> 00:35:37,190 And the answer is 465.13, around. 393 00:35:40,050 --> 00:35:45,550 Maybe I took two minutes to figure the answer, the 394 00:35:45,550 --> 00:35:49,350 right answer. But there is another one, another 395 00:35:49,350 --> 00:35:52,790 idea, maybe shorter. It'll take shorter time. 396 00:35:56,890 --> 00:36:00,230 It's correct, but straightforward. Just compute 397 00:36:00,230 --> 00:36:05,790 the lower and upper limits. And the width is the 398 00:36:05,790 --> 00:36:07,190 difference between these two values. 399 00:36:11,370 --> 00:36:16,050 If you look carefully at this equation, difference 400 00:36:16,050 --> 00:36:21,560 between these two values gives the width. Now 401 00:36:21,560 --> 00:36:25,880 let's imagine that the lower limit equals x bar 402 00:36:25,880 --> 00:36:28,920 minus 403 00:36:28,920 --> 00:36:36,340 the error term. And upper limit is also x bar plus 404 00:36:36,340 --> 00:36:37,960 the error term. 405 00:36:41,460 --> 00:36:46,580 Now if we add this, or if we subtract 2 from 1, 406 00:36:47,900 --> 00:36:52,560 you will get upper limit minus lower limit equals 407 00:36:52,560 --> 00:36:55,740 x 408 00:36:55,740 --> 00:37:07,280 bar cancels with 2x bar. If you subtract, w minus 409 00:37:07,280 --> 00:37:10,960 equals 2e. 410 00:37:12,520 --> 00:37:18,060 Upper limit minus lower limit is the width. So w, 411 00:37:18,760 --> 00:37:24,800 the width is twice the sampling error. So we have 412 00:37:24,800 --> 00:37:29,980 this new information, W equals twice of the margin 413 00:37:29,980 --> 00:37:36,400 of error. If we add 1 and 2, that will give lower 414 00:37:36,400 --> 00:37:41,120 limit plus upper limit equals to x bar. That means 415 00:37:41,120 --> 00:37:45,800 x bar equals lower limit plus upper limit divided 416 00:37:45,800 --> 00:37:46,220 by 2. 417 00:37:53,970 --> 00:37:59,790 the error, and X bar is the average of lower and 418 00:37:59,790 --> 00:38:05,310 upper limits. So by using this idea now, if we 419 00:38:05,310 --> 00:38:12,490 compute the error term, E equals Z sigma over root 420 00:38:12,490 --> 00:38:13,630 N, this quantity. 421 00:38:17,350 --> 00:38:25,260 And again, Z is 1645. Sigma is 1000. Divide by 422 00:38:25,260 --> 00:38:34,960 root 50. This gives 232.6. This is the error tier, 423 00:38:36,300 --> 00:38:40,040 or the margin of error. As we know, that's called 424 00:38:40,040 --> 00:38:46,400 margin of error or sampling error. 425 00:38:50,580 --> 00:38:56,190 So the error is this amount. The width is twice 426 00:38:56,190 --> 00:39:04,490 this value. So W equals 2 times the error. And the 427 00:39:04,490 --> 00:39:10,830 answer should be the same as the one we just... So 428 00:39:10,830 --> 00:39:13,450 we end with the same result. Now which one is 429 00:39:13,450 --> 00:39:17,370 shorter? Forget about my explanation up to this 430 00:39:17,370 --> 00:39:22,570 point. We started from this one. We just computed 431 00:39:22,570 --> 00:39:27,390 the error tier. I mean this amount. Then we found 432 00:39:27,390 --> 00:39:32,950 the error to be this 232 multiply this by 2 will 433 00:39:32,950 --> 00:39:35,550 give the sampling error or the, I'm sorry, will 434 00:39:35,550 --> 00:39:39,630 give the weight of the interval. Now imagine for 435 00:39:39,630 --> 00:39:43,370 this problem, the income, the average income is 436 00:39:43,370 --> 00:39:43,830 not given. 