1 00:00:09,320 --> 00:00:15,210 The second material exam. Question number one. a 2 00:00:15,210 --> 00:00:20,170 corporation randomly selected or selects 150 3 00:00:20,170 --> 00:00:25,430 salespeople and finds that 66% who have never 4 00:00:25,430 --> 00:00:29,930 taken self-improvement course would like such a 5 00:00:29,930 --> 00:00:35,350 course. So in this case, currently, they select 6 00:00:35,350 --> 00:00:52,170 150 salespeople and find that 66% would 7 00:00:52,170 --> 00:00:59,130 like or who have never taken this course. The firm 8 00:00:59,130 --> 00:01:05,110 did a similar study 10 years ago in which 60% of a 9 00:01:05,110 --> 00:01:10,290 random sample of 160 salespeople wanted a self 10 00:01:10,290 --> 00:01:11,830 -improvement course. 11 00:01:15,620 --> 00:01:21,520 They select a random sample of 160 and tell that 12 00:01:21,520 --> 00:01:30,260 60% would like to take this course. So we have 13 00:01:30,260 --> 00:01:34,120 here two information about previous study and 14 00:01:34,120 --> 00:01:37,440 currently. So currently we have this information. 15 00:01:39,660 --> 00:01:44,860 The sample size was 150, with a proportion 66% for 16 00:01:44,860 --> 00:01:47,920 the people who would like to attend or take this 17 00:01:47,920 --> 00:01:51,280 course. Mid-Paiwan and Pai Tu represent the true 18 00:01:51,280 --> 00:01:55,260 proportion, it means the population proportion, of 19 00:01:55,260 --> 00:01:57,800 workers who would like to attend a self 20 00:01:57,800 --> 00:02:01,140 -improvement course in the recent study and the 21 00:02:01,140 --> 00:02:05,400 past studies in Taiwan. So recent, Paiwan. 22 00:02:07,740 --> 00:02:12,100 And Pi 2 is the previous study. This weather, this 23 00:02:12,100 --> 00:02:17,100 proportion has changed from the previous study by 24 00:02:17,100 --> 00:02:21,580 using two approaches. Critical value approach and 25 00:02:21,580 --> 00:02:26,920 B value approach. So here we are talking about Pi 26 00:02:26,920 --> 00:02:32,650 1 equals Pi 2. Since the problem says that The 27 00:02:32,650 --> 00:02:35,870 proportion has changed. You don't know the exact 28 00:02:35,870 --> 00:02:39,610 direction, either greater than or smaller than. So 29 00:02:39,610 --> 00:02:45,810 this one should be Y1 does not equal Y2. So step 30 00:02:45,810 --> 00:02:49,050 one, you have to state the appropriate null and 31 00:02:49,050 --> 00:02:50,930 alternative hypothesis. 32 00:02:53,330 --> 00:02:58,150 Second step, compute the value of the test 33 00:02:58,150 --> 00:03:01,510 statistic. In this case, your Z statistic should 34 00:03:01,510 --> 00:03:09,910 be P1 minus P2 minus Pi 1 minus Pi 2, under the 35 00:03:09,910 --> 00:03:17,310 square root of P dash 1 minus P dash times 1 over 36 00:03:17,310 --> 00:03:24,550 N1 plus 1 over N1. Now, P1 and P2 are given under 37 00:03:24,550 --> 00:03:29,730 the null hypothesis Pi 1 minus Pi 2 is 0. So here 38 00:03:29,730 --> 00:03:32,770 we have to compute P dash, which is the overall 39 00:03:35,350 --> 00:03:40,710 B dash equals x1 plus x2 divided by n1 plus n2. 40 00:03:42,170 --> 00:03:45,150 Now these x's, I mean the number of successes are 41 00:03:45,150 --> 00:03:49,450 not given directly in this problem, but we can 42 00:03:49,450 --> 00:03:54,050 figure out the values of x1 and x2 by using this 43 00:03:54,050 --> 00:03:58,610 information, which is n1 equals 150 and b1 equals 44 00:03:58,610 --> 00:04:03,210 66%. Because we know that b1 equals x1 over n1. 