1 00:00:09,320 --> 00:00:15,760 Last time we discussed hypothesis test for 2 00:00:15,760 --> 00:00:19,440 two population proportions. And we mentioned that 3 00:00:19,440 --> 00:00:25,750 the assumptions are for the first sample. n times 4 00:00:25,750 --> 00:00:28,910 pi should be at least 5, and also n times 1 minus 5 00:00:28,910 --> 00:00:33,050 pi is also at least 5. The same for the second 6 00:00:33,050 --> 00:00:37,570 sample, n 2 times pi 2 is at least 5, as well as n 7 00:00:37,570 --> 00:00:42,860 times 1 minus pi 2 is also at least 5. Also, we 8 00:00:42,860 --> 00:00:46,000 discussed that the point estimate for the 9 00:00:46,000 --> 00:00:51,700 difference of Pi 1 minus Pi 2 is given by P1 minus 10 00:00:51,700 --> 00:00:57,160 P2. That means this difference is unbiased point 11 00:00:57,160 --> 00:01:03,160 estimate of Pi 1 minus Pi 2. Similarly, P2 minus 12 00:01:03,160 --> 00:01:06,700 P1 is the point estimate of the difference Pi 2 13 00:01:06,700 --> 00:01:08,160 minus Pi 1. 14 00:01:11,260 --> 00:01:16,140 We also discussed that the bold estimate for the 15 00:01:16,140 --> 00:01:20,900 overall proportion is given by this equation. So B 16 00:01:20,900 --> 00:01:25,980 dash is called the bold estimate for the overall 17 00:01:25,980 --> 00:01:31,740 proportion. X1 and X2 are the number of items of 18 00:01:31,740 --> 00:01:35,170 interest. And the two samples that you have in one 19 00:01:35,170 --> 00:01:39,150 and two, where in one and two are the sample sizes 20 00:01:39,150 --> 00:01:42,110 for the first and the second sample respectively. 21 00:01:43,470 --> 00:01:46,830 The appropriate statistic in this course is given 22 00:01:46,830 --> 00:01:52,160 by this equation. Z-score or Z-statistic is the 23 00:01:52,160 --> 00:01:56,340 point estimate of the difference pi 1 minus pi 2 24 00:01:56,340 --> 00:02:00,620 minus the hypothesized value under if 0, I mean if 25 00:02:00,620 --> 00:02:05,200 0 is true, most of the time this term equals 0, 26 00:02:05,320 --> 00:02:10,480 divided by this quantity is called the standard 27 00:02:10,480 --> 00:02:14,100 error of the estimate, which is square root of B 28 00:02:14,100 --> 00:02:17,660 dash 1 minus B dash times 1 over N1 plus 1 over 29 00:02:17,660 --> 00:02:22,160 N2. So this is your Z statistic. The critical 30 00:02:22,160 --> 00:02:27,980 regions. I'm sorry, first, the appropriate null 31 00:02:27,980 --> 00:02:32,200 and alternative hypothesis are given by three 32 00:02:32,200 --> 00:02:38,280 cases we have. Either two-tailed test or one 33 00:02:38,280 --> 00:02:42,540 -tailed and it has either upper or lower tail. So 34 00:02:42,540 --> 00:02:46,140 for example, for lower-tailed test, We are going 35 00:02:46,140 --> 00:02:51,500 to test to see if a proportion 1 is smaller than a 36 00:02:51,500 --> 00:02:54,560 proportion 2. This one can be written as pi 1 37 00:02:54,560 --> 00:02:59,080 smaller than pi 2 under H1, or the difference 38 00:02:59,080 --> 00:03:01,160 between these two population proportions is 39 00:03:01,160 --> 00:03:04,940 negative, is smaller than 0. So either you may 40 00:03:04,940 --> 00:03:08,660 write the alternative as pi 1 smaller than pi 2, 41 00:03:09,180 --> 00:03:11,860 or the difference, which is pi 1 minus pi 2 42 00:03:11,860 --> 00:03:15,730 smaller than 0. For sure, the null hypothesis is 43 00:03:15,730 --> 00:03:18,830 the opposite of the alternative hypothesis. So if 44 00:03:18,830 --> 00:03:22,310 this is one by one smaller than by two, so the 45 00:03:22,310 --> 00:03:24,710 opposite by one is greater than or equal to two. 46 00:03:25,090 --> 00:03:27,670 Similarly, but the opposite side here, we are 47 00:03:27,670 --> 00:03:31,530 talking about the upper tail of probability. So 48 00:03:31,530 --> 00:03:33,910 under the alternative hypothesis, by one is 49 00:03:33,910 --> 00:03:37,870 greater than by two. Or it could be written as by 50 00:03:37,870 --> 00:03:40,150 one minus by two is positive, that means greater 51 00:03:40,150 --> 00:03:45,970 than zero. While for the two-tailed test, for the 52 00:03:45,970 --> 00:03:49,310 alternative hypothesis, we have Y1 does not equal 53 00:03:49,310 --> 00:03:51,870 Y2. In this case, we are saying there is no 54 00:03:51,870 --> 00:03:55,950 difference under H0, and there is a difference. 55 00:03:56,920 --> 00:03:59,680 should be under each one. Difference means either 56 00:03:59,680 --> 00:04:03,220 greater than or smaller than. So we have this not 57 00:04:03,220 --> 00:04:06,800 equal sign. So by one does not equal by two. Or it 58 00:04:06,800 --> 00:04:08,980 could be written as by one minus by two is not 59 00:04:08,980 --> 00:04:12,320 equal to zero. It's the same as the one we have 60 00:04:12,320 --> 00:04:15,100 discussed when we are talking about comparison of 61 00:04:15,100 --> 00:04:19,500 two population means. We just replaced these by's 62 00:04:19,500 --> 00:04:24,960 by mus. Finally, the rejection regions are given 63 00:04:24,960 --> 00:04:30,000 by three different charts here for the lower tail 64 00:04:30,000 --> 00:04:35,500 test. We reject the null hypothesis if the value 65 00:04:35,500 --> 00:04:37,500 of the test statistic fall in the rejection 66 00:04:37,500 --> 00:04:40,940 region, which is in the left side. So that means 67 00:04:40,940 --> 00:04:44,040 we reject zero if this statistic is smaller than 68 00:04:44,040 --> 00:04:49,440 negative zero. That's for lower tail test. On the 69 00:04:49,440 --> 00:04:51,620 other hand, for other tailed tests, your rejection 70 00:04:51,620 --> 00:04:54,800 region is the right side, so you reject the null 71 00:04:54,800 --> 00:04:57,160 hypothesis if this statistic is greater than Z 72 00:04:57,160 --> 00:05:01,700 alpha. In addition, for two-tailed tests, there 73 00:05:01,700 --> 00:05:04,300 are two rejection regions. One is on the right 74 00:05:04,300 --> 00:05:07,000 side, the other on the left side. Here, alpha is 75 00:05:07,000 --> 00:05:10,960 