437 00:39:47,590 --> 00:39:55,450 Suppose x bar is not given. Now the question is, 438 00:39:55,550 --> 00:40:00,030 can you find the answer by using this idea? But 439 00:40:00,030 --> 00:40:04,390 here, without using x bar, we computed the 440 00:40:04,390 --> 00:40:07,810 sampling error to multiply this value by 2 and get 441 00:40:07,810 --> 00:40:08,130 the answer. 442 00:40:11,810 --> 00:40:13,810 So that's for number 12. 443 00:40:16,830 --> 00:40:20,550 Again, for this particular example, there are two 444 00:40:20,550 --> 00:40:25,610 ways to solve this problem. The first one, you 445 00:40:25,610 --> 00:40:28,390 have to construct the confidence interval, then 446 00:40:28,390 --> 00:40:32,910 subtract upper limit from the lower limit, you 447 00:40:32,910 --> 00:40:38,030 will get the width of the interval. The other way, 448 00:40:38,610 --> 00:40:42,150 just compute the error and multiply the answer by 449 00:40:42,150 --> 00:40:48,210 2, you will get the same result. Number 13. 450 00:40:56,020 --> 00:41:00,980 13th says that the head librarian at the Library 451 00:41:00,980 --> 00:41:04,780 of Congress has asked her assistant for an 452 00:41:04,780 --> 00:41:07,980 interval estimate of a mean number of books 453 00:41:07,980 --> 00:41:12,720 checked out each day. The assistant provides the 454 00:41:12,720 --> 00:41:23,000 following interval estimate. From 740 to 920 books 455 00:41:23,000 --> 00:41:28,360 per day. If the head librarian knows that the 456 00:41:28,360 --> 00:41:33,880 population standard deviation is 150 books shipped 457 00:41:33,880 --> 00:41:37,420 outwardly, approximately how large a sample did 458 00:41:37,420 --> 00:41:40,200 her assistant use to determine the interval 459 00:41:40,200 --> 00:41:46,540 estimate? So the information we have is the 460 00:41:46,540 --> 00:41:50,860 following. We have information about the 461 00:41:50,860 --> 00:41:51,700 confidence interval. 462 00:42:01,440 --> 00:42:02,800 920 books. 463 00:42:05,940 --> 00:42:08,700 And sigma is known to be 150. 464 00:42:12,980 --> 00:42:17,980 That's all we have. He asked about how large a 465 00:42:17,980 --> 00:42:20,880 sample did Herelsen's conclusion determine the 466 00:42:20,880 --> 00:42:21,820 interval estimate. 467 00:42:26,740 --> 00:42:31,850 Look at the answers. A is 2. B is 3, C is 12, it 468 00:42:31,850 --> 00:42:33,950 cannot be determined from the information given. 469 00:42:37,190 --> 00:42:43,190 Now, in order to find the number, the sample, we 470 00:42:43,190 --> 00:42:48,350 need Sigma or E squared. Confidence is not given. 471 00:42:50,550 --> 00:43:00,140 Sigma is okay. We can find the error. The error is 472 00:43:00,140 --> 00:43:07,940 just W divided by 2. So the error is fine. I mean, 473 00:43:08,100 --> 00:43:12,200 E is fine. E is B minus A or upper limit minus 474 00:43:12,200 --> 00:43:17,100 lower limit divided by 2. So width divided by 2. 475 00:43:17,240 --> 00:43:20,740 So this is fine. But you don't have information 476 00:43:20,740 --> 00:43:21,780 about Z. 477 00:43:25,020 --> 00:43:29,550 We are looking for N. So Z is not I mean, cannot 478 00:43:29,550 --> 00:43:32,810 be computed because the confidence level is not 479 00:43:32,810 --> 00:43:39,830 given. So the information is determined 480 00:43:39,830 --> 00:43:46,170 from the information given. Make sense? So we 481 00:43:46,170 --> 00:43:50,790 cannot compute this score. Z is fine. Z is 150. 