WEBVTT 1 00:00:03.510 --> 00:00:04.590 Hello, students. 2 00:00:04.590 --> 00:00:07.350 This is problem three, D. 3 00:00:07.350 --> 00:00:11.250 Now, this is not a problem in our text, 4 00:00:11.250 --> 00:00:13.920 but it's important for us to learn 5 00:00:13.920 --> 00:00:17.790 how to calculate percentile of a normal distribution. 6 00:00:17.790 --> 00:00:20.310 So we are extending problem number three 7 00:00:20.310 --> 00:00:24.030 and we want to know now, based on the values provided, 8 00:00:24.030 --> 00:00:26.760 what would be the total cholesterol of children 9 00:00:26.760 --> 00:00:30.570 in this population at the 95th percentile? 10 00:00:30.570 --> 00:00:35.570 In other words, what is the x value at the 95th percentile? 11 00:00:35.760 --> 00:00:39.990 So similar to the previous one, I am going to start 12 00:00:39.990 --> 00:00:42.813 with the values that were provided to us. 13 00:00:45.030 --> 00:00:48.040 So, our x 14 00:00:49.050 --> 00:00:50.842 at 95th percentile 15 00:00:50.842 --> 00:00:55.842 is what we are trying to figure out. 16 00:01:00.471 --> 00:01:02.400 That is what we are trying to figure out. 17 00:01:02.400 --> 00:01:05.520 The mu here is, again, 191. 18 00:01:05.520 --> 00:01:09.263 Sigma is 22.4. Okay. 19 00:01:13.680 --> 00:01:17.880 So next I am going to draw 20 00:01:17.880 --> 00:01:20.013 the normal distribution. 21 00:01:28.860 --> 00:01:32.400 And again, we need that z transformation, 22 00:01:32.400 --> 00:01:35.010 otherwise we cannot really do anything. 23 00:01:35.010 --> 00:01:35.843 And 24 00:01:44.182 --> 00:01:46.099 what we are looking for 25 00:01:50.640 --> 00:01:52.830 is this x, okay? 26 00:01:52.830 --> 00:01:55.277 So basically, this is the 5% 27 00:01:59.070 --> 00:01:59.903 and 28 00:02:05.070 --> 00:02:07.673 this is the 95%. 29 00:02:16.800 --> 00:02:18.900 And what is that x value? 30 00:02:18.900 --> 00:02:22.050 So basically, here we are kind of flipping the process 31 00:02:22.050 --> 00:02:26.220 because previously we looked at x less than some value 32 00:02:26.220 --> 00:02:30.510 transformed it to the z value or the z distribution 33 00:02:30.510 --> 00:02:33.720 or we looked at x above some value 34 00:02:33.720 --> 00:02:35.070 and, again, transformed it 35 00:02:35.070 --> 00:02:37.500 to the g value or the z distribution. 36 00:02:37.500 --> 00:02:40.680 And then we identified some value 37 00:02:40.680 --> 00:02:45.680 that was for z less than as per our table. 38 00:02:45.900 --> 00:02:50.900 And then we found out the probability of the z value 39 00:02:51.540 --> 00:02:53.460 being less than some value. 40 00:02:53.460 --> 00:02:55.473 Now, here what we are doing is, 41 00:02:56.760 --> 00:02:58.980 again, kind of flipping the process. 42 00:02:58.980 --> 00:03:01.590 Here we are saying that the probability of z 43 00:03:01.590 --> 00:03:05.253 less than some value is 0.95, okay? 44 00:03:06.630 --> 00:03:08.550 Or in the 95 percentile. 45 00:03:08.550 --> 00:03:10.920 So now, let's go to the back of our book 46 00:03:10.920 --> 00:03:14.373 and we are going to see what is the z value for 0.95. 47 00:03:16.590 --> 00:03:18.210 So basically, now we are going to go 48 00:03:18.210 --> 00:03:19.950 to the middle of our table, 49 00:03:19.950 --> 00:03:23.640 not on the columns or the rows, in the middle of our table, 50 00:03:23.640 --> 00:03:26.403 and we are going to look for the value 0.95. 51 00:03:27.660 --> 00:03:29.010 So what do we find here? 52 00:03:29.010 --> 00:03:32.343 We find that we don't have an exact 0.95 value. 53 00:03:33.570 --> 00:03:35.170 But what we do have is a value 54 00:03:35.170 --> 00:03:40.170 for 0.9495 and 0.9505. 55 00:03:40.800 --> 00:03:45.800 So in the back of our book, we have a value for zero. 56 00:03:46.620 --> 00:03:49.383 Oops, sorry, I gotta change this to black now. 57 00:03:50.910 --> 00:03:52.533 So let me go back to black. 58 00:03:53.670 --> 00:03:57.790 And we have the value 4.9495 59 00:04:00.770 --> 00:04:05.627 and per 0.9505. 60 00:04:07.102 --> 00:04:08.910 Okay? So what are these values? 