1 00:00:07,390 --> 00:00:09,890 So last time, we discussed how to find the 2 00:00:09,890 --> 00:00:14,570 probabilities underneath the normal curve for 3 00:00:14,570 --> 00:00:20,450 three cases. If the point lies in the lower tail, 4 00:00:21,690 --> 00:00:28,150 as this one, or upper tail, or between. So we can 5 00:00:28,150 --> 00:00:32,290 do the computations for this kind of 6 00:00:32,290 --> 00:00:36,090 probabilities. And I think we gave two examples. 7 00:00:36,620 --> 00:00:40,400 For example, if we are looking for probability of 8 00:00:40,400 --> 00:00:45,020 Z is smaller than 0.1, and Z, as we mentioned last 9 00:00:45,020 --> 00:00:48,660 time, is the standardized normal distribution that 10 00:00:48,660 --> 00:00:53,120 has mean of 0 and sigma is 1. In this case, just 11 00:00:53,120 --> 00:00:57,540 go to the table you have. Now we are looking for 0 12 00:00:57,540 --> 00:01:03,700 .12, for example. So here we have 0.1. Under 2, we 13 00:01:03,700 --> 00:01:07,260 get this result. This value is the probability of 14 00:01:07,260 --> 00:01:10,480 Z smaller than 0.5. But you have to keep in mind 15 00:01:10,480 --> 00:01:14,460 that we have to transform first from normal 16 00:01:14,460 --> 00:01:17,580 distribution to standardized normal distribution. 17 00:01:18,620 --> 00:01:21,220 For this specific example, if we are looking for 18 00:01:21,220 --> 00:01:26,960 mean of 8 and standard deviation of 5, that's for 19 00:01:26,960 --> 00:01:30,020 the normal distribution. In this case, the z-score 20 00:01:30,020 --> 00:01:32,920 is given by this equation, which is x minus 3 21 00:01:32,920 --> 00:01:37,060 divided by sigma. So 8.6 minus 8 divided by 5 22 00:01:37,060 --> 00:01:41,360 gives 0.12. In this case, we can use the 23 00:01:41,360 --> 00:01:46,480 standardized normal table. Now, for the other 24 00:01:46,480 --> 00:01:48,700 case, we are looking for the probability of x 25 00:01:48,700 --> 00:01:51,880 greater than 8.6. So we are looking for the upper 26 00:01:51,880 --> 00:01:56,900 tail probability. The table we have gives the area 27 00:01:56,900 --> 00:02:01,640 to the left side. And we know that the total area 28 00:02:01,640 --> 00:02:04,760 underneath the normal curve is 1. So the 29 00:02:04,760 --> 00:02:08,980 probability of x greater than 8.6 is the same as 1 30 00:02:08,980 --> 00:02:13,720 minus the probability of x smaller than x less 31 00:02:13,720 --> 00:02:19,100 than 8.6. So here, for 8.6, we got z squared to be 32 00:02:19,100 --> 00:02:24,300 0.12. So the probability of z greater than 0.12 is 33 00:02:24,300 --> 00:02:27,620 the same as 1 minus z of z is smaller than 0.12. 34 00:02:28,850 --> 00:02:33,050 So 1 minus the answer we just got from part A will 35 00:02:33,050 --> 00:02:35,930 get us to something. So this is the probability of 36 00:02:35,930 --> 00:02:41,170 X is greater than 8.6. So all of the time, if you 37 00:02:41,170 --> 00:02:45,390 are asking about computing the probability in the 38 00:02:45,390 --> 00:02:48,290 upper tail, you have to first find the probability 39 00:02:48,290 --> 00:02:52,630 in the lower tail, then subtract that value from 40 00:02:52,630 --> 00:02:56,710 1. So that's the probability for the upper tail. 41 00:02:58,110 --> 00:03:02,470 The last one, we are looking for probability 42 00:03:02,470 --> 00:03:06,970 between two values. For example, x. What's the 43 00:03:06,970 --> 00:03:10,190 probability of x greater than 8 and smaller than 8 44 00:03:10,190 --> 00:03:15,670 .6? So we are looking for this area, the red area. 45 00:03:16,250 --> 00:03:18,710 So the probability of x between these two values 46 00:03:18,710 --> 00:03:21,850 actually equals the probability of x smaller than. 47 00:03:23,040 --> 00:03:27,560 8.6 minus the probability of X smaller than 8. 