45 00:04:06,860 --> 00:04:14,360 So, by using this equation, X1 equals N1 times V1. 46 00:04:16,920 --> 00:04:28,100 N1 150 times 66 percent, that will give 150 times 47 00:04:28,100 --> 00:04:35,940 66, so that's 99. So 150 times, it's 99. 48 00:04:41,690 --> 00:04:49,670 Similarly, X2 equals N2 times V2. N2 is given by 49 00:04:49,670 --> 00:04:53,770 160, so 160 times 60 percent, 50 00:04:55,750 --> 00:05:02,550 96. So the number of successes are 96 for the 51 00:05:02,550 --> 00:05:06,070 second, for the previous. Nine nine. 52 00:05:11,270 --> 00:05:16,410 So B dash equals x199 53 00:05:16,410 --> 00:05:28,330 plus 96 divided by n1 plus n2, 350. And that will 54 00:05:28,330 --> 00:05:34,850 give the overall proportions divided by 310, 0 55 00:05:34,850 --> 00:05:35,510 .629. 56 00:05:40,870 --> 00:05:44,570 So, this is the value of the overall proportion. 57 00:05:45,390 --> 00:05:50,650 Now, B dash equals 1.629. So, 1 times 1 minus B 58 00:05:50,650 --> 00:05:54,970 dash is 1 minus this value times 1 over N1, 1 over 59 00:05:54,970 --> 00:06:00,550 150 plus 1 over 160. Simple calculation will give 60 00:06:01,460 --> 00:06:07,280 The value of z, which is in this case 1.093. 61 00:06:07,780 --> 00:06:10,620 So just plug this information into this equation, 62 00:06:11,340 --> 00:06:19,320 you will get z value, which is 1.093. He asked to 63 00:06:19,320 --> 00:06:21,780 do this problem by using two approaches, critical 64 00:06:21,780 --> 00:06:25,180 value and b value. Let's start with the first one, 65 00:06:26,780 --> 00:06:27,780 b value approach. 66 00:06:32,710 --> 00:06:36,330 Now your B value or critical value, start with 67 00:06:36,330 --> 00:06:37,050 critical value. 68 00:06:40,850 --> 00:06:46,490 Now since we are taking about a two-sided test, so 69 00:06:46,490 --> 00:06:50,170 there are two critical values which are plus or 70 00:06:50,170 --> 00:06:54,670 minus Z alpha over. Alpha is given by five 71 00:06:54,670 --> 00:06:56,990 percent, so in this case 72 00:06:59,630 --> 00:07:03,370 is equal to plus or minus 1.96. 73 00:07:05,930 --> 00:07:10,010 Now, does this value, I mean does the value of 74 00:07:10,010 --> 00:07:14,910 this statistic which is 1.093 fall in the critical 75 00:07:14,910 --> 00:07:22,730 region? Now, my critical regions are above 196 or 76 00:07:22,730 --> 00:07:28,130 below negative 1.96. Now this value actually falls 77 00:07:29,300 --> 00:07:32,420 In the non-rejection region, so we don't reject 78 00:07:32,420 --> 00:07:36,160 the null hypothesis. So my decision, don't reject 79 00:07:36,160 --> 00:07:39,980 the null hypothesis. That means there is not 80 00:07:39,980 --> 00:07:43,420 sufficient evidence to support the alternative 81 00:07:43,420 --> 00:07:46,960 which states that the proportion has changed from 82 00:07:46,960 --> 00:07:51,290 the previous study. So we don't reject the null 83 00:07:51,290 --> 00:07:54,010 hypothesis. It means there is not sufficient 84 00:07:54,010 --> 00:07:58,050 evidence to support the alternative hypothesis. 85 00:07:58,270 --> 00:08:02,010 That means you cannot say that the proportion has 86 00:08:02,010 --> 00:08:05,530 changed from the previous study. That by using 87 00:08:05,530 --> 00:08:09,650 critical value approach. Now what's about p-value? 88 00:08:11,830 --> 00:08:16,170 In order to determine the p-value, 89 00:08:19,460 --> 00:08:23,320 We have to find the probability that the Z 90 00:08:23,320 --> 00:08:28,060 statistic fall in the rejection regions. So that 91 00:08:28,060 --> 00:08:36,260 means Z greater than my values 1093 or 92 00:08:36,260 --> 00:08:41,060 Z smaller than negative 1.093. 