split into two halves, alpha over two to the 76 00:05:10,960 --> 00:05:14,060 right, similarly alpha over two to the left side. 77 00:05:14,640 --> 00:05:16,900 Here, we reject the null hypothesis if your Z 78 00:05:16,900 --> 00:05:20,900 statistic falls in the rejection region here, that 79 00:05:20,900 --> 00:05:24,820 means z is smaller than negative z alpha over 2 or 80 00:05:24,820 --> 00:05:30,360 z is greater than z alpha over 2. Now this one, I 81 00:05:30,360 --> 00:05:33,980 mean the rejection regions are the same for either 82 00:05:33,980 --> 00:05:38,540 one sample t-test or two sample t-test, either for 83 00:05:38,540 --> 00:05:41,560 the population proportion or the population mean. 84 00:05:42,180 --> 00:05:46,120 We have the same rejection regions. Sometimes we 85 00:05:46,120 --> 00:05:49,800 replace z by t. It depends if we are talking about 86 00:05:49,800 --> 00:05:54,760 small samples and sigmas unknown. So that's the 87 00:05:54,760 --> 00:05:58,160 basic concepts about testing or hypothesis testing 88 00:05:58,160 --> 00:06:01,200 for the comparison between two population 89 00:06:01,200 --> 00:06:05,140 proportions. And we stopped at this point. I will 90 00:06:05,140 --> 00:06:08,780 give three examples, three examples for testing 91 00:06:08,780 --> 00:06:11,660 about two population proportions. The first one is 92 00:06:11,660 --> 00:06:17,050 given here. It says that, is there a significant 93 00:06:17,050 --> 00:06:20,490 difference between the proportion of men and the 94 00:06:20,490 --> 00:06:24,170 proportion of women who will vote yes on a 95 00:06:24,170 --> 00:06:24,630 proposition? 96 00:06:28,220 --> 00:06:30,480 In this case, we are talking about a proportion. 97 00:06:30,840 --> 00:06:34,520 So this problem tests for a proportion. We have 98 00:06:34,520 --> 00:06:38,980 two proportions here because we have two samples 99 00:06:38,980 --> 00:06:43,800 for two population spheres, men and women. So 100 00:06:43,800 --> 00:06:46,600 there are two populations. So we are talking about 101 00:06:46,600 --> 00:06:50,620 two population proportions. Now, we have to state 102 00:06:50,620 --> 00:06:53,440 carefully now an alternative hypothesis. So for 103 00:06:53,440 --> 00:06:57,640 example, let's say that phi 1 is the population 104 00:06:57,640 --> 00:07:07,140 proportion, proportion of men who will vote for a 105 00:07:07,140 --> 00:07:11,740 proposition A for example, for vote yes, for vote 106 00:07:11,740 --> 00:07:13,300 yes for proposition A. 107 00:07:30,860 --> 00:07:36,460 is the same but of men, of women, I'm sorry. So 108 00:07:36,460 --> 00:07:42,160 the first one for men and the other of 109 00:07:42,160 --> 00:07:48,400 women. Now, in a random, so in this case, we are 110 00:07:48,400 --> 00:07:51,020 talking about difference between two population 111 00:07:51,020 --> 00:07:52,940 proportions, so by one equals by two. 112 00:07:56,920 --> 00:08:00,820 Your alternate hypothesis should be, since the 113 00:08:00,820 --> 00:08:03,220 problem talks about, is there a significant 114 00:08:03,220 --> 00:08:07,140 difference? Difference means two tails. So it 115 00:08:07,140 --> 00:08:12,740 should be pi 1 does not equal pi 2. Pi 1 does not 116 00:08:12,740 --> 00:08:17,400 equal pi 2. So there's still one state null and 117 00:08:17,400 --> 00:08:20,680 alternate hypothesis. Now, in a random sample of 118 00:08:20,680 --> 00:08:28,880 36 out of 72 men, And 31 of 50 women indicated 119 00:08:28,880 --> 00:08:33,380 they would vote yes. So for example, if X1 120 00:08:33,380 --> 00:08:39,000 represents number of men who would vote yes, that 121 00:08:39,000 --> 00:08:45,720 means X1 equals 36 in 122 00:08:45,720 --> 00:08:54,950 172. So that's for men. Now for women. 31 out of 123 00:08:54,950 --> 00:08:59,370 50. So 50 is the sample size for the second 124 00:08:59,370 --> 00:09:05,890 sample. Now it's ask about this test about the 125 00:09:05,890 --> 00:09:08,230 difference between the two population proportion 126 00:09:08,230 --> 00:09:13,890 at 5% level of significance. So alpha is given to 127 00:09:13,890 --> 00:09:19,390 be 5%. So that's all the information you have in 128 00:09:19,390 --> 00:09:23,740 order to answer this question. So based on this 129 00:09:23,740 --> 00:09:27,220 statement, we state null and alternative 130 00:09:27,220 --> 00:09:30,160 hypothesis. Now based on this information, we can 131 00:09:30,160 --> 00:09:32,220 solve the problem by using three different 132 00:09:32,220 --> 00:09:39,220 approaches. Critical value approach, B value, and 133 00:09:39,220 --> 00:09:42,320 confidence interval approach. Because we can use 134 00:09:42,320 --> 00:09:44,220 confidence interval approach because we are 135 00:09:44,220 --> 00:09:47,380 talking about two-tailed test. So let's start with 136 00:09:47,380 --> 00:09:50,240 the basic one, critical value approach. So 137 00:09:50,240 --> 00:09:50,980 approach A. 138 00:10:01,140 --> 00:10:03,400 Now since we are talking about two-tailed test, 139 00:10:04,340 --> 00:10:08,120 your critical value should be plus or minus z 140 00:10:08,120 --> 00:10:12,780 alpha over 2. And since alpha is 5% so the 141 00:10:12,780 --> 00:10:18,420 critical values are z 142 00:10:18,420 --> 00:10:26,650 plus or minus 0.25 which is 196. Or you may use 143 00:10:26,650 --> 00:10:30,050 the standard normal table in order to find the 144 00:10:30,050 --> 00:10:33,330 critical values. Or just if you remember that 145 00:10:33,330 --> 00:10:37,150 values from previous time. So the critical regions 146 00:10:37,150 --> 00:10:47,030 are above 196 or smaller than negative 196. I have 147 00:10:47,030 --> 00:10:51,090 to compute the Z statistic. Now Z statistic is 148 00:10:51,090 --> 00:10:55,290 given by this equation. Z stat equals B1 minus B2. 149 00:10:55,730 --> 00:11:03,010 minus Pi 1 minus Pi 2. This quantity divided by P 150 00:11:03,010 --> 00:11:09,690 dash 1 minus P dash multiplied by 1 over N1 plus 1 151 00:11:09,690 --> 00:11:17,950 over N1. Here we have to find B1, B2. So B1 equals 152 00:11:17,950 --> 00:11:21,910 X1 over N1. X1 is given. 