48 00:03:27,880 --> 00:03:31,140 And that, in this case, you have to compute two 49 00:03:31,140 --> 00:03:34,780 values of this score. One for the first value, 50 00:03:34,880 --> 00:03:40,020 which is A. This value is zero because the mean is 51 00:03:40,020 --> 00:03:45,240 zero. And we know that Z can be negative. If X is 52 00:03:45,240 --> 00:03:49,460 smaller than Mu, Z can be positive if X is greater 53 00:03:49,460 --> 00:03:55,220 than Mu and equals zero only if X equals Mu. In 54 00:03:55,220 --> 00:03:59,340 this case, X equals Mu, so Z score is zero. The 55 00:03:59,340 --> 00:04:03,220 other one as we got before is 0.12. So now, we 56 00:04:03,220 --> 00:04:07,340 transform actually the probability of X between 57 00:04:08,870 --> 00:04:13,870 8 and 8.6 to z-score between 0 and 0.12. In this 58 00:04:13,870 --> 00:04:17,170 case, we can use the normal theorem. Now, this 59 00:04:17,170 --> 00:04:21,130 area is, as we mentioned, just b of x smaller than 60 00:04:21,130 --> 00:04:28,850 0.12 minus b of x, b of z smaller than 0. This 61 00:04:28,850 --> 00:04:34,330 probability, we know that, 0.54878. Now, the 62 00:04:34,330 --> 00:04:38,170 probability of z smaller than 0 is one-half. 63 00:04:38,860 --> 00:04:41,380 Because the total area underneath the normal curve 64 00:04:41,380 --> 00:04:45,860 is 1, and 0 divided the curve into two equally 65 00:04:45,860 --> 00:04:49,100 parts. So the area to the right of 0 is the same 66 00:04:49,100 --> 00:04:52,100 as the area to the left of 0. So in this case, 67 00:04:52,880 --> 00:04:57,160 minus 0.5. So this is your answer. So the 68 00:04:57,160 --> 00:05:04,040 probability of Z between 8 and 8.6 is around 0.0478. 69 00:05:04,320 --> 00:05:07,020 I think we stopped last time at this point. 70 00:05:09,860 --> 00:05:17,760 This is another example to compute the probability 71 00:05:17,760 --> 00:05:24,020 of X greater than 7.4 and 8. Also, it's the same 72 00:05:24,020 --> 00:05:29,700 idea here, just find these scores for the two 73 00:05:29,700 --> 00:05:30,160 values. 74 00:05:33,850 --> 00:05:38,670 L with B of Z greater than minus 0.12 up to 0. Now 75 00:05:38,670 --> 00:05:47,010 this red area equals the area below 0. I mean B of 76 00:05:47,010 --> 00:05:50,330 Z smaller than 0 minus the probability of Z 77 00:05:50,330 --> 00:05:57,090 smaller than minus 0.12. Now by using symmetric 78 00:05:57,090 --> 00:06:00,470 probability of the normal distribution, we know 79 00:06:00,470 --> 00:06:04,620 that the probability of Z smaller than minus 0.12 80 00:06:04,620 --> 00:06:11,420 equals the probability of Z greater than 0.12. 81 00:06:11,780 --> 00:06:15,520 Because this area, if we have Z smaller than minus 82 00:06:15,520 --> 00:06:19,300 0.12, the area to the left, that equals the area 83 00:06:19,300 --> 00:06:21,560 to the right of the same point, because of 84 00:06:21,560 --> 00:06:25,480 symmetry. And finally, you will end with this 85 00:06:25,480 --> 00:06:25,740 result. 86 00:06:30,670 --> 00:06:36,730 D of z minus 0.12, all the way up to 0, this area, 87 00:06:37,630 --> 00:06:43,130 is the same as the area from 0 up to 0.5. So this 88 00:06:43,130 --> 00:06:46,530 area actually is the same as D of z between 0 and 89 00:06:46,530 --> 00:06:51,270 0.5. So if you have a negative sign, and then take 90 00:06:51,270 --> 00:06:54,390 the opposite one, the answer will be the same 91 00:06:54,390 --> 00:06:57,610 because the normal distribution is symmetric 92 00:06:57,610 --> 00:07:04,690 around 0. The questions, I think we stopped 93 00:07:04,690 --> 00:07:09,330 here. And also we talked about empirical rules. 94 00:07:10,930 --> 00:07:13,250 The one we mentioned in chapter three, in chapter 95 00:07:13,250 --> 00:07:16,170 three. And we know that, as we mentioned before, 96 00:07:16,310 --> 00:07:25,550 that is 68.16% of the observations fall within one 97 00:07:25,550 --> 00:07:30,660 standard deviation around the mean. So this area 98 00:07:30,660 --> 00:07:35,380 from mu minus one sigma up to mu plus sigma, this 99 00:07:35,380 --> 00:07:42,080 area covers around 68%. Also 100 00:07:42,080 --> 00:07:50,700 95% or actually 95.44% of the data falls within 101 00:07:50,700 --> 00:07:54,840 two standard deviations of the mean. And finally, 102 