93 00:08:45,450 --> 00:08:49,730 1093 is the same as the left of negative, so they 94 00:08:49,730 --> 00:08:52,810 are the same because of symmetry. So just take 1 95 00:08:52,810 --> 00:08:54,050 and multiply by 2. 96 00:08:58,430 --> 00:09:03,070 Now simple calculation will give the value of 0 97 00:09:03,070 --> 00:09:09,950 .276 in chapter 6. So go back to chapter 6 to 98 00:09:09,950 --> 00:09:13,290 figure out how can we calculate the probability of 99 00:09:13,290 --> 00:09:19,830 Z greater than 1.0938. Now my B value is 0.276, 100 00:09:20,030 --> 00:09:25,190 always we reject the null hypothesis if my B value 101 00:09:25,190 --> 00:09:29,050 is smaller than alpha. Now this value is much much 102 00:09:29,050 --> 00:09:31,210 bigger than alpha, so we don't reject the null 103 00:09:31,210 --> 00:09:36,710 hypothesis. So since my B value is much greater 104 00:09:36,710 --> 00:09:42,650 than alpha, that means we don't reject the null 105 00:09:42,650 --> 00:09:46,810 hypothesis, so we reach the same conclusion, that 106 00:09:46,810 --> 00:09:49,270 there is not sufficient evidence to support the 107 00:09:49,270 --> 00:09:55,270 alternative. Also, we can perform the test by 108 00:09:55,270 --> 00:09:59,810 using confidence interval approach, because here 109 00:09:59,810 --> 00:10:02,850 we are talking about two-tailed test. Your 110 00:10:02,850 --> 00:10:06,670 confidence interval is given by 111 00:10:10,620 --> 00:10:17,280 B1 minus B2 plus 112 00:10:17,280 --> 00:10:23,720 or minus Z alpha over 2 times B 113 00:10:23,720 --> 00:10:30,120 dash 1 minus B dash multiplied by 1 over N1 plus 1 114 00:10:30,120 --> 00:10:37,520 over N2. By the way, this one 115 00:10:37,520 --> 00:10:43,320 called the margin of error. So z times square root 116 00:10:43,320 --> 00:10:45,940 of this sequence is called the margin of error, 117 00:10:46,940 --> 00:10:52,280 and the square root itself is called the standard 118 00:10:52,280 --> 00:10:59,560 error of the point estimate of pi 1 minus pi 2, 119 00:10:59,720 --> 00:11:04,430 which is P1 minus P2. So square root of b dash 1 120 00:11:04,430 --> 00:11:07,650 minus b dash multiplied by 1 over n1 plus 1 over 121 00:11:07,650 --> 00:11:12,270 n2 is called the standard error of the estimate of 122 00:11:12,270 --> 00:11:15,910 pi 1 minus pi 2. So this is standard estimate of 123 00:11:15,910 --> 00:11:21,750 b1 minus b2. Simply, you will get the confidence 124 00:11:21,750 --> 00:11:26,470 interval to be between pi 1 minus the difference 125 00:11:26,470 --> 00:11:32,620 between the two proportions, 4 between negative. 0 126 00:11:32,620 --> 00:11:37,160 .5 and 127 00:11:37,160 --> 00:11:38,940 0.7. 128 00:11:44,060 --> 00:11:48,400 Now this interval actually contains 129 00:11:50,230 --> 00:11:54,250 The value of 0, that means we don't reject the 130 00:11:54,250 --> 00:11:57,570 null hypothesis. So since this interval starts 131 00:11:57,570 --> 00:12:01,870 from negative, lower bound is negative 0.5, upper 132 00:12:01,870 --> 00:12:06,190 bound is 0.17, that means 0 inside this interval, 133 00:12:06,750 --> 00:12:09,130 I mean the confidence captures the value of 0, 134 00:12:09,610 --> 00:12:13,810 that means we don't reject the null hypothesis. So 135 00:12:13,810 --> 00:12:17,110 by using three different approaches, we end with 136 00:12:17,110 --> 00:12:20,930 the same decision and conclusion. That