153 00:11:27,180 --> 00:11:32,160 to that means 50%. Similarly, 154 00:11:32,920 --> 00:11:39,840 B2 is A equals X2 over into X to the third power 155 00:11:39,840 --> 00:11:48,380 over 50, so that's 60%. Also, we have to compute 156 00:11:48,380 --> 00:11:55,500 the bold estimate of the overall proportion of B 157 00:11:55,500 --> 00:11:55,860 dash 158 00:12:01,890 --> 00:12:07,130 What are the sample sizes we have? X1 and X2. 36 159 00:12:07,130 --> 00:12:14,550 plus 31. Over 72 plus 7. 72 plus 7. So that means 160 00:12:14,550 --> 00:12:22,310 67 over 152.549. 161 00:12:24,690 --> 00:12:25,610 120. 162 00:12:30,400 --> 00:12:34,620 So simple calculations give B1 and B2, as well as 163 00:12:34,620 --> 00:12:39,340 B dash. Now, plug these values on the Z-state 164 00:12:39,340 --> 00:12:43,540 formula, we get the value that is this. So first, 165 00:12:44,600 --> 00:12:47,560 state null and alternative hypothesis, pi 1 minus 166 00:12:47,560 --> 00:12:50,080 pi 2 equals 0. That means the two populations are 167 00:12:50,080 --> 00:12:55,290 equal. We are going to test this one against Pi 1 168 00:12:55,290 --> 00:12:58,570 minus Pi 2 is not zero. That means there is a 169 00:12:58,570 --> 00:13:02,430 significant difference between proportions. Now 170 00:13:02,430 --> 00:13:06,290 for men, we got proportion of 50%. That's for the 171 00:13:06,290 --> 00:13:09,370 similar proportion. And similar proportion for 172 00:13:09,370 --> 00:13:15,390 women who will vote yes for position A is 62%. The 173 00:13:15,390 --> 00:13:19,530 pooled estimate for the overall proportion equals 174 00:13:19,530 --> 00:13:24,530 0.549. Now, based on this information, we can 175 00:13:24,530 --> 00:13:27,610 calculate the Z statistic. Straightforward 176 00:13:27,610 --> 00:13:33,470 calculation, you will end with this result. So, Z 177 00:13:33,470 --> 00:13:39,350 start negative 1.31. 178 00:13:41,790 --> 00:13:44,950 So, we have to compute this one before either 179 00:13:44,950 --> 00:13:47,650 before using any of the approaches we have. 180 00:13:50,940 --> 00:13:52,960 If we are going to use their critical value 181 00:13:52,960 --> 00:13:55,140 approach, we have to find Z alpha over 2 which is 182 00:13:55,140 --> 00:13:59,320 1 more than 6. Now the question is, is this value 183 00:13:59,320 --> 00:14:05,140 falling the rejection regions right or left? it's 184 00:14:05,140 --> 00:14:10,660 clear that this value, negative 1.31, lies in the 185 00:14:10,660 --> 00:14:12,960 non-rejection region, so we don't reject a null 186 00:14:12,960 --> 00:14:17,900 hypothesis. So my decision is don't reject H0. My 187 00:14:17,900 --> 00:14:22,580 conclusion is there is not significant evidence of 188 00:14:22,580 --> 00:14:25,160 a difference in proportions who will vote yes 189 00:14:25,160 --> 00:14:31,300 between men and women. Even it seems to me that 190 00:14:31,300 --> 00:14:34,550 there is a difference between Similar proportions, 191 00:14:34,790 --> 00:14:38,290 50% and 62%. Still, this difference is not 192 00:14:38,290 --> 00:14:41,670 significant in order to say that there is 193 00:14:41,670 --> 00:14:44,730 significant difference between the proportions of 194 00:14:44 223 00:16:48,920 --> 00:16:53,940 for pi 1 minus pi 2 is given by this equation. Now 224 00:16:53,940 --> 00:16:58,250 let's see how can we use the other two approaches 225 00:16:58,250 --> 00:17:01,570 in order to test if there is a significant 226 00:17:01,570 --> 00:17:04,230 difference between the proportions of men and 227 00:17:04,230 --> 00:17:07,910 women. I'm sure you don't have this slide for 228 00:17:07,910 --> 00:17:12,730 computing B value and confidence interval. 229 00:17:30,230 --> 00:17:35,050 Now since we are talking about two-tails, your B 230 00:17:35,050 --> 00:17:37,670 value should be the probability of Z greater than 231 00:17:37,670 --> 00:17:45,430 1.31 and smaller than negative 1.31. So my B value 232 00:17:45,430 --> 00:17:53,330 in this case equals Z greater than 1.31 plus Z 233 00:17:55,430 --> 00:17:59,570 smaller than negative 1.31. Since we are talking 234 00:17:59,570 --> 00:18:03,810 about two-tailed tests, so there are two rejection 235 00:18:03,810 --> 00:18:08,910 regions. My Z statistic is 1.31, so it should be 236 00:18:08,910 --> 00:18:14,990 here 1.31 to the right, and negative 1.31 to the left. Now, what's 237 00:18:14,990 --> 00:18:20,150 the probability that the Z statistic will fall in 238 00:18:20,150 --> 00:18:23,330 the rejection regions, right or left? So we have 239 00:18:23,330 --> 00:18:27,650 to add. B of Z greater than 1.31 and B of Z 240 00:18:27,650 --> 00:18:30,750 smaller than negative 1.31. Now the two areas to the 241 00:18:30,750 --> 00:18:34,790 right of 1.31 and to the left of negative 1.31 are 242 00:18:34,790 --> 00:18:38,110 equal because of symmetry. So just compute one and 243 00:18:38,110 --> 00:18:43,030 multiply that by two, you will get the B value. So 244 00:18:43,030 --> 00:18:47,110 two times. Now by using the concept in chapter 245 00:18:47,110 --> 00:18:50,550 six, easily you can compute either this one or the 246 00:18:50,550 --> 00:18:53,030 other one. The other one directly from the 247 00:18:53,030 --> 00:18:55,870 negative z-score table. The other one you should 248 00:18:55,870 --> 00:18:58,710 have the complement 1 minus, because it's smaller 249 00:18:58,710 --> 00:19:02,170 than 1.1. And either way you will get this result. 