482 00:43:51,330 --> 00:43:54,310 The margin of error, we can compute the margin by 483 00:43:54,310 --> 00:43:59,090 using this interval, the width. Divide by two 484 00:43:59,090 --> 00:44:05,790 gives the same result. Now for number 14, we have 485 00:44:05,790 --> 00:44:11,330 the same information. But here, 486 00:44:14,450 --> 00:44:22,030 she asked her assistant to use 25 days. So now, n 487 00:44:22,030 --> 00:44:24,990 is 25. We have the same information about the 488 00:44:24,990 --> 00:44:25,310 interval. 489 00:44:32,020 --> 00:44:33,300 And sigma is 150. 490 00:44:36,300 --> 00:44:40,800 So she asked her assistant to use 25 days of data 491 00:44:40,800 --> 00:44:43,860 to construct the interval estimate. So n is 25. 492 00:44:44,980 --> 00:44:48,300 What confidence level can she attach to the 493 00:44:48,300 --> 00:44:53,500 interval estimate? Now in this case, we are asking 494 00:44:53,500 --> 00:44:56,240 about confidence, not z. 495 00:45:00,930 --> 00:45:03,530 You have to distinguish between confidence level 496 00:45:03,530 --> 00:45:08,130 and z. We use z, I'm sorry, we use z level to 497 00:45:08,130 --> 00:45:13,350 compute the z score. Now, which one is correct? 99 498 00:45:13,350 --> 00:45:21,670 .7, 99, 98, 95.4. Let's see. Now, what's the 499 00:45:21,670 --> 00:45:25,070 average? I'm sorry, the formula is x bar plus or 500 00:45:25,070 --> 00:45:29,270 minus z sigma over root n. What's the average? In 501 00:45:29,270 --> 00:45:34,710 this case, this is the formula we have. We are 502 00:45:34,710 --> 00:45:38,770 looking about this one. Now, also there are two 503 00:45:38,770 --> 00:45:43,250 ways to solve this problem. Either focus on the 504 00:45:43,250 --> 00:45:47,950 aortia, or just find a continuous interval by 505 00:45:47,950 --> 00:45:55,830 itself. So let's focus on this one. Z sigma over 506 00:45:55,830 --> 00:45:56,230 root of. 507 00:45:59,620 --> 00:46:05,380 And we have x bar. What's the value of x bar? x 508 00:46:05,380 --> 00:46:15,240 bar 740 plus x 509 00:46:15,240 --> 00:46:16,400 bar 830. 510 00:46:25,380 --> 00:46:31,740 1660 divided by 2, 830. Now, z equals, I don't 511 00:46:31,740 --> 00:46:40,660 know, sigma, sigma is 150, n is 5. So here we have 512 00:46:40,660 --> 00:46:41,600 30 sigma. 513 00:46:44,980 --> 00:46:51,560 Now, what's the value of sigma? 36, so we have x 514 00:46:51,560 --> 00:46:54,900 bar, now the value of x bar. 515 00:47:02,330 --> 00:47:10,530 So we have x bar 830 plus or minus 30 there. 516 00:47:15,290 --> 00:47:18,030 Now, if you look carefully at this equation, 517 00:47:19,550 --> 00:47:24,570 what's the value of z in order to have this 518 00:47:24,570 --> 00:47:29,630 confidence interval, which is 740 and 920? 519 00:47:36,170 --> 00:47:40,730 So, Z should be... 520 00:47:40,730 --> 00:47:46,290 What's the value of Z? Now, 830 minus 3Z equals 521 00:47:46,290 --> 00:47:46,870 this value. 522 00:47:49,830 --> 00:47:53,390 So, Z equals... 523 00:47:53,390 --> 00:47:56,450 3. 524 00:47:56,830 --> 00:48:03,540 So, Z is 3. That's why. Now, Z is 3. What do you 525 00:48:03,540 --> 00:48:05,180 think the corresponding C level? 