is, we 137 00:12:20,930 --> 00:12:25,370 don't reject null hypotheses. That's all for 138 00:12:25,370 --> 00:12:26,110 number one. 139 00:12:31,450 --> 00:12:32,910 Question number two. 140 00:12:36,170 --> 00:12:40,450 The excellent drug company claims its aspirin 141 00:12:40,450 --> 00:12:43,610 tablets will relieve headaches faster than any 142 00:12:43,610 --> 00:12:47,470 other aspirin on the market. So they believe that 143 00:12:48,440 --> 00:12:52,220 Their drug is better than the other drug in the 144 00:12:52,220 --> 00:12:57,180 market. To determine whether Excellence claim is 145 00:12:57,180 --> 00:13:04,260 valid, random samples of size 15 are chosen from 146 00:13:04,260 --> 00:13:07,080 aspirins made by Excellence and the sample drug 147 00:13:07,080 --> 00:13:12,300 combined. So sample sizes of 15 are chosen from 148 00:13:12,300 --> 00:13:16,260 each. So that means N1 equals 15 and N2 also 149 00:13:16,260 --> 00:13:21,160 equals 15. And aspirin is given to each of the 30 150 00:13:21,160 --> 00:13:23,520 randomly selected persons suffering from 151 00:13:23,520 --> 00:13:27,220 headaches. So the total sample size is 30, because 152 00:13:27,220 --> 00:13:30,780 15 from the first company, and the second for the 153 00:13:30,780 --> 00:13:36,860 simple company. So they are 30 selected persons 154 00:13:36,860 --> 00:13:40,280 who are suffering from headaches. So we have 155 00:13:40,280 --> 00:13:43,380 information about number of minutes required for 156 00:13:43,380 --> 00:13:47,720 each to recover from the headache. is recorded, 157 00:13:48,200 --> 00:13:51,500 the sample results are. So here we have two 158 00:13:51,500 --> 00:13:56,260 groups, two populations. Company is called 159 00:13:56,260 --> 00:13:58,420 excellent company and other one simple company. 160 00:13:59,120 --> 00:14:04,320 The information we have, the sample means are 8.4 161 00:14:04,320 --> 00:14:08,260 for the excellent and 8.9 for the simple company. 162 00:14:09,040 --> 00:14:13,280 With the standard deviations for the sample are 2 163 00:14:13,280 --> 00:14:18,340 .05 and 2.14 respectively for excellent and simple 164 00:14:18,340 --> 00:14:21,480 and as we mentioned the sample sizes are the same 165 00:14:21,480 --> 00:14:26,380 are equal 15 and 15. Now we are going to test at 166 00:14:26,380 --> 00:14:32,540 five percent level of significance test whether to 167 00:14:32,540 --> 00:14:35,560 determine whether excellence aspirin cure 168 00:14:35,560 --> 00:14:39,140 headaches significantly faster than simple 169 00:14:39,140 --> 00:14:46,420 aspirin. Now faster it means Better. Better it 170 00:14:46,420 --> 00:14:49,480 means the time required to relieve headache is 171 00:14:49,480 --> 00:14:53,920 smaller there. So you have to be careful in this 172 00:14:53,920 --> 00:15:00,800 case. If we assume that Mu1 is the mean time 173 00:15:00,800 --> 00:15:05,120 required for excellent aspirin. So Mu1 for 174 00:15:05,120 --> 00:15:05,500 excellent. 175 00:15:17,260 --> 00:15:21,540 So Me1, mean time required for excellence aspirin, 176 00:15:22,780 --> 00:15:28,860 and Me2, mean time required for simple aspirin. So 177 00:15:28,860 --> 00:15:32,760 each one, Me1, is smaller than Me3. 