250 00:19:05,110 --> 00:19:11,750 Now my p-value is around 19%. Always we reject the 251 00:19:11,750 --> 00:19:14,930 null hypothesis. If your B value is smaller than 252 00:19:14,930 --> 00:19:20,410 alpha, that always we reject null hypothesis, if my 253 00:19:20,410 --> 00:19:25,950 B value is smaller than alpha, alpha is given 5% 254 00:19:25,950 --> 00:19:31,830 since B value equals 255 00:19:31,830 --> 00:19:36,910 19%, which is much bigger than 5%, so we don't reject our analysis. So my 256 00:19:36,910 --> 00:19:41,170 decision is we don't reject at zero. So the same 257 00:19:41,170 --> 00:19:48,390 conclusion as we reached by using critical 258 00:19:48,390 --> 00:19:52,690 value approach. So again, by using B value, we have to 259 00:19:52,690 --> 00:19:57,850 compute the probability that your Z statistic 260 00:19:57,850 --> 00:20:00,770 falls in the rejection regions. I end with this 261 00:20:00,770 --> 00:20:05,320 result, my B value is around 19%. As we mentioned 262 00:20:05,320 --> 00:20:10,600 before, we reject null hypothesis if my B value is 263 00:20:10,600 --> 00:20:14,180 smaller than alpha. Now, my B value in this case 264 00:20:14,180 --> 00:20:17,920 is much, much bigger than 5%, so my decision is 265 00:20:17,920 --> 00:20:22,740 don't reject null hypothesis. Any questions? 267 00:20:36,140 --> 00:20:41,160 The other approach, the third one, confidence 268 00:20:41,160 --> 00:20:42,520 interval approach. 269 00:20:46,260 --> 00:20:48,980 Now, for the confidence interval approach, we have 270 00:20:48,980 --> 00:20:53,960 this equation, b1 minus b2. Again, the point 271 00:20:53,960 --> 00:21:03,760 estimate, plus or minus z square root b1 times 1 272 00:21:03,760 --> 00:21:09,810 minus b1 divided by a1. B2 times 1 minus B2 273 00:21:09,810 --> 00:21:11,650 divided by N2. 274 00:21:13,850 --> 00:21:20,730 Now we have B1 and B2, so 0.5 minus 0.62. That's 275 00:21:20,730 --> 00:21:25,170 your calculations from previous information we 276 00:21:25,170 --> 00:21:28,470 have. Plus or minus Z alpha over 2, the critical 277 00:21:28,470 --> 00:21:35,030 value again is 1.96 times Square root of P1, 0.5 278 00:21:35,030 --> 00:21:41,090 times 1 minus 0.5 divided by N1 plus P2, 62 percent 279 00:21:41,090 --> 00:21:46,550 times 1 minus P2 divided by N2. 0.5 minus 62 280 00:21:46,550 --> 00:21:50,650 percent is negative 12 percent plus or minus the 281 00:21:50,650 --> 00:21:53,090 margin of error. This amount is again as we 282 00:21:53,090 --> 00:21:56,730 mentioned before, is the margin of error, 0.177. 283 00:21:57,530 --> 00:21:59,830 Now simple calculation will end with this result 284 00:21:59,830 --> 00:22:03,300 that is the difference between the two proportions 285 00:22:03,300 --> 00:22:09,820 lie between negative 0.296 and 0.057. That means 286 00:22:09,820 --> 00:22:14,580 we are 95% confident that the difference between 287 00:22:14,580 --> 00:22:19,100 the proportions of men who will vote yes for a 288 00:22:19,100 --> 00:22:27,640 position A and women equals negative 0.297 up to 0 289 00:22:27,640 --> 00:22:31,680 .057. Now the question is, since we are testing 290 00:22:31,680 --> 00:22:37,380 if the difference between p1 and p2 equals zero, the 291 00:22:37,380 --> 00:22:41,700 question is does this interval contain zero or 292 00:22:41,700 --> 00:22:47,680 capture zero? Now since we start here from 293 00:22:47,680 --> 00:22:51,230 negative and end with positive, I mean the lower 294 00:22:51,230 --> 00:22:55,330 bound is negative 0.297 and the upper bound is 0 295 00:22:55,330 --> 00:23:00,610 .057. So zero is inside the interval, I mean the 296 00:23:00,610 --> 00:23:03,870 confidence interval contains zero in this case, so 297 00:23:03,870 --> 00:23:06,650 we don't reject the null hypothesis because maybe 298 00:23:06,650 --> 00:23:11,780 the difference equals zero. So since this interval 299 00:23:11,780 --> 00:23:16,300 does contain the hypothesized difference of zero, so we 300 00:23:16,300 --> 00:23:21,100 don't reject null hypothesis at 5% level. So the 301 00:23:21,100 --> 00:23:24,880 same conclusion as we got before by using critical 302 00:23:24,880 --> 00:23:27,460 value approach and p-value approach. So either 303 00:23:27,460 --> 00:23:32,100 one will end with the same decision. Either reject 304 00:23:32,100 --> 00:23:37,020 or fail to reject, it depends on the test itself. 305 00:23:38,760 --> 00:23:43,820 That's all. Do you have any question? Any 306 00:23:43,820 --> 00:23:47,540 question? So again, there are three different 307 00:23:47,540 --> 00:23:51,600 approaches in order to solve this problem. One is 308 00:23:51,600 --> 00:23:55,680 critical value approach, the standard one. The 309 00:23:55,680 --> 00:23:58,900 other two are the p-value approach and confidence 310 00:23:58,900 --> 00:24:02,140 interval. One more time, confidence interval is 311 00:24:02,140 --> 00:24:07,080 only valid for 312 00:24:08,770 --> 00:24:13,110 two-tailed tests. Because the confidence interval 313 00:24:13,110 --> 00:24:16,430 we have is just for two-tailed tests, so it could 314 00:24:16,430 --> 00:24:20,210 be used only for testing about two-tailed tests. 315 00:24:23,350 --> 00:24:25,990 As we mentioned before, I'm going to skip 316 00:24:25,990 --> 00:24:32,390 hypothesis for variances as well as ANOVA test. So 317 00:24:32,390 --> 00:24:36,410 that's all for chapter ten. 318 00:24:37,670 --> 00:24:42,390 But now I'm going to do some of the practice 319 00:24:42,390 --> 00:24:43,730 problems. 320 00:24:46,750 --> 00:24:52,630 Chapter 10. To practice, let's start with some 321 00:24:52,630 --> 00:24:55,270 practice problems for Chapter 10. 322 00:24:59,270 --> 00:25:03,770 A few years ago, Pepsi invited consumers to take 323 00:25:03,770 --> 00:25:08,870 the Pepsi challenge. Consumers were asked to 324 00:25:08,870 --> 00:25:13,790 decide which of two sodas, Coke or Pepsi, they 325 00:25:13,790 --> 00:25:17,930 preferred in a blind taste test. Pepsi was 326 00:25:17,930 --> 00:25:21,930 interested in determining what factors played a 327 00:25:21,930 --> 00:25:25,930 role in people's taste preferences. One of the 328 00:25:25,930 --> 00:25:28,630 factors studied was the gender of the consumer. 329 00:25:29,650 --> 00:25:32,350 Below are the results of the analysis comparing 330 00:25:32,350 --> 00:25:36,870 the taste preferences of men and women with the 331 00:25:36,870 --> 00:25:41,630 proportions depicting preference in or for Pepsi. 