526 00:48:11,460 --> 00:48:16,560 99.7% If 527 00:48:16,560 --> 00:48:27,080 you remember for the 68 empirical rule 68, 95, 99 528 00:48:27,080 --> 00:48:33,760 .7% In chapter 6 we said that 99.7% of the data 529 00:48:33,760 --> 00:48:37,220 falls within three standard deviations of the 530 00:48:37,220 --> 00:48:41,980 mean. So if these three, I am sure that we are 531 00:48:41,980 --> 00:48:50,340 using 99.7% for the confidence level. So for this 532 00:48:50,340 --> 00:48:53,280 particular example here, we have new information 533 00:48:53,280 --> 00:48:57,280 about the sample size. So N is 25. 534 00:49:01,630 --> 00:49:06,190 So just simple calculation x bar as I mentioned 535 00:49:06,190 --> 00:49:11,510 here. The average is lower limit plus upper limit 536 00:49:11,510 --> 00:49:18,270 divided by 2. So x bar equals 830. So now your 537 00:49:18,270 --> 00:49:25,130 confidence interval is x bar plus or minus z sigma 538 00:49:25,130 --> 00:49:31,070 over root n. z sigma over root n, z is unknown, 539 00:49:32,190 --> 00:49:37,030 sigma is 150, n is 25, which is 5, square root of 540 00:49:37,030 --> 00:49:48,390 it, so we'll have 3z. So now x bar 830 minus 3z, 541 00:49:49,610 --> 00:49:53,870 this is the lower limit, upper limit 830 plus 3z. 542 00:49:55,480 --> 00:49:59,000 Now, the confidence interval is given by 740 and 543 00:49:59,000 --> 00:50:09,020 920. Just use the lower limit. 830 minus 3z equals 544 00:50:09,020 --> 00:50:10,820 740. 545 00:50:12,300 --> 00:50:18,280 Simple calculation here. 830 minus 740 is 90, 546 00:50:18,660 --> 00:50:22,340 equals 3z. That means z equals 3. 547 00:50:26,070 --> 00:50:29,750 Now the z value is 3, it means the confidence is 548 00:50:29,750 --> 00:50:33,530 9917, so the correct answer is A. 549 00:50:44,690 --> 00:50:49,390 The other way, you can use that one, by using the 550 00:50:53,010 --> 00:50:55,830 Margin of error, which is the difference between 551 00:50:55,830 --> 00:50:58,270 these two divided by two, you will get the same 552 00:50:58,270 --> 00:51:02,630 result. So there are two methods, one of these 553 00:51:02,630 --> 00:51:05,830 straightforward one. The other one, as you 554 00:51:05,830 --> 00:51:13,550 mentioned, is the error term. It's B minus upper 555 00:51:13,550 --> 00:51:19,550 limit minus lower limit divided by two. Upper 556 00:51:19,550 --> 00:51:27,450 limit is 920. Minus 74 divided by 2. What's the 557 00:51:27,450 --> 00:51:28,370 value for this one? 558 00:51:34,570 --> 00:51:40,610 90. So the margin of error is 90. So E equals E. 559 00:51:41,070 --> 00:51:43,790 Sigma or N equals? 560 00:51:47,110 --> 00:51:50,810 All squared. So by using this equation you can get 561 00:51:50,810 --> 00:51:56,860 your result. So, N is 25, Z is unknown, Sigma is 562 00:51:56,860 --> 00:52:05,520 150, R is 90. This one squared. You will get the 563 00:52:05,520 --> 00:52:10,020 same Z-score. Make sense? 564 00:52:17,770 --> 00:52:21,810 Because if you take z to be three times one-fifth 565 00:52:21,810 --> 00:52:25,150 divided by nine squared, you will get the same 566 00:52:25,150 --> 00:52:30,790 result for z value. So both will give the same 567 00:52:30,790 --> 00:52:35,790 result. So that's for the multiple choice 568 00:52:35,790 --> 00:52:42,430 problems. Any question? Let's move to the section 569 00:52:42,430 --> 00:52:46,370 number two, true or false problems. 