178 00:15:41,140 --> 00:15:45,960 Since Me1 represents the time required to relieve 179 00:15:45,960 --> 00:15:51,500 headache by using excellent aspirin and this one 180 00:15:51,500 --> 00:15:55,460 is faster faster it means it takes less time in 181 00:15:55,460 --> 00:15:59,620 order to recover from headache so mu1 should be 182 00:15:59,620 --> 00:16:06,400 smaller than mu2 we are going to use T T is x1 bar 183 00:16:06,400 --> 00:16:11,380 minus x2 bar minus the difference between the two 184 00:16:11,380 --> 00:16:14,720 population proportions divided by 185 00:16:17,550 --> 00:16:22,070 S squared B times 1 over N1 plus 1 over N2. 186 00:16:25,130 --> 00:16:30,470 S squared B N1 187 00:16:30,470 --> 00:16:35,330 minus 1 S1 squared plus N2 minus 1 S2 squared 188 00:16:35,330 --> 00:16:41,990 divided by N1 plus N2 minus 1. Now, a simple 189 00:16:41,990 --> 00:16:44,030 calculation will give the following results. 190 00:16:59,660 --> 00:17:03,080 So again, we have this data. Just plug this 191 00:17:03,080 --> 00:17:06,620 information here to get the value of S square B. 192 00:17:07,740 --> 00:17:13,120 And finally, you will end with this result. 193 00:17:18,220 --> 00:17:24,920 S squared B equals 2 194 00:17:24,920 --> 00:17:27,240 .095 squared. 195 00:17:30,140 --> 00:17:35,920 Your T statistic equals negative 196 00:17:42,790 --> 00:17:48,370 So that's your T-statistic value. So just plug the 197 00:17:48,370 --> 00:17:51,210 values in 1 and 2, this 1 squared and this 2 198 00:17:51,210 --> 00:17:53,350 squared into this equation, you will get this 199 00:17:53,350 --> 00:18:02,970 value. So 2.059 squared, that is 4.239. 200 00:18:07,670 --> 00:18:10,690 Here you can use either the critical value 201 00:18:10,690 --> 00:18:17,200 approach, Or B value. Let's do a critical value. 202 00:18:21,920 --> 00:18:27,460 Since the alternative is the lower tail, one-sided 203 00:18:27,460 --> 00:18:31,820 lower tail, so your B value, your critical value 204 00:18:31,820 --> 00:18:37,630 is negative, T alpha, and there is a freedom. So 205 00:18:37,630 --> 00:18:47,630 this is equal to negative T, 5% with 28 degrees of 206 00:18:47,630 --> 00:18:55,270 freedom. By using the table you have 28, 207 00:18:56,030 --> 00:19:00,070 28 208 00:19:00,070 --> 00:19:12,790 under 5%, so 28 under 5%, so 209 00:19:12,790 --> 00:19:20,870 1.701, negative 1.701. 210 00:19:23,750 --> 00:19:28,290 Now, we reject the null hypothesis if 211 00:19:33,770 --> 00:19:42,890 region. Now again, since it's lower TL, so your 212 00:19:42,890 --> 00:19:48,830 rejection region is below negative 1.701. 213 00:19:51,230 --> 00:19:55,630 Now, does this value fall in the rejection region? 214 00:19:56,510 --> 00:20:02,350 It falls in the non-rejection region. So the 215 00:20:02,350 --> 00:20:08,040 answer is Don't reject the null hypothesis. That 216 00:20:08,040 --> 00:20:11,380 means we don't have sufficient evidence to support 217 00:20:11,380 --> 00:20:16,300 the excellent drug company claim which states that 218 00:20:16,300 --> 00:20:21,380 their aspirin tablets relieve headaches faster 219 00:20:21,380 --> 00:20:28,540 than the simple one. So that's by using a critical 220 00:20:28,540 --> 00:20:33,230 value approach because this value falls in the non 221 00:20:33,230 --> 00:20:36,450 -rejection region, so we don't reject the null 222 00:20:36,450 --> 00:20:36,890 hypothesis. 223 00:20:44,130 --> 00:20:48,930 Or you maybe use the B-value approach. 224 00:20:53,070 --> 00:20:56,850 Now, since the alternative is µ1 smaller than µ2, 225 00:20:57,640 --> 00:21:03,260 So B value is probability of T smaller than 226 00:21:03,260 --> 00:21:08,820 negative 0 227 00:21:08,820 --> 00:21:12,400 .653. 228 00:21:14,300 --> 00:21:18,420 So we are looking for this probability B of Z 229 00:21:18,420 --> 00:21:21,340 smaller than negative 0.653. 