332 00:25:42,810 --> 00:25:49,190 For men, the sample size 333 00:25:49,190 --> 00:25:57,990 is 109. So that's your N1. And the proportion 334 00:26:00,480 --> 00:26:09,100 for men is around 42%. For women, 335 00:26:11,640 --> 00:26:25,720 N2 equals 52, and the proportion of females, 25%. The 336 00:26:25,720 --> 00:26:29,870 difference between the proportions of men and women or 337 00:26:29,870 --> 00:26:35,590 males and females is 0.172, around 0.172. And this 338 00:26:35,590 --> 00:26:41,530 statistic is given by 2.118, so approximately 2 339 00:26:41,530 --> 00:26:47,170 .12. Now, based on this result, based on this 340 00:26:47,170 --> 00:26:49,090 information, question number one, 341 00:26:53,910 --> 00:26:58,690 To determine if a difference exists in the taste 342 00:26:58,690 --> 00:27:04,490 preferences of men and women, give the correct 343 00:27:04,490 --> 00:27:06,970 alternative hypothesis that will guide the test. 344 00:27:08,830 --> 00:27:15,830 A, B, Why B? Because the test defines between the 345 00:27:15,830 --> 00:27:18,650 new form A and the new form B. Because if we say 346 00:27:18,650 --> 00:27:21,910 that H1 is equal to U1 minus M equals F, 347 00:27:28,970 --> 00:27:34,190 So the correct answer is B? B. So that's 348 00:27:34,190 --> 00:27:40,830 incorrect. C. Why? Why C is the correct answer? 349 00:27:45,470 --> 00:27:46,070 Because 350 00:27:52,720 --> 00:27:56,500 p1 is not equal because we have difference. So 351 00:27:56,500 --> 00:27:59,380 since we have difference here, it should be not 352 00:27:59,380 --> 00:28:02,240 equal to. And since we are talking about 353 00:28:02,240 --> 00:28:06,120 proportions, so you have to ignore A and B. So A 354 00:28:06,120 --> 00:28:10,020 and B should be ignored first. Then you either 355 00:28:10,020 --> 00:28:15,220 choose C or D. C is the correct answer. So C is 356 00:28:15,220 --> 00:28:20,440 the correct answer. That's for number one. Part 357 00:28:20,440 --> 00:28:27,100 two. Now suppose Pepsi wanted to test to determine 358 00:28:27,100 --> 00:28:35,680 if men preferred Pepsi more than women. Using 359 00:28:35,680 --> 00:28:38,400 the test statistic given, compute the appropriate 360 00:28:38,400 --> 00:28:43,940 p-value for the test. Let's assume that pi 1 is 361 00:28:43,940 --> 00:28:48,640 the population proportion for men who preferred 362 00:28:48,640 --> 00:28:56,440 Pepsi, and pi 2 for women who prefer Pepsi. Now 363 00:28:56,440 --> 00:29:00,140 he asks about suppose the company wanted to test 364 00:29:00,140 --> 00:29:02,760 to determine if males prefer Pepsi more than 365 00:29:02,760 --> 00:29:08,080 females. Using again the statistic given, which is 366 00:29:08,080 --> 00:29:13,400 2.12 for example, compute the appropriate p-value. Now 367 00:29:13,400 --> 00:29:18,160 let's state first H0 and H1. 368 00:29:27,450 --> 00:29:31,970 H1, pi 1 369 00:29:31,970 --> 00:29:34,410 minus pi 2 is greater than zero. 370 00:29:37,980 --> 00:29:42,740 Because it says that men prefer Pepsi more than 371 00:29:42,740 --> 00:29:46,940 women. pi 1 for men, pi 2 for women. So I 372 00:29:46,940 --> 00:29:50,800 should have pi 1 greater than pi 2, or pi 1 minus 373 00:29:50,800 --> 00:29:54,940 pi 2 is positive. So it's upper-tailed. Now, in this 374 00:29:54,940 --> 00:30:01,940 case, my p-value, its probability, is p. 375 00:30:05,680 --> 00:30:07,320 It's around this value. 376 00:30:12,410 --> 00:30:18,230 1 minus p of z smaller than 2.12. So 1 minus, 377 00:30:18,350 --> 00:30:21,530 now by using the table or the z table we have. 378 00:30:25,510 --> 00:30:29,370 Since we are talking about 2.12, so 379 00:30:29,370 --> 00:30:34,670 the answer is .983. So 380 00:30:34,670 --> 00:30:40,590 1 minus .983, so the answer is 0.017. So my p value 381 00:30:43,430 --> 00:30:49,890 equals 0.017. So A is the correct answer. Now if 382 00:30:49,890 --> 00:30:53,970 the problem is a two-tailed test, it should be 383 00:30:53,970 --> 00:30:57,450 multiplied by 2. So the answer, the correct one, should 384 00:30:57,450 --> 00:31:02,230 be B. So you have A and B. If it is one-tailed, 385 00:31:02,390 --> 00:31:06,310 your correct answer is A. If it is two-tailed, I 386 00:31:06,310 --> 00:31:10,550 mean, if we are testing to determine if a 387 00:31:10,550 --> 00:31:13,890 difference exists, then you have to multiply this 388 00:31:13,890 --> 00:31:19,030 one by two. So that's your p value. Any questions? 389 00:31:23,010 --> 00:31:27,550 Number three. Suppose Pepsi wanted to test to 390 00:31:27,550 --> 00:31:33,230 determine if men prefer Pepsi less than 391 00:31:33,230 --> 00:31:36,810 women, using the statistic given, compute the 392 00:31:36,810 --> 00:31:42,990 appropriate p-value. Now, H1 in this case, p1 is 393 00:31:42,990 --> 00:31:48,490 smaller than p2, p1 smaller than p2. Now your 394 00:31:48,490 --> 00:31:54,490 p-value, z is smaller than, because here it is 395 00:31:54,490 --> 00:31:58,050 smaller than my statistic 2.12. 396 00:32:01,570 --> 00:32:04,790 We don't write a negative sign. Because the value of 397 00:32:04,790 --> 00:32:08,150 the statistic is 2.12. But here we are going to 398 00:32:08,150 --> 00:32:11,790 test a lower-tailed test. So my p-value is p of Z 399 00:32:11,790 --> 00:32:15,250 smaller than. So smaller comes from the 400 00:32:15,250 --> 00:32:17,730 alternative. This is the sign under the alternative. 