00:07:55,040 --> 00:08:00,220 around most of the data, around 99.73% of the data 103 00:08:00,220 --> 00:08:06,860 falls within three subdivisions of the population 104 00:08:06,860 --> 00:08:07,160 mean. 105 00:08:11,550 --> 00:08:14,970 And now the new topic is how can we find the X 106 00:08:14,970 --> 00:08:18,510 value if the probability is given. It's vice 107 00:08:18,510 --> 00:08:21,810 versa. In the previous questions, we were asking 108 00:08:21,810 --> 00:08:26,130 about find the probability, for example, if X is 109 00:08:26,130 --> 00:08:30,450 smaller than a certain number. Now suppose this 110 00:08:30,450 --> 00:08:34,070 probability is given, and we are looking to find 111 00:08:34,070 --> 00:08:38,710 this value. I mean, for example, suppose in the 112 00:08:38,710 --> 00:08:39,850 previous examples here, 113 00:08:43,380 --> 00:08:46,760 Suppose we know this probability. So the 114 00:08:46,760 --> 00:08:50,220 probability is given. The question is, how can we 115 00:08:50,220 --> 00:08:54,140 find this value? It's the opposite, sometimes 116 00:08:54,140 --> 00:08:57,180 called backward normal calculations. 117 00:09:01,660 --> 00:09:05,580 There are actually two steps to find the x value 118 00:09:05,580 --> 00:09:10,040 for a certain probability or for a given or for a 119 00:09:10,040 --> 00:09:12,900 known probability the first step we have to find 120 00:09:12,900 --> 00:09:20,540 the z score then use this equation to find the x 121 00:09:20,540 --> 00:09:25,100 value corresponding to the z score you have and x 122 00:09:25,100 --> 00:09:30,120 is just mu plus sigma times mu so first step you 123 00:09:30,120 --> 00:09:31,740 have to find the z score 124 00:09:35,350 --> 00:09:38,690 corresponding to the probability we have. So find 125 00:09:38,690 --> 00:09:43,870 the z value for the non-probability, then use that 126 00:09:43,870 --> 00:09:47,910 z score to find the value of x by using this 127 00:09:47,910 --> 00:09:52,010 equation. So x equals mu plus z sigma. z could be 128 00:09:52,010 --> 00:09:54,490 negative, could be positive, depends on the 129 00:09:54,490 --> 00:09:59,170 probability you have. If the probability is above 130 00:09:59,170 --> 00:10:03,390 0.5, I mean 0.5 and greater than 0.5, this 131 00:10:03,390 --> 00:10:08,880 corresponds to z positive. But if z-score is negative, I'm 132 00:10:08,880 --> 00:10:10,680 sorry, if z-score is negative, then the 133 00:10:10,680 --> 00:10:14,880 probability should be smaller than 0.5. So if the 134 00:10:14,880 --> 00:10:19,360 probability is given less than 0.5, then your z 135 00:10:19,360 --> 00:10:21,100 -score should be negative, otherwise should be 136 00:10:21,100 --> 00:10:23,720 positive. So you have to be careful in this case. 137 00:10:25,700 --> 00:10:31,240 Now look at this example. Let x represent the time 138 00:10:31,240 --> 00:10:35,770 it takes in seconds to download an image file 139 00:10:35,770 --> 00:10:39,730 from the internet. The same example as we did 140 00:10:39,730 --> 00:10:43,590 before. And here we assume that x is normal 141 00:10:43,590 --> 00:10:46,330 distribution with mean of 8 and standard deviation 142 00:10:46,330 --> 00:10:51,710 of 5. Now, let's see how can we find the value of 143 00:10:51,710 --> 00:10:58,050 x such that 20% of download times are smaller than 144 00:10:58,050 --> 00:10:58,410 x. 145 00:11:01,060 --> 00:11:04,580 So, this probability is a fraction. Also, always 146 00:11:04,580 --> 00:11:07,840 the probability is between 0 and 1. So, the 147 00:11:07,840 --> 00:11:11,820 probability here is 20%. In this case, your z 148 00:11:11,820 --> 00:11:15,560 -score should be negative. Because 20% is more 149 00:11:15,560 --> 00:11:18,660 than 0.5. So, z-score should be in this side, in 150 00:11:18,660 --> 00:11:19,320 the left side. 151 00:11:22,340 --> 00:11:26,380 So, again, he asks about finding x-value such that 152 00:11:26,380 --> 00:11:27,140 20%. 