570 00:52:47,810 --> 00:52:48,790 Number one, 571 00:52:51,850 --> 00:52:57,950 a race car driver 572 00:52:57,950 --> 00:53:03,670 tested his car for time from 0 to 60 mileage per 573 00:53:03,670 --> 00:53:09,390 hour. And in 20 tests, obtained an average of 4.85 574 00:53:09,390 --> 00:53:16,660 seconds, with some deviation of 1.47 seconds. 95 575 00:53:16,660 --> 00:53:23,440 confidence interval for the 0 to 60 time is 4.62 576 00:53:23,440 --> 00:53:29,540 seconds up to 5.18. I think straightforward. Just 577 00:53:29,540 --> 00:53:33,440 simple calculation, it will give the right answer. 578 00:53:36,660 --> 00:53:40,640 x bar n, 579 00:53:41,360 --> 00:53:43,620 so we have to use this equation. 580 00:53:48,220 --> 00:53:54,020 You can do it. So it says the answer is false. You 581 00:53:54,020 --> 00:53:58,340 have to check this result. So it's 4.5 plus or 582 00:53:58,340 --> 00:54:03,460 minus T. We have to find T. S is given to be 147 583 00:54:03,460 --> 00:54:10,120 divided by root 20. Now, to find T, we have to use 584 00:54:10,120 --> 00:54:18,480 0 to 5 and 19. By this value here, you'll get the 585 00:54:18,480 --> 00:54:22,160 exact answer. Part number two. 586 00:54:24,980 --> 00:54:32,380 Given a sample mean of 2.1. So x bar is 2.1. 587 00:54:33,680 --> 00:54:34,680 Excuse me? 588 00:54:38,500 --> 00:54:45,920 Because n is small. Now, this sample, This sample 589 00:54:45,920 --> 00:54:52,220 gives an average of 4.85, and standard deviation 590 00:54:52,220 --> 00:55:02,420 based on this sample. So S, so X bar, 4.85, and S 591 00:55:02,420 --> 00:55:09,640 is equal to 1.47. So this is not sigma, because it 592 00:55:09,640 --> 00:55:15,210 says that 20 tests, so random sample is 20. This 593 00:55:15,210 --> 00:55:19,390 sample gives an average of this amount and 594 00:55:19,390 --> 00:55:21,350 standard deviation of this amount. 595 00:55:29,710 --> 00:55:34,610 We are looking for the 596 00:55:34,610 --> 00:55:40,470 continence interval, and we have two cases. First 597 00:55:40,470 --> 00:55:43,630 case, if sigma is known, 598 00:55:47,220 --> 00:55:48,240 Sigma is unknown. 599 00:55:51,520 --> 00:55:58,440 Now for this example, sigma is unknown. So since 600 00:55:58,440 --> 00:56:05,740 sigma is unknown, we have to use T distribution if 601 00:56:05,740 --> 00:56:09,940 the distribution is normal or if N is large 602 00:56:09,940 --> 00:56:14,380 enough. Now for this example, N is 20. So we have 603 00:56:14,380 --> 00:56:17,860 to assume that the population is approximately 604 00:56:17,860 --> 00:56:23,660 normal. So we have to use t. So my confidence 605 00:56:23,660 --> 00:56:26,100 interval should be x bar plus or minus 3s over 606 00:56:26,100 --> 00:56:32,560 root n. Now, number two. Given a sample mean of 2 607 00:56:32,560 --> 00:56:36,180 .1 and a population standard deviation. I 608 00:56:36,180 --> 00:56:38,720 mentioned that population standard deviation is 609 00:56:38,720 --> 00:56:43,900 given. So sigma is 0.7. So sigma is known in this 610 00:56:43,900 --> 00:56:49,170 example. So in part two, sigma is given. Now, from 611 00:56:49,170 --> 00:56:50,890 a sample of 10 data points, 612 00:56:53,730 --> 00:56:56,190 we are looking for 90% confidence interval. 