401 00:32:18,910 --> 00:32:21,810 And you have to take the value of the Z statistic 402 00:32:21,810 --> 00:32:22,510 as it is. 403 00:32:25,610 --> 00:32:34,100 So p of Z is smaller than 2.12. So they need, if 404 00:32:34,100 --> 00:32:38,060 you got a correct answer, D is the correct one. If p is 405 00:32:38,060 --> 00:32:40,420 the correct answer, you will get .9996 406 00:32:40,420 --> 00:32:47,620 .6, that's the incorrect answer. Any questions? The 407 00:32:47,620 --> 00:32:53,920 correct answer is D, number 408 00:32:53,920 --> 00:32:57,620 four. Suppose 409 00:32:57,620 --> 00:33:03,650 that Now, for example, forget the information we 410 00:33:03,650 --> 00:33:07,390 have so far for p-value. Suppose that the two 411 00:33:07,390 --> 00:33:11,910 -tailed p-value was really 445 00:35:55,720 --> 00:35:58,800 we if we reject it means that we have sufficient 446 00:35:58,800 --> 00:36:02,700 evidence to support the alternative so D is 447 00:36:02,700 --> 00:36:07,470 incorrect Now what's about C at five percent Five, 448 00:36:07,830 --> 00:36:10,570 so this value is greater than five, so we don't 449 00:36:10,570 --> 00:36:13,270 reject. So that's incorrect. 450 00:36:21,370 --> 00:36:28,030 B. At five, at 10% now, there is sufficient 451 00:36:28,030 --> 00:36:34,550 evidence. Sufficient means we reject. We reject. 452 00:36:35,220 --> 00:36:40,440 Since this B value, 0.7, is smaller than alpha. 7% 453 00:36:40,440 --> 00:36:44,240 is smaller than 10%. So we reject. That means you 454 00:36:44,240 --> 00:36:46,960 have to read carefully. There is sufficient 455 00:36:46,960 --> 00:36:50,280 evidence to include, to indicate the proportion of 456 00:36:50,280 --> 00:36:54,820 males preferring Pepsi differs from the proportion 457 00:36:54,820 --> 00:36:58,660 of females. That's correct. So B is the correct 458 00:36:58,660 --> 00:37:05,570 state. Now look at A. A, at 5% there is sufficient 459 00:37:05,570 --> 00:37:09,710 evidence? No, because this value is greater than 460 00:37:09,710 --> 00:37:16,970 alpha, so we don't reject. For this one. Here we 461 00:37:16,970 --> 00:37:21,050 reject because at 10% we reject. So B is the 462 00:37:21,050 --> 00:37:27,670 correct answer. Make sense? Yeah, exactly, for 463 00:37:27,670 --> 00:37:31,850 10%. If this value is 5%, then B is incorrect. 464 00:37:34,190 --> 00:37:38,690 Again, if we change this one to be 5%, still this 465 00:37:38,690 --> 00:37:39,870 statement is false. 466 00:37:43,050 --> 00:37:48,670 It should be smaller than alpha in order to reject 467 00:37:48,670 --> 00:37:53,770 the null hypothesis. So, B is the correct 468 00:37:53,770 --> 00:37:56,350 statement. 469 00:37:58,180 --> 00:38:02,080 Always insufficient means you don't reject null 470 00:38:02,080 --> 00:38:06,000 hypothesis. Now for D, we reject null hypothesis 471 00:38:06,000 --> 00:38:10,500 at 8%. Since this value 0.7 is smaller than alpha, 472 00:38:10,740 --> 00:38:14,700 so we reject. So this is incorrect. Now for C, be 473 00:38:14,700 --> 00:38:19,440 careful. At 5%, if this, if we change this one 474 00:38:19,440 --> 00:38:23,560 little bit, there is insufficient evidence. What 475 00:38:23,560 --> 00:38:32,320 do you think? About C. If we change part C as at 5 476 00:38:32,320 --> 00:38:36,540 % there is insufficient evidence to indicate the 477 00:38:36,540 --> 00:38:39,840 proportion of males preferring Pepsi equals. 478 00:38:44,600 --> 00:38:49,940 You cannot say equal because this one maybe yes 479 00:38:49,940 --> 00:38:53,200 maybe no you don't know the exact answer. So if we 480 00:38:53,200 --> 00:38:56,380 don't reject the null hypothesis then you don't 481 00:38:56,380 --> 00:38:58,780 have sufficient evidence in order to support each 482 00:38:58,780 --> 00:39:03,800 one. So, don't reject the zero as we mentioned 483 00:39:03,800 --> 00:39:10,660 before. Don't reject the zero does not imply 484 00:39:10,660 --> 00:39:16,840 if zero is true. It means the evidence, the data 485 00:39:16,840 --> 00:39:19,500 you have is not sufficient to support the 486 00:39:19,500 --> 00:39:25,260 alternative evidence. So, don't say equal to. So 487 00:39:25,260 --> 00:39:30,560 say don't reject rather than saying accept. So V 488 00:39:30,560 --> 00:39:31,460 is the correct answer. 489 00:39:35,940 --> 00:39:43,020 Six, seven, and eight. Construct 90% confidence 490 00:39:43,020 --> 00:39:48,380 interval, construct 95, construct 99. It's 491 00:39:48,380 --> 00:39:52,700 similar, just the critical value will be changed. 492 00:39:53,620 --> 00:39:58,380 Now my question is, which is the widest confidence 493 00:39:58,380 --> 00:40:03,080 interval in this case? 99. The last one is the 494 00:40:03,080 --> 00:40:08,040 widest because here 99 is the largest confidence 495 00:40:08,040 --> 00:40:11,160 limit. So that means the width of the interval is 496 00:40:11,160 --> 00:40:12,620 the largest in this case. 497 00:40:17,960 --> 00:40:23,770 For 5, 6 and 7. The question is construct either 498 00:40:23,770 --> 00:40:30,930 90%, 95% or 99% for the same question. Simple 499 00:40:30,930 --> 00:40:33,510 calculation will give the confidence interval for 500 00:40:33,510 --> 00:40:38,590 each one. My question was, which one is the widest 501 00:40:38,590 --> 00:40:43,630 confidence interval? Based on the C level, 99% 502 00:40:43,630 --> 00:40:47,350 gives the widest confidence interval comparing to 503 00:40:47,350 --> 00:41:02,100 90% and 95%. The exact answers for 5, 6 and 7, 0.5 504 00:41:02,100 --> 00:41:08,900 to 30 percent. For 95 percent, 0.2 to 32 percent. 