153 00:11:31,740 --> 00:11:35,400 So here again we are looking for this value, for 154 00:11:35,400 --> 00:11:40,760 the value of x, which is smaller than the area to 155 00:11:40,760 --> 00:11:45,680 the left of this x, equals 0.2. 156 00:11:47,480 --> 00:11:51,100 Now, the first step, we have to find the z-score. 157 00:11:52,650 --> 00:11:56,430 It's backward, z-score first, then x. Find a z 158 00:11:56,430 --> 00:12:00,450 -score corresponding to the probability of 0.2. 159 00:12:02,510 --> 00:12:07,710 The approximate one, the near value, I mean, to 160 00:12:07,710 --> 00:12:12,190 the 0.2 is 0.2005. Sometimes you have the exact 161 00:12:12,190 --> 00:12:16,570 value from the table you have, but most of the 162 00:12:16,570 --> 00:12:19,050 time you don't have it. So you have to look at the 163 00:12:19,050 --> 00:12:21,790 approximate value, which is very close to the one 164 00:12:21,790 --> 00:12:25,840 you have. So here, we are looking for 0.2. The 165 00:12:25,840 --> 00:12:30,660 closest value to 0.2 is 0.2005. Now, the 166 00:12:30,660 --> 00:12:34,720 corresponding value to this probability is minus 0 167 00:12:34,720 --> 00:12:40,120 .8 all the way up to 4. So your z-score is 168 00:12:40,120 --> 00:12:47,840 negative 0.84. So this is the first step. Any 169 00:12:47,840 --> 00:12:51,120 question? Again. 170 00:12:53,950 --> 00:12:57,050 Now if we just go back to this equation, 171 00:12:59,930 --> 00:13:03,670 z equals x minus mu over sigma. A cross 172 00:13:03,670 --> 00:13:07,810 multiplication, I mean if you multiply both sides 173 00:13:07,810 --> 00:13:16,110 by sigma, you will get sigma times z equals x 174 00:13:16,110 --> 00:13:17,510 minus mu. 175 00:13:32,120 --> 00:13:35,500 Now, in this question, 176 00:13:37,960 --> 00:13:43,160 he asks about, find the value of x such that 20% 177 00:13:43,160 --> 00:13:46,560 of download times are less than x. 178 00:13:50,740 --> 00:13:54,080 Now the probability is less than 0.5, so your z 179 00:13:54,080 --> 00:13:57,780 -score should be on the left side. So here we need 180 00:13:57,780 --> 00:14:03,660 to find the value of z first. Go back to the 181 00:14:03,660 --> 00:14:04,640 normal table you have. 182 00:14:07,680 --> 00:14:08,860 This is the normal table. 183 00:14:16,800 --> 00:14:21,250 We are looking for minus 0.2. I'm sorry, we are 184 00:14:21,250 --> 00:14:28,910 looking for 0.2. So the closest value to 0.2 is 185 00:14:28,910 --> 00:14:34,750 this one, 0.2005. So this is the closest value. 186 00:14:49,630 --> 00:14:54,470 So the exact answer is sometimes not given. So the 187 00:14:54,470 --> 00:14:59,190 approximate one, minus 0.8, all the way up to 4. 188 00:15:00,030 --> 00:15:06,410 So z-score minus 0.8. Any question? 189 00:15:10,330 --> 00:15:15,330 So the value of z-score is minus 0.84. So my 190 00:15: 223 00:19:18,560 --> 00:19:22,660 So z-score is 1. So we are looking for the 224 00:19:22,660 --> 00:19:24,700 probability of z greater than 1. 225 00:19:28,340 --> 00:19:35,220 1.5. 1.2. 1.5. 1.5. 1.5. So I'm looking for the 226 00:19:35,220 --> 00:19:37,760 probability of x of z greater than 227 00:19:40,980 --> 00:19:48,700 1 minus P 228 00:19:48,700 --> 00:19:52,700 of Z less than or equal to 1.5. Now go back to the 229 00:19:52,700 --> 00:19:53,040 table. 230 00:20:01,540 --> 00:20:07,260 Now 1.5 under 0. It's 0.9332. 231 00:20:11,410 --> 00:20:19,750 So, 1 minus this probability gives 0.0668. 232 00:20:21,210 --> 00:20:24,350 That's the probability of X greater than 4.4. 233 00:20:28,130 --> 00:20:31,590 So, the answer is 0.0668. 234 00:20:34,870 --> 00:20:37,550 Now, for the same question. 235 00:20:41,320 --> 00:20:44,380 What's the probability that a randomly selected 236 00:20:44,380 --> 00:20:47,080 fish will weigh between 3 and 5 pounds? 237 00:20:49,660 --> 00:20:50,400 3? 238 00:20:52,940 --> 00:21:01,040 Up to 5. So first we have to find the z-score for 3 239 00:21:01,040 --> 00:21:11,260 out of 5. For it to be just 3 minus 3.2. Divide by 240 00:21:11,260 --> 00:21:17,380 0.8 is the first z value. Negative 0.2 divided by 241 00:21:17,380 --> 00:21:30,360 0.8 minus 0.25. The other one, 5 minus 3.2 divided 242 00:21:30,360 --> 00:21:36,680 by 0.8. 