613 00:56:58,790 --> 00:57:07,230 90% confidence interval will have a width of 2.36. 614 00:57:16,460 --> 00:57:19,180 What is two times the assembling error? 615 00:57:22,500 --> 00:57:28,040 So the answer is given. So the error here, error A 616 00:57:28,040 --> 00:57:30,160 equals W. 617 00:57:32,060 --> 00:57:34,120 So the answer is 1.16. 618 00:57:40,520 --> 00:57:45,220 So he asked about given a sample, 90% confidence 619 00:57:45,220 --> 00:57:50,540 interval will have a width of 2.36. Let's see if 620 00:57:50,540 --> 00:57:54,780 the exact width is 2.36 or not. Now we have x bar 621 00:57:54,780 --> 00:58:03,240 plus or minus z, sigma 1.8. x bar is 2.1, plus or 622 00:58:03,240 --> 00:58:08,660 minus. Now what's the error? 1.18. 623 00:58:11,230 --> 00:58:16,370 this amount without calculation or you just use 624 00:58:16,370 --> 00:58:19,590 this straightforward calculation here we are 625 00:58:19,590 --> 00:58:23,530 talking about z about 90 percent so this amount 1 626 00:58:23,530 --> 00:58:30,330 .645 times sigma divided by root n for sure this 627 00:58:30,330 --> 00:58:35,430 quantity equals 1.18 But you don't need to do that 628 00:58:35,430 --> 00:58:40,570 because the width is given to be 2.36. So E is 1 629 00:58:40,570 --> 00:58:46,430 .18. So that saves time in order to compute the 630 00:58:46,430 --> 00:58:55,190 error term. So now 2.1 minus 1.8. 2.1 plus 1.8. 631 00:58:56,350 --> 00:58:59,070 That F, the width, is 2.36. 632 00:59:02,010 --> 00:59:04,170 that if the width equals this value. 633 00:59:10,410 --> 00:59:15,270 2.36. So I solved the problem if the width. But he 634 00:59:15,270 --> 00:59:18,430 asked about, do you know this value? I don't know 635 00:59:18,430 --> 00:59:21,230 that one, so we have to compute the exact answer 636 00:59:21,230 --> 00:59:28,230 now. So x bar 2.1 plus 1645 sigma 637 00:59:34,480 --> 00:59:38,600 My calculator can find the error now. What's the 638 00:59:38,600 --> 00:59:41,260 value for this amount? My calculator. 639 00:59:50,700 --> 00:59:56,440 It's 5.75. 5.75. 640 00:59:57,640 --> 01:00:01,830 So this is your error. So E equals this amount. So 641 01:00:01,830 --> 01:00:05,370 W equals 2 plus 4. 642 01:00:08,350 --> 01:00:17,050 So the error is 5.74. So what's the width? The 643 01:00:17,050 --> 01:00:20,210 width equals 2 times E. 644 01:00:25,590 --> 01:00:30,340 Again. This value, 1.645 times 1.7 divided by root 645 01:00:30,340 --> 01:00:30,580 10. 646 01:00:37,280 --> 01:00:38,280 Three point. 647 01:00:44,020 --> 01:00:49,700 So again, arrow is 3.64. So what's the width? 648 01:00:51,160 --> 01:00:53,540 Twice this value, so two times this one. 649 01:00:57,880 --> 01:00:59,560 7.28. 650 01:01:02,120 --> 01:01:07,180 Now it says in the problem here we have width of 2 651 01:01:07,180 --> 01:01:09,200 .36. So it's incorrect. 652 01:01:11,840 --> 01:01:16,320 So just simple calculation gives width of 7.28, 653 01:01:17,180 --> 01:01:23,900 not 2.36. Number three. 654 01:01:27,950 --> 01:01:32,850 Look at number four. Other things be equal. As the 655 01:01:32,850 --> 01:01:35,550 confidence level for a confidence interval 656 01:01:35,550 --> 01:01:41,250 increases, the width of the interval increases. As 657 01:01:41,250 --> 01:01:45,310 the confidence level increases, confidence 658 01:01:45,310 --> 01:01:47,650 interval increases, the width of the interval 659 01:01:47,650 --> 01:01:52,750 increases. Correct. So that's true. Let's do 660 01:01:52,750 --> 01:01:53,790 number seven. 