505 00:41:10,750 --> 00:41:16,030 For 99, negative 0.3 to 0.37. So this is the 506 00:41:16,030 --> 00:41:21,970 widest. Because here we start from 5 to 30. Here 507 00:41:21,970 --> 00:41:26,030 we start from lower than 5, 2%, up to upper, for 508 00:41:26,030 --> 00:41:31,190 greater than 30, 32. Here we start from negative 3 509 00:41:31,190 --> 00:41:35,330 % up to 37. So this is the widest confidence 510 00:41:35,330 --> 00:41:41,950 interval. Number six. Number six. number six five 511 00:41:41,950 --> 00:41:44,850 six and seven are the same except we just share 512 00:41:44,850 --> 00:41:49,710 the confidence level z so here we have one nine 513 00:41:49,710 --> 00:41:54,070 six instead of one six four and two point five 514 00:41:54,070 --> 00:42:01,170 seven it's our seven six next read the table e 515 00:42:12,610 --> 00:42:19,330 Table A. Corporation randomly selects 150 516 00:42:19,330 --> 00:42:25,830 salespeople and finds that 66% who have never 517 00:42:25,830 --> 00:42:29,070 taken self-improvement course would like such a 518 00:42:29,070 --> 00:42:33,830 course. So currently, or in recent, 519 00:42:37,660 --> 00:42:46,940 It says that out of 150 sales people, find that 66 520 00:42:46,940 --> 00:42:51,000 % would 521 00:42:51,000 --> 00:42:56,720 like to take such course. The firm did a similar 522 00:42:56,720 --> 00:43:01,480 study 10 years ago. So in the past, they had the 523 00:43:01,480 --> 00:43:07,430 same study in which 60% of a random sample of 160 524 00:43:07,430 --> 00:43:12,430 salespeople wanted a self-improvement course. So 525 00:43:12,430 --> 00:43:13,710 in the past, 526 00:43:16,430 --> 00:43:25,230 into 160, and proportion is 60%. The groups are 527 00:43:25,230 --> 00:43:29,690 assumed to be independent random samples. Let Pi 1 528 00:43:29,690 --> 00:43:32,890 and Pi 2 represent the true proportion of workers 529 00:43:32,890 --> 00:43:36,030 who would like to attend a self-improvement course 530 00:43:36,030 --> 00:43:39,550 in the recent study and the past study 531 00:43:39,550 --> 00:43:44,490 respectively. So suppose Pi 1 and Pi 2. Pi 1 for 532 00:43:44,490 --> 00:43:49,470 recent study and Pi 2 for the past study. So 533 00:43:49,470 --> 00:43:53,590 that's the question. Now, question number one. 534 00:43:56,580 --> 00:44:00,220 If the firm wanted to test whether this proportion 535 00:44:00,220 --> 00:44:06,800 has changed from the previous study, which 536 00:44:06,800 --> 00:44:09,100 represents the relevant hypothesis? 537 00:44:14,160 --> 00:44:18,540 Again, the firm wanted to test whether this 538 00:44:18,540 --> 00:44:21,740 proportion has changed. From the previous study, 539 00:44:22,160 --> 00:44:25,900 which represents the relevant hypothesis in this 540 00:44:25,900 --> 00:44:26,140 case? 541 00:44:33,560 --> 00:44:40,120 Which is the correct? A is 542 00:44:40,120 --> 00:44:44,500 the correct answer. Why A is the correct answer? 543 00:44:45,000 --> 00:44:48,040 Since we are talking about proportions, so it 544 00:44:48,040 --> 00:44:51,750 should have pi. It changed, it means does not 545 00:44:51,750 --> 00:44:55,410 equal 2. So A is the correct answer. Now B is 546 00:44:55,410 --> 00:45:00,850 incorrect because why B is incorrect? Exactly 547 00:45:00,850 --> 00:45:03,770 because under H0 we have pi 1 minus pi 2 does not 548 00:45:03,770 --> 00:45:08,570 equal 0. Always equal sign appears only under the 549 00:45:08,570 --> 00:45:14,950 null hypothesis. So it's the opposite here. Now C 550 00:45:14,950 --> 00:45:21,190 and D talking about Upper tier or lower tier, but 551 00:45:21,190 --> 00:45:23,890 here we're talking about two-tiered test, so A is 552 00:45:23,890 --> 00:45:24,750 the correct answer. 553 00:45:29,490 --> 00:45:33,090 This sign null hypothesis states incorrectly, 554 00:45:34,030 --> 00:45:38,010 because under H0 should have equal sign, and for 555 00:45:38,010 --> 00:45:39,730 alternate it should be not equal to. 556 00:45:42,770 --> 00:45:43,630 Number two. 557 00:45:47,860 --> 00:45:51,840 If the firm wanted to test whether a greater 558 00:45:51,840 --> 00:45:56,680 proportion of workers would currently like to 559 00:45:56,680 --> 00:46:00,180 attend a self-improvement course than in the past, 560 00:46:00,900 --> 00:46:05,840 currently, the proportion is greater than in the 561 00:46:05,840 --> 00:46:13,680 past. Which represents the relevant hypothesis? C 562 00:46:13,680 --> 00:46:18,180 is the correct answer. Because it says a greater 563 00:46:18,180 --> 00:46:22,340 proportion of workers work currently. So by one, 564 00:46:22,420 --> 00:46:26,340 greater than by two. So C is the correct answer. 565 00:46:31,340 --> 00:46:40,140 It says that the firm wanted to test proportion of 566 00:46:40,140 --> 00:46:46,640 workers currently study 567 00:46:46,640 --> 00:46:50,320 or recent study by one represents the proportion 568 00:46:50,320 --> 00:46:55,140 of workers who would like to attend the course so 569 00:46:55,140 --> 00:46:58,080 that's by one greater than 570 00:47:01,730 --> 00:47:05,350 In the past. So it means by one is greater than by 571 00:47:05,350 --> 00:47:11,870 two. It means by one minus by two is positive. So 572 00:47:11,870 --> 00:47:14,590 the alternative is by one minus two by two is 573 00:47:14,590 --> 00:47:16,430 positive. So this one is the correct answer. 574 00:47:21,530 --> 00:47:26,910 Exactly. If if here we have what in the past 575 00:47:26,910 --> 00:47:30,430 should be it should be the correct answer. 576 00:47:34,690 --> 00:47:40,450 That's to three. Any question for going to number 577 00:47:40,450 --> 00:47:49,590 three? Any question for number two? Three. What is 578 00:47:49,590 --> 00:47:52,790 the unbiased point estimate for the difference 579 00:47:52,790 --> 00:47:54,410 between the two population proportions? 580 00:47:58,960 --> 00:48:04,360 B1 minus B2 which is straight forward calculation 581 00:48:04,360 --> 00:48:06,980 gives A the correct answer. Because the point 582 00:48:06,980 --> 00:48:13,320 estimate in this case is B1 minus B2. B1 is 66 583 00:48:13,320 --> 00:48:18,560 percent, B2 is 60 percent, so the answer is 6 584 00:48:18,560 --> 00:48:26,190 percent. So B1 minus B2 which is 6 percent. I 585 00:48:26,190 --> 00:48:32,450 think three is straightforward. Number four, what 586 00:48:32,450 --> 00:48:38,450 is or are the critical values which, when 587 00:48:38,450 --> 00:48:41,870 performing a z-test on whether population 588 00:48:41,870 --> 00:48:46,570 proportions are different at 5%. Here, yes, we are 589 00:48:46,570 --> 00:48:52,250 talking about two-tailed test, and alpha is 5%. So 590 00:48:52,250 --> 00:48:55,550 my critical values, they are two critical values, 591 00:48:55,630 --> 00:48:55,830 actually. 