1 minus 0.8 divided by 0.8 equals 243 00:21:42,680 --> 00:21:44,120 2.25. 244 00:21:50,840 --> 00:21:57,020 Just double check this result. So here, the 245 00:21:57,020 --> 00:22:04,520 probability of X between 3 and 5 equals minus 0 246 00:22:04,520 --> 00:22:08,200 .25, smaller than Z, smaller than 2.5. 247 00:22:12,650 --> 00:22:17,210 So it's transformed from normal distribution to 248 00:22:17,210 --> 00:22:20,750 standardized normal distribution. So here instead 249 00:22:20,750 --> 00:22:23,530 of computing the probability of X between three 250 00:22:23,530 --> 00:22:26,070 and five, we are looking for the probability 251 00:22:26,070 --> 00:22:31,350 between Z between actually minus. It's minus 252 00:22:31,350 --> 00:22:34,590 because your value here is smaller than the 253 00:22:34,590 --> 00:22:37,790 average. The average was 3.2, so it should be 254 00:22:37,790 --> 00:22:42,630 negative. So z score between minus 0.25 all the 255 00:22:42,630 --> 00:22:48,150 way up to 2.25. So now, this is the probability we 256 00:22:48,150 --> 00:22:56,590 are looking for. Zero in the middle minus one 257 00:22:56,590 --> 00:23:03,610 -fourth to the left of z of zero, mu of zero. And 258 00:23:03,610 --> 00:23:08,560 this is the value of 2.25. Now we are looking 259 00:23:08,560 --> 00:23:09,940 actually for this probability. 260 00:23:12,960 --> 00:23:18,360 The area between minus 0.25 all the way up to 2.5. 261 00:23:19,980 --> 00:23:25,200 So this area equals the 262 00:23:25,200 --> 00:23:29,000 probability of Z less than 2.25 minus. 263 00:23:34,280 --> 00:23:38,780 And again, use the normal. table to give this 264 00:23:38,780 --> 00:23:39,860 value and another one. 265 00:23:42,980 --> 00:23:52,400 Any questions? So first step here, we compute the 266 00:23:52,400 --> 00:23:56,880 z-score for each value x. So the problem is 267 00:23:56,880 --> 00:24:01,380 transformed from normal distribution to 268 00:24:01,380 --> 00:24:05,060 standardized normal distribution. So it becomes z 269 00:24:05,060 --> 00:24:11,500 between minus 1.25 up to 2.25. Now, this area, 270 00:24:11,960 --> 00:24:19,900 this dashed area equals the area below 2.25 minus 271 00:24:19,900 --> 00:24:25,000 the area below minus 1.25. Now, by using the 272 00:24:25,000 --> 00:24:27,760 similar way we did before, you will compute the 273 00:24:27,760 --> 00:24:30,960 value of z. The probability of z is smaller than 2 274 00:24:30,960 --> 00:24:39,580 .25 by using The normal table. So here, 2.2 up to 275 00:24:39,580 --> 00:24:47,900 5. So 9, 8, 7, 8. 9, 8, 7, 8. 276 00:24:53,900 --> 00:25:00,260 So the area below 2.25, 2.2, this value. All the 277 00:25:00,260 --> 00:25:03,940 way up to 5 gives 987. 278 00:25:05,540 --> 00:25:08,860 Now, what's about the probability of Z smaller 279 00:25:08,860 --> 00:25:15,320 than minus 0.25? If you go back to the Z table, 280 00:25:15,380 --> 00:25:18,960 but for the other one, the negative one. 281 00:25:23,120 --> 00:25:28,540 Minus 2 minus 0.2 up 282 00:25:28,540 --> 00:25:34,780 to 5. 0.4013 minus, 283 00:25:36,620 --> 00:25:43,100 that will give the probability between three and 284 00:25:43,100 --> 00:25:43,280 five. 285 00:25:46,180 --> 00:25:48,900 This is the second part. 286 00:25:51,120 --> 00:25:52,460 So the final answer. 287 00:26:00,630 --> 00:26:05,450 So this is the probability that the selected fish 288 00:26:05,450 --> 00:26:10,650 will weigh between three and five pounds. 289 00:26:11,810 --> 00:26:20,770 Now, other question is, for the same problem, you 290 00:26:20,770 --> 00:26:28,020 said a citation Catfish should be one of the top 2 291 00:26:28,020 --> 00:26:33,860 % in the weight. Assuming the weights of catfish 292 00:26:33,860 --> 00:26:38,660 are normally distributed, at what weight in bounds 293 00:26:38,660 --> 00:26:43,680 should the citation, the notation be established? 294 00:26:45,800 --> 00:26:50,600 So in this board, he asked about what's the value 295 00:26:50,600 --> 00:26:52,120 of x, for example. 