230 00:21:23,210 --> 00:21:27,050 The table you have gives the area in the upper 231 00:21:27,050 --> 00:21:33,190 tail. So this is the same as beauty greater than. 232 00:21:37,790 --> 00:21:44,350 Because the area to the right of 0.653 is the same 233 00:21:44,350 --> 00:21:48,070 as the area to the left of negative 0.75. Because 234 00:21:48,070 --> 00:21:52,970 of symmetry. Just look at the tea table. Now, 235 00:21:53,070 --> 00:22:00,810 smaller than negative, means this area is actually 236 00:22:00,810 --> 00:22:02,690 the same as the area to the right of the same 237 00:22:02,690 --> 00:22:07,330 value, but on the other side. So these two areas 238 00:22:07,330 --> 00:22:11,890 are the same. So it's the same as D of T greater 239 00:22:11,890 --> 00:22:17,710 than 0.653. If you look at the table for 28 240 00:22:17,710 --> 00:22:19,150 degrees of freedom, 241 00:22:22,300 --> 00:22:23,520 That's your 28. 242 00:22:27,580 --> 00:22:32,720 I am looking for the value of 0.653. The first 243 00:22:32,720 --> 00:22:38,420 value here is 0.683. The other one is 0.8. It 244 00:22:38,420 --> 00:22:43,600 means my value is below this one. If you go back 245 00:22:43,600 --> 00:22:46,600 here, 246 00:22:46,700 --> 00:22:52,610 so it should be to the left of this value. Now 247 00:22:52,610 --> 00:22:57,170 here 25, then 20, 20, 15 and so on. So it should 248 00:22:57,170 --> 00:23:01,930 be greater than 25. So your B value actually is 249 00:23:01,930 --> 00:23:08,570 greater than 25%. As we mentioned before, T table 250 00:23:08,570 --> 00:23:12,010 does not give the exact B value. So approximately 251 00:23:12,010 --> 00:23:17,290 my B value is greater than 25%. This value 252 00:23:17,290 --> 00:23:22,400 actually is much bigger than 5%. So again, we 253 00:23:22,400 --> 00:23:27,480 reject, we don't reject the null hypothesis. So 254 00:23:27,480 --> 00:23:30,600 again, to compute the B value, it's probability of 255 00:23:30,600 --> 00:23:37,320 T smaller than the value of the statistic, which 256 00:23:37,320 --> 00:23:42,040 is negative 0.653. The table you have gives the 257 00:23:42,040 --> 00:23:43,040 area to the right. 258 00:23:46,980 --> 00:23:50,700 So this probability is the same as B of T greater 259 00:23:50,700 --> 00:23:55,920 than 0.653. So by using this table, you will get 260 00:23:55,920 --> 00:24:00,100 approximate value of B, which is greater than 25%. 261 00:24:00,100 --> 00:24:02,960 Always, as we mentioned, we reject the null 262 00:24:02,960 --> 00:24:06,660 hypothesis if my B value is smaller than alpha. In 263 00:24:06,660 --> 00:24:08,920 this case, this value is greater than alpha, so we 264 00:24:08,920 --> 00:24:11,480 don't reject the null. So we reach the same 265 00:24:11,480 --> 00:24:15,640 decision as by using the critical value approach. 266 00:24:17,040 --> 00:24:23,360 Any question? So that's for number two. Question 267 00:24:23,360 --> 00:24:24,040 number three. 268 00:24:32,120 --> 00:24:35,820 To test the effectiveness of a business school 269 00:24:35,820 --> 00:24:41,640 preparation course, eight students took a general 270 00:24:41,640 --> 00:24:47,210 business test before and after the course. Let X1 271 00:24:47,210 --> 00:24:50,330 denote before, 272 00:24:53,010 --> 00:24:55,450 and X2 after. 273 00:24:59,630 --> 00:25:04,630 And the difference is X2 minus X1. 