661 01:01:56,840 --> 01:02:02,300 A point estimate consists 662 01:02:02,300 --> 01:02:06,700 of a single sample statistic that is used to 663 01:02:06,700 --> 01:02:11,000 estimate the true population parameter. That's 664 01:02:11,000 --> 01:02:15,940 correct because any point estimate, for example x 665 01:02:15,940 --> 01:02:21,360 bar, is used to determine the confidence interval 666 01:02:21,360 --> 01:02:25,600 for the unknown parameter mu. So a single 667 01:02:25,600 --> 01:02:30,580 statistic can be used to estimate the true 668 01:02:30,580 --> 01:02:33,400 population parameter, either X bar as a point 669 01:02:33,400 --> 01:02:34,900 estimate or P. 670 01:02:41,380 --> 01:02:48,000 So that's true. Number eight. The T distribution 671 01:02:48,000 --> 01:02:51,100 is used to develop a confidence interval estimate 672 01:02:51,100 --> 01:02:54,240 of the population mean when the population 673 01:02:54,240 --> 01:02:57,200 standard deviation is unknown. That's correct 674 01:02:57,200 --> 01:03:01,240 because we are using T distribution if sigma is 675 01:03:01,240 --> 01:03:03,740 not given and here we have to assume the 676 01:03:03,740 --> 01:03:07,960 population is normal. 9. 677 01:03:11,540 --> 01:03:15,180 The standardized normal distribution is used to 678 01:03:15,180 --> 01:03:17,340 develop a confidence interval estimate of the 679 01:03:17,340 --> 01:03:20,700 population proportion when the sample size is 680 01:03:20,700 --> 01:03:22,820 large enough or sufficiently large. 681 01:03:28,640 --> 01:03:32,640 The width 682 01:03:32,640 --> 01:03:37,720 of a confidence interval equals twice the sampling 683 01:03:37,720 --> 01:03:42,570 error. The weight equals twice the sample, so 684 01:03:42,570 --> 01:03:46,370 that's true. A population parameter is used to 685 01:03:46,370 --> 01:03:50,650 estimate a confidence interval? No way. Because we 686 01:03:50,650 --> 01:03:53,570 use statistics to estimate the confidence 687 01:03:53,570 --> 01:03:58,130 interval. These are statistics. So we are using 688 01:03:58,130 --> 01:04:02,390 statistics to construct the confidence interval. 689 01:04:04,190 --> 01:04:10,080 Number 12. Holding the sample size fixed. In 690 01:04:10,080 --> 01:04:14,560 increasing level, the level of confidence in a 691 01:04:14,560 --> 01:04:17,520 confidence interval will necessarily lead to wider 692 01:04:17,520 --> 01:04:20,500 confidence interval. That's true. Because as C 693 01:04:20,500 --> 01:04:24,840 level increases, Z becomes large, so we have large 694 01:04:24,840 --> 01:04:29,670 width, so the confidence becomes wider. Last one, 695 01:04:30,550 --> 01:04:33,150 holding the weight of a confidence interval fixed 696 01:04:33,150 --> 01:04:36,190 and increasing the level of confidence can be 697 01:04:36,190 --> 01:04:40,090 achieved with lower sample size with large sample 698 01:04:40,090 --> 01:04:44,830 size. So it's false. So that's for section two. 699 01:04:46,230 --> 01:04:49,970 One section is left, free response problems or 700 01:04:49,970 --> 01:04:52,990 questions, you can do it at home. So next time, 701 01:04:53,070 --> 01:04:57,530 inshallah, we'll start chapter nine. That's all.