296 00:26:57,160 --> 00:27:01,680 is greater than what value here. And this 297 00:27:01,680 --> 00:27:07,880 probability equals 2%. Because you said the 298 00:27:07,880 --> 00:27:14,720 citation catfish should be one of the top 2%. So 299 00:27:14,720 --> 00:27:23,560 the area in the right here, this area is 2%. 300 00:27:26,000 --> 00:27:34,080 What's the value of x in this case? So here, the 301 00:27:34,080 --> 00:27:38,420 value of x greater than a equals 0.02, and we are 302 00:27:38,420 --> 00:27:39,460 looking for this value. 303 00:27:45,750 --> 00:27:49,170 gives the area to the left side. So this 304 00:27:49,170 --> 00:27:54,490 probability actually, the area to the right is 2%, 305 00:27:54,490 --> 00:28:00,810 so the area to the left is 98%. So this is the 306 00:28:00,810 --> 00:28:01,410 same as, 307 00:28:04,610 --> 00:28:07,930 as we know, the equal sign does not matter because 308 00:28:07,930 --> 00:28:09,090 we have continuous distribution. 309 00:28:12,050 --> 00:28:14,650 continuous distribution, so equal sign does not 310 00:28:14,650 --> 00:28:18,510 matter. So now, if you ask about P of X greater 311 00:28:18,510 --> 00:28:22,190 than a certain value equals a probability of, for 312 00:28:22,190 --> 00:28:27,330 example, 0.02, you have to find the probability to 313 00:28:27,330 --> 00:28:31,890 the left, which is 0.98, because our table gives 314 00:28:31,890 --> 00:28:36,470 the area to the left. Now, we have to find the 315 00:28:36,470 --> 00:28:40,820 value of A such that Probability of X is more than 316 00:28:40,820 --> 00:28:44,820 or equal to 0.98. So again, we have to look at the 317 00:28:44,820 --> 00:28:50,140 normal table, but backwards, because this value is 318 00:28:50,140 --> 00:28:53,720 given. If the probability is given, we have to 319 00:28:53,720 --> 00:28:58,900 look inside the body of the table in order to find 320 00:28:58,900 --> 00:28:59,580 the z-score. 321 00:29:03,350 --> 00:29:07,850 x equals mu plus z sigma in order to find the 322 00:29:07,850 --> 00:29:12,290 corresponding value x. So again, go back to the 323 00:29:12,290 --> 00:29:22,010 normal table, and we are looking for 98%. The 324 00:29:22,010 --> 00:29:27,930 closest value to 98%, look here, if you stop here 325 00:29:27,930 --> 00:29:30,850 at 2, go to the right, 326 00:29:33,660 --> 00:29:39,380 Here we have 9798 or 327 00:29:39,380 --> 00:29:41,480 9803. 328 00:29:42,640 --> 00:29:50,460 So the answer might be your z-score could be 2.05 329 00:29:50,460 --> 00:29:59,440 or 2.06. So again, in this case, the table does 330 00:29:59,440 --> 00:30:04,400 not give the exact. So the approximate one might 331 00:30:04,400 --> 00:30:08,920 be between them exactly. Or just take one of 332 00:30:08,920 --> 00:30:13,140 these. So either you can take 9798, which is 333 00:30:13,140 --> 00:30:19,500 closer to 98% than 9803, because it's three 334 00:30:19,500 --> 00:30:23,780 distant apart. So maybe we can take this value. 335 00:30:24,500 --> 00:30:27,640 Again, if you take the other one, you will be OK. 336 00:30:28,540 --> 00:30:36,730 So you take either 2.05. or 2.06. So let's take 337 00:30:36,730 --> 00:30:45,270 the first value, for example. So my x equals mu, z 338 00:30:45,270 --> 00:30:56,570 is 2.05, times sigma, 0.8. Multiply 2.05 by 8, 0 339 00:30:56,570 --> 00:31:03,610 .8, then add 3.2, you will get What's your answer? 340 00:31:08,390 --> 00:31:17,450 3.2 plus 2 point... So around 4.8. So your answer 341 00:31:17,450 --> 00:31:18,890 is 4.84. 342 00:31:23,810 --> 00:31:29,470 Now if you go back to the problem, and suppose you 343 00:31:29,470 --> 00:31:30,830 know the value of x. 344 00:31:34,250 --> 00:31:36,010 So the probability of X. 345 00:31:42,990 --> 00:31:45,110 Double check to the answer. 