274 00:25:14,780 --> 00:25:19,540 The mean of the difference equals 50. And the 275 00:25:19,540 --> 00:25:25,540 standard deviation of the difference is 65.03. So 276 00:25:25,540 --> 00:25:28,900 sample statistics are sample mean for the 277 00:25:28,900 --> 00:25:32,040 difference and sample standard deviation of the 278 00:25:32,040 --> 00:25:36,860 difference. So these two values are given. Test to 279 00:25:36,860 --> 00:25:40,200 determine the effectiveness of a business school 280 00:25:40,200 --> 00:25:45,960 preparation course. So what's your goal? An 281 00:25:45,960 --> 00:25:48,120 alternative, null equals zero. An alternative 282 00:25:48,120 --> 00:25:52,340 should 283 00:25:52,340 --> 00:25:58,360 be greater than zero. Because D is X2 minus X1. So 284 00:25:58,360 --> 00:26:02,840 effective, it means after is better than before. 285 00:26:03,680 --> 00:26:08,420 So my score after taking the course is better than 286 00:26:08,420 --> 00:26:12,080 before taking the course. So X in UD is positive. 287 00:26:19,090 --> 00:26:27,510 T is D bar minus 0 divided by SD over square root 288 00:26:27,510 --> 00:26:41,090 of A. D bar is 50 divided by 65 divided 289 00:26:41,090 --> 00:26:54,490 by Square root of 8. So 50 divided by square 290 00:26:54,490 --> 00:26:57,910 root of 8, 2.17. 291 00:27:04,070 --> 00:27:09,570 Now Yumi used the critical value approach. So my 292 00:27:09,570 --> 00:27:10,930 critical value is T alpha. 293 00:27:13,680 --> 00:27:20,140 And degrees of freedom is 7. It's upper 10. So 294 00:27:20,140 --> 00:27:27,300 it's plus. So it's T alpha 0, 5. And DF is 7, 295 00:27:27,320 --> 00:27:33,820 because N equals 8. Now by using the table, at 7 296 00:27:33,820 --> 00:27:34,680 degrees of freedom, 297 00:27:38,220 --> 00:27:39,340 so at 7, 298 00:27:53,560 --> 00:28:03,380 So my T value is greater than the 299 00:28:03,380 --> 00:28:07,020 critical region, so we reject the null hypothesis. 300 00:28:10,740 --> 00:28:17,700 The rejection region starts from 1.9895 and this 301 00:28:17,700 --> 00:28:24,800 value actually greater than 1.8. So since it falls 302 00:28:24,800 --> 00:28:30,320 in the rejection region, then we reject the null 303 00:28:30,320 --> 00:28:35,060 hypothesis. It means that taking the course, 304 00:28:36,370 --> 00:28:39,690 improves your score. So we have sufficient 305 00:28:39,690 --> 00:28:43,010 evidence to support the alternative hypothesis. 306 00:28:44,330 --> 00:28:50,650 That's for number three. The other part, the other 307 00:28:50,650 --> 00:28:51,130 part. 308 00:28:54,290 --> 00:28:58,550 A statistician selected a sample of 16 receivable 309 00:28:58,550 --> 00:29:03,530 accounts. He reported that the sample information 310 00:29:04,690 --> 00:29:07,790 indicated the mean of the population ranges from 311 00:29:07,790 --> 00:29:12,730 these two values. So we have lower and upper 312 00:29:12,730 --> 00:29:21,910 limits, which are given by 4739. 313 00:29:36,500 --> 00:29:42,400 So the mean of the population ranges between these 314 00:29:42,400 --> 00:29:47,880 two values. And in addition to that, we have 315 00:29:47,880 --> 00:29:55,920 information about the sample standard deviation is 316 00:29:55,920 --> 00:29:56,340 400. 317 00:29:59,500 --> 00:30:03,260 The statistician neglected to report what 318 00:30:03,260 --> 00:30:07,440 confidence level he had used. So we don't know C 319 00:30:07,440 --> 00:30:14,180 level. So C level is unknown, which actually is 1 320 00:30:14,180 --> 00:30:14,760 minus alpha. 