592 00:49:27,080 --> 00:49:31,000 What is or are the critical values when testing 593 00:49:31,000 --> 00:49:34,260 whether population proportions are different at 10 594 00:49:34,260 --> 00:49:39,240 %? The same instead here we have 10 instead of 5%. 595 00:49:40,920 --> 00:49:45,100 So A is the correct answer. So just use the table. 596 00:49:47,340 --> 00:49:51,440 Now for the previous one, we have 0 to 5, 0 to 5. 597 00:49:51,980 --> 00:49:57,740 The other one, alpha is 10%. So 0, 5 to the right, 598 00:49:57,880 --> 00:50:03,580 the same as to the left. So plus or minus 164. 599 00:50:06,700 --> 00:50:11,580 So 4 and 5 by using the z table. 600 00:50:20,560 --> 00:50:25,280 So exactly, since alpha here is 1, 0, 2, 5, so the 601 00:50:25,280 --> 00:50:27,880 area becomes smaller than, so it should be z 602 00:50:27,880 --> 00:50:32,380 greater than. So 1.106, the other one 1.645, 603 00:50:32,800 --> 00:50:38,030 number 6. What is or are? The critical value in 604 00:50:38,030 --> 00:50:42,450 testing whether the current population is higher 605 00:50:42,450 --> 00:50:50,990 than. Higher means above. Above 10. Above 10, 5%. 606 00:50:50,990 --> 00:50:55,870 So which? B. 607 00:50:58,470 --> 00:51:00,810 B is the correct. Z alpha. 608 00:51:06,700 --> 00:51:08,440 So, B is the correct answer. 609 00:51:11,200 --> 00:51:11,840 7. 610 00:51:14,740 --> 00:51:21,320 7 and 8 we should have to calculate number 1. 7 611 00:51:21,320 --> 00:51:25,880 was the estimated standard error of the difference 612 00:51:25,880 --> 00:51:29,660 between the two sample proportions. We should have 613 00:51:29,660 --> 00:51:30,740 a standard error. 614 00:51:34,620 --> 00:51:40,320 Square root, B dash 1 minus B dash multiplied by 1 615 00:51:40,320 --> 00:51:45,300 over N1 plus 1 over N2. And we have to find B dash 616 00:51:45,300 --> 00:51:49,220 here. Let's see how can we find B dash. 617 00:51:52,720 --> 00:51:59,700 B dash 618 00:51:59,700 --> 00:52:05,800 equal x1 plus x2. Now what's the value of X1? 619 00:52:10,400 --> 00:52:16,220 Exactly. Since B1 is X1 over N1. So that means X1 620 00:52:16,220 --> 00:52:26,600 is N1 times B1. So N1 is 150 times 60%. So that's 621 00:52:26,600 --> 00:52:35,980 99. And similarly, X2 N2, which is 160, times 60% 622 00:52:35,980 --> 00:52:48,420 gives 96. So your B dash is x1 plus x2 divided by 623 00:52:48,420 --> 00:52:55,200 N1 plus N2, which is 150 plus 310. So complete B 624 00:52:55,200 --> 00:52:58,760 dash versus the bold estimate of overall 625 00:52:58,760 --> 00:53:03,570 proportion So 9 and 9 plus 9 is 6. 626 00:53:06,390 --> 00:53:07,730 That's just B-. 627 00:53:13,210 --> 00:53:14,290 6 to 9. 628 00:53:17,150 --> 00:53:23,190 6 to 9. So this is not your answer. It's just B-. 629 00:53:23,770 --> 00:53:29,030 Now take this value and the square root of 6 to 9. 630 00:53:30,060 --> 00:53:36,280 times 1.629 multiplied by 1 over N1 which is 150 631 00:53:36,280 --> 00:53:44,980 plus 160. That's your standard error. B dash is 632 00:53:44,980 --> 00:53:49,080 not standard error. B dash is the bold estimate of 633 00:53:49,080 --> 00:53:53,740 overall 667 00:56:53,150 --> 00:56:58,230 critical regions are 1.96 and above or smaller 668 00:56:58,230 --> 00:57:07,550 than minus 1.96. Now, my z statistic is 1.903. Now 669 00:57:07,550 --> 00:57:12,610 this value falls in the non-rejection region. So 670 00:57:12,610 --> 00:57:14,310 we don't reject the null hypothesis. 671 00:57:16,900 --> 00:57:21,400 Ignore A and C, so the answer is either B or D. 672 00:57:22,260 --> 00:57:26,360 Now let's read B. Don't reject the null and 673 00:57:26,360 --> 00:57:28,820 conclude that the proportion of employees who are 674 00:57:28,820 --> 00:57:31,600 interested in self-improvement course has not 675 00:57:31,600 --> 00:57:32,100 changed. 676 00:57:37,040 --> 00:57:40,060 That's correct. Because we don't reject the null 677 00:57:40,060 --> 00:57:42,900 hypothesis. It means there is no significant 678 00:57:42,900 --> 00:57:45,760 difference. So it has not changed. Now, D, don't 679 00:57:45,760 --> 00:57:47,540 reject the null hypothesis and conclude the 680 00:57:47,540 --> 00:57:49,760 proportion of Obliques who are interested in a 681 00:57:49,760 --> 00:57:52,700 certain point has increased, which is incorrect. 682 00:57:53,640 --> 00:57:57,960 So B is the correct answer. So again, since my Z 683 00:57:57,960 --> 00:58:01,080 statistic falls in the non-rejection region, we 684 00:58:01,080 --> 00:58:04,380 don't reject the null hypothesis. So either B or D 685 00:58:04,380 --> 00:58:07,350 is the correct answer. But here we are talking 686 00:58:07,350 --> 00:58:12,190 about none or don't reject the null hypothesis. 687 00:58:12,470 --> 00:58:14,310 That means we don't have sufficient evidence 688 00:58:14,310 --> 00:58:17,610 support that there is significant change between 689 00:58:17,610 --> 00:58:20,670 the two proportions. So there is no difference. So 690 00:58:20,670 --> 00:58:23,270 it has not changed. It's the correct one. So you 691 00:58:23,270 --> 00:58:29,890 have to choose B. So B is the most correct answer. 692 00:58:30,830 --> 00:58:35,600 Now, 10, 11, and 12. Talking about constructing 693 00:58:35,600 --> 00:58:41,700 confidence interval 99, 95, and 90%. It's similar. 694 00:58:42,620 --> 00:58:46,140 And as we mentioned before, 99% will give the 695 00:58:46,140 --> 00:58:50,940 widest confidence interval. And the answers for 696 00:58:50,940 --> 00:59:04,300 these are 14, 11, 14, is negative 0.8 to 20%. For 697 00:59:04,300 --> 00:59:11,720 11, 0.5, negative 0.5 to 17. For 90%, negative 0.3 698 00:59:11,720 --> 00:59:15,420 to 0.15. So this is the widest confidence 699 00:59:15,420 --> 00:59:22,220 interval, which was for 99%. So similar as the 700 00:59:22,220 --> 00:59:26,360 previous one we had discussed. So for 99, always 701 00:59:26,360 --> 00:59:32,230 we get The widest confidence interval. Any 702 00:59:32,230 --> 00:59:37,490 question? That's all. Next time shall start 703 00:59:37,490 --> 00:59:41,350 chapter 12, Chi-square test of independence.