346 00:31:49,490 --> 00:31:58,010 4.84. Just check. V of X greater than this value 347 00:31:58,010 --> 00:32:03,290 should 348 00:32:03,290 --> 00:32:09,500 be Two percent. Two percent. So the probability of 349 00:32:09,500 --> 00:32:13,440 X greater than this value should be equal to one 350 00:32:13,440 --> 00:32:18,980 zero. So this problem is called backward normal 351 00:32:18,980 --> 00:32:24,960 calculation because here first step we find the 352 00:32:24,960 --> 00:32:28,040 value of this score corresponding to this 353 00:32:28,040 --> 00:32:33,420 probability. Be careful. The probability of X 354 00:32:33,420 --> 00:32:38,740 greater than 2 is 0.02. So my value here should be 355 00:32:38,740 --> 00:32:44,980 to the right. Because it says greater than A is 356 00:32:44,980 --> 00:32:51,240 just 2%. If you switch the position of A, for 357 00:32:51,240 --> 00:32:57,130 example, if A is on this side, And he asked about 358 00:32:57,130 --> 00:33:02,850 E of X greater than E is 2%. This area is not 2%. 359 00:33:02,850 --> 00:33:10,050 From A up to infinity here, this area is not 2% 360 00:33:10,050 --> 00:33:14,070 because at least it's greater than 0.5. Make 361 00:33:14,070 --> 00:33:16,610 sense? So your A should be to the right side. 362 00:33:17,590 --> 00:33:21,810 Because the value of X greater than E, 2% is on 363 00:33:21,810 --> 00:33:26,400 the other side. Let's move to the next one. 364 00:33:36,180 --> 00:33:37,660 For the same question. 365 00:33:52,790 --> 00:33:57,150 Again, the owner of Catfish Market determined the 366 00:33:57,150 --> 00:34:00,930 average weight of a catfish 3.2 with 367 00:34:00,930 --> 00:34:04,670 standardization 0.8 and we are assuming the 368 00:34:04,670 --> 00:34:08,270 weights of catfish are normally distributed, kiosk 369 00:34:08,270 --> 00:34:17,390 above. Above what weight? Do 89.8% of the weights 370 00:34:17,390 --> 00:34:18,070 care? 371 00:34:20,630 --> 00:34:27,980 Above? And above, so x greater than. X minus. And 372 00:34:27,980 --> 00:34:34,900 98, 89, sorry, 89. So this is a percentage he's 373 00:34:34,900 --> 00:34:45,860 looking for. 89.8%. Now what's the value of A? Or 374 00:34:45,860 --> 00:34:51,560 above what weight? Do 89.8% of the weights occur? 375 00:34:57,730 --> 00:35:02,670 You just make the normal curve in order to 376 00:35:02,670 --> 00:35:07,010 understand the probability. Now, A should be to 377 00:35:07,010 --> 00:35:11,550 the right or to the left side? Imagine A in the 378 00:35:11,550 --> 00:35:15,990 right side here. Do you think the area above A is 379 00:35:15,990 --> 00:35:21,670 89%? It's smaller than 0.5 for sure. So it should 380 00:35:21,670 --> 00:35:29,950 be the other side. So this is your 8. Now, this 381 00:35:29,950 --> 00:35:36,690 area makes sense that it's above 0.5. It's 0.8980. 382 00:35:38,750 --> 00:35:42,850 Now, B of X greater than equals this value. And 383 00:35:42,850 --> 00:35:46,990 again, the table gives the area to the left. So 384 00:35:46,990 --> 00:35:53,740 this is actually X less than A, 1 minus this 385 00:35:53,740 --> 00:35:56,600 value, equals 0.1020. 386 00:36:01,480 --> 00:36:08,760 Now go back to 387 00:36:08,760 --> 00:36:14,400 the normal table. Here it's 0.1020. So it should 388 00:36:14,400 --> 00:36:17,500 be negative. I mean, your z-scope should be 389 00:36:17,500 --> 00:36:17,800 negative. 390 00:36:22,000 --> 00:36:25,640 Now look at 0.102. 391 00:36:28,560 --> 00:36:38,680 Exactly this value. 0.102 is minus 1.2 up to 7. So 392 00:36:38,680 --> 00:36:39,900 minus 1.27. 393 00:36:49,120 --> 00:36:57,900 Minus 1.2. All the way up to 7 is 0.102. So the 394 00:36:57,900 --> 00:37:04,160 corresponding z-score is minus 1.17. Now x again 395 00:37:04,160 --> 00:37:05,980 equals mu plus z sigma. 396 00:37:10,280 --> 00:37:19,960 Mu is 3.2 plus z is negative 1.17 times sigma. 397 00:37:24,250 --> 00:37:34,390 So it's equal to 3.2 minus 127 times 0.3. By 398 00:37:34,390 --> 00:37:36,510 calculator, you'll get the final result. 399 00:37:51,120 --> 00:37:56,180 If the probability is smaller than 0.5, then this 400 00:37:56,180 --> 00:38:00,480 one is negative. Go to the other one. If the 401 00:38:00,480 --> 00:38:04,040 probability is above 0.5, then use the positive z 402 00:38:04,040 --> 00:38:09,600 -score. So what's the answer? 