321 00:30:20,980 --> 00:30:25,360 Based on the above information, what's the 322 00:30:25,360 --> 00:30:28,380 confidence level? So we are looking for C level. 323 00:30:29,380 --> 00:30:34,160 Now just keep in mind the confidence interval is 324 00:30:34,160 --> 00:30:38,200 given and we are looking for C level. 325 00:30:42,920 --> 00:30:46,600 So this area actually is alpha over 2 and other 326 00:30:46,600 --> 00:30:49,940 one is alpha over 2, so the area between is 1 327 00:30:49,940 --> 00:30:50,440 minus alpha. 328 00:30:53,340 --> 00:30:58,620 Now since the sample size equal 329 00:31:01,950 --> 00:31:10,010 16, N equals 16, so N equals 16, so your 330 00:31:10,010 --> 00:31:12,490 confidence interval should be X bar plus or minus 331 00:31:12,490 --> 00:31:14,610 T, S over root N. 332 00:31:19,350 --> 00:31:26,390 Now, C level can be determined by T, and we know 333 00:31:26,390 --> 00:31:28,130 that this quantity, 334 00:31:30,730 --> 00:31:36,970 represents the margin of error. So, E equals TS 335 00:31:36,970 --> 00:31:42,950 over root N. Now, since the confidence interval is 336 00:31:42,950 --> 00:31:50,270 given, we know from previous chapters that the 337 00:31:50,270 --> 00:31:53,970 margin equals the difference between upper and 338 00:31:53,970 --> 00:31:59,560 lower divided by two. So, half distance of lower 339 00:31:59,560 --> 00:32:06,320 and upper gives the margin. So that will give 260 340 00:32:06,320 --> 00:32:17,620 .2. So that's E. So now E is known to be 260.2 341 00:32:17,620 --> 00:32:24,320 equals to S is given by 400 and N is 16. 342 00:32:26,800 --> 00:32:29,420 Now, simple calculation will give the value of T, 343 00:32:30,060 --> 00:32:31,340 which is the critical value. 344 00:32:35,280 --> 00:32:38,160 So, my T equals 2.60. 345 00:32:41,960 --> 00:32:47,220 Actually, this is T alpha over 2. Now, the value 346 00:32:47,220 --> 00:32:52,400 of the critical value is known to be 2.602. What's 347 00:32:52,400 --> 00:32:56,520 the corresponding alpha over 2? Now look at the 348 00:32:56,520 --> 00:32:59,660 table, at 15 degrees of freedom, 349 00:33:02,720 --> 00:33:10,680 look at 15, at this value 2.602, at this value. 350 00:33:12,640 --> 00:33:19,880 So, 15 degrees of freedom, 2.602, so the 351 00:33:19,880 --> 00:33:21,940 corresponding alpha over 2, not alpha. 352 00:33:24,610 --> 00:33:31,830 it's 1% so my alpha over 2 is 353 00:33:31,830 --> 00:33:43,110 1% so alpha is 2% so the confidence level is 1 354 00:33:43,110 --> 00:33:50,510 minus alpha so 1 minus alpha is 90% so c level is 355 00:33:50,510 --> 00:33:59,410 98% so that's level or the confidence level. So 356 00:33:59,410 --> 00:34:03,990 again, maybe this is a tricky question. 357 00:34:07,330 --> 00:34:10,530 But at least you know that if the confidence 358 00:34:10,530 --> 00:34:15,270 interval is given, you can determine the margin of 359 00:34:15,270 --> 00:34:18,930 error by the difference between lower and upper 360 00:34:18,930 --> 00:34:23,310 divided by two. Then we know this term represents 361 00:34:23,310 --> 00:34:27,150 this margin. So by using this equation, we can 362 00:34:27,150 --> 00:34:29,770 compute the value of T, I mean the critical value. 363 00:34:30,670 --> 00:34:35,290 So since the critical value is given or is 364 00:34:35,290 --> 00:34:38,590 computed, we can determine the corresponding alpha 365 00:34:38,590 --> 00:34:45,390 over 2. So alpha over 2 is 1%. So your alpha is 366 00:34:45,390 --> 00:34:51,710 2%. So my C level is 98%. That's 367 00:34:51,710 --> 00:34:56,180 all. Any questions? We're done, Muhammad.