2.18. 403 00:38:12,680 --> 00:38:19,850 Be careful. In the previous one, We had a 404 00:38:19,850 --> 00:38:28,870 probability of X greater than A equal 2%. In 405 00:38:28,870 --> 00:38:33,070 this case, the value of A, for example, is located 406 00:38:33,070 --> 00:38:35,930 in the upper tail here. 407 00:38:40,210 --> 00:38:45,530 For this part, you ask about B of X greater than A 408 00:38:45,530 --> 00:38:51,090 equal 0.89. It's here more than 0.5 445 00:42:54,950 --> 00:42:59,010 Below minus 3, I mean smaller than minus 3, or 446 00:42:59,010 --> 00:43:04,010 above 3, these points are suspected to be 447 00:43:04,010 --> 00:43:04,750 outliers. 448 00:43:09,230 --> 00:43:16,230 So any point, any data value smaller than minus 3 449 00:43:16,230 --> 00:43:22,330 in this form, or above plus 3 is considered to be 450 00:43:22,330 --> 00:43:30,250 an outlier. If we compute the lower limit, which 451 00:43:30,250 --> 00:43:37,310 is Q1 minus 1.5 IQR over the upper limit. 452 00:43:40,170 --> 00:43:47,490 So we 453 00:43:47,490 --> 00:43:51,690 have lower limit, upper limit. So lower limit. 454 00:43:55,400 --> 00:43:56,480 And upper limit. 455 00:43:59,980 --> 00:44:08,420 Any data point below lower limit or above upper 456 00:44:08,420 --> 00:44:13,080 limit is considered to be a type. So therefore, we 457 00:44:13,080 --> 00:44:18,320 have two methods to determine or to examine if the 458 00:44:18,320 --> 00:44:20,960 observation is enough there or not. One by using 459 00:44:20,960 --> 00:44:24,060 this score, straightforward. And the other one, we 460 00:44:24,060 --> 00:44:27,420 have to look for the lower limit and upper limit. 461 00:44:28,960 --> 00:44:34,200 The other method by using software and later on 462 00:44:34,200 --> 00:44:39,380 you will take SPSS in order to determine if the 463 00:44:39,380 --> 00:44:43,100 data is normally distributed by using something 464 00:44:43,100 --> 00:44:52,540 called QQ plot or normal probability plot. So I'm 465 00:44:52,540 --> 00:44:56,710 going to skip this part. Because data is taken by 466 00:44:56,710 --> 00:45:00,970 using software. But in general, 467 00:45:04,330 --> 00:45:11,750 you may look at this graph. Generally speaking, if 468 00:45:11,750 --> 00:45:17,730 you have a probability plot of a data, and the 469 00:45:17,730 --> 00:45:25,530 points lie on a straight line, or close to it, in 470 00:45:25,530 --> 00:45:29,650 this case, the distribution is normal. It's hard 471 00:45:29,650 --> 00:45:33,390 to make this graph manual. It's better to use 472 00:45:33,390 --> 00:45:38,510 software. But at least if we have this graph, and 473 00:45:38,510 --> 00:45:42,150 the points are close to the straight line. I mean, 474 00:45:42,250 --> 00:45:45,950 the points are either on the straight line, lies 475 00:45:45,950 --> 00:45:49,550 on the straight line, or close to it. In this case, 476 00:45:50,870 --> 00:45:55,030 the data is normally distributed. If the data 477 00:45:55,030 --> 00:46:00,870 points scattered away of the straight line, then 478 00:46:00,870 --> 00:46:04,390 the distribution is not normal, either skewed to 479 00:46:04,390 --> 00:46:08,690 the right or skewed to the left. So for this 480 00:46:08,690 --> 00:46:14,150 specific graph, the plot is normally distributed, 481 00:46:14,290 --> 00:46:17,550 approximately normally distributed. Because most 482 00:46:17,550 --> 00:46:23,230 of the points here lie close to the line and few 483 00:46:24,260 --> 00:46:29,660 are scattered away. Or it means that there are few 484 00:46:29,660 --> 00:46:31,940 outliers in this case, we can ignore these values. 485 00:46:33,100 --> 00:46:35,620 So here the plot is approximately a straight line 486 00:46:35,620 --> 00:46:40,360 except for a few outliers at the low and the 487 00:46:40,360 --> 00:46:44,780 right, those points. So generally speaking, the 488 00:46:44,780 --> 00:46:51,200 distribution is normal distribution. That's all 489 00:46:51,200 --> 00:46:53,700 for this chapter.