61 00:04:08.910 --> 00:04:10.800 This value is 1.64. 62 00:04:12.884 --> 00:04:15.888 This value is 1.65. 63 00:04:15.888 --> 00:04:20.430 So if 0.95 were to be placed in our table, 64 00:04:20.430 --> 00:04:21.660 where would it be? 65 00:04:21.660 --> 00:04:23.040 It would be in between. 66 00:04:23.040 --> 00:04:24.060 So what do we do? 67 00:04:24.060 --> 00:04:28.080 We take these two numbers, add them up, 68 00:04:28.080 --> 00:04:30.210 divide them by two, and get the average. 69 00:04:30.210 --> 00:04:33.120 And the average, what would it that be? 70 00:04:33.120 --> 00:04:37.650 It would be 1.645. 71 00:04:37.650 --> 00:04:39.180 Now, if you do not want 72 00:04:39.180 --> 00:04:41.670 to go through all the complicated process, 73 00:04:41.670 --> 00:04:44.850 then you can really go to the next page 74 00:04:44.850 --> 00:04:47.190 and you can look at table 1A. 75 00:04:47.190 --> 00:04:49.020 And here we have the z values 76 00:04:49.020 --> 00:04:51.810 for some commonly used percentile. 77 00:04:51.810 --> 00:04:55.830 And for the 95th percentile, that is 1.645. 78 00:04:55.830 --> 00:05:00.420 Again, very important to use the z table 79 00:05:00.420 --> 00:05:02.370 at the back of our book. 80 00:05:02.370 --> 00:05:05.610 Now, we are going to the actual process. 81 00:05:05.610 --> 00:05:09.270 What we are going to do is here, again, use the same formula 82 00:05:09.270 --> 00:05:12.390 but the transformation is somewhat reverse. 83 00:05:12.390 --> 00:05:15.483 So here z, again, the same formula, 84 00:05:17.520 --> 00:05:20.523 x minus mu divided by sigma. 85 00:05:22.200 --> 00:05:25.503 The z value here is 1.645. 86 00:05:28.140 --> 00:05:30.240 Now, we have to do a little algebra here. 87 00:05:30.240 --> 00:05:31.920 And I'm going to show you the algebra 88 00:05:31.920 --> 00:05:34.200 before I actually plug in the numbers. 89 00:05:34.200 --> 00:05:38.670 So it's going to be z multiplied by sigma 90 00:05:38.670 --> 00:05:41.160 because that's going under the reverse side 91 00:05:41.160 --> 00:05:45.300 and then it's going to be plus mu equal to x. 92 00:05:45.300 --> 00:05:48.120 Because remember, we are looking at the x value here. 93 00:05:48.120 --> 00:05:51.240 So when we insert the numbers here, 94 00:05:51.240 --> 00:05:55.563 z here is 1.645. 95 00:05:56.940 --> 00:05:59.970 Sigma again is 22.4. 96 00:05:59.970 --> 00:06:03.270 Our mu again is 191, 97 00:06:03.270 --> 00:06:05.400 and that will give our x value. 98 00:06:05.400 --> 00:06:07.110 So what is our X value going to be? 99 00:06:07.110 --> 00:06:08.777 It's going to be 227.85, 100 00:06:15.810 --> 00:06:19.830 which is kind of 227.9, we can say. 101 00:06:19.830 --> 00:06:21.360 So what does this mean? 102 00:06:21.360 --> 00:06:25.140 So this means that given the values provided, 103 00:06:25.140 --> 00:06:29.100 the 95th percentile of cholesterol for children, 104 00:06:29.100 --> 00:06:31.020 10 to 15 years of age, 105 00:06:31.020 --> 00:06:33.830 in this population is 227.9. 106 00:06:38.190 --> 00:06:42.030 So basically, we can say that 5% of the children, 107 00:06:42.030 --> 00:06:43.380 and I'm gonna write this here, 108 00:06:43.380 --> 00:06:47.373 5% of the children, given the values, 109 00:06:56.400 --> 00:06:57.610 given the values 110 00:06:59.550 --> 00:07:01.720 and between the ages of 111 00:07:07.860 --> 00:07:09.813 10 to 15 years of age, 112 00:07:19.140 --> 00:07:21.190 will have a cholesterol above 113 00:07:22.710 --> 00:07:25.950 or basically total cholesterol will have a, 114 00:07:25.950 --> 00:07:28.990 and I'll abbreviate, total cholesterol above 115 00:07:31.830 --> 00:07:35.643 227.9, okay? 116 00:07:36.720 --> 00:07:37.553 All right. 117 00:07:37.553 --> 00:07:40.260 So again, please feel free to reach out 118 00:07:40.260 --> 00:07:43.620 if this is not clear, if you're having any confusion. 119 00:07:43.620 --> 00:07:46.290 Very important that you understand the process 120 00:07:46.290 --> 00:07:50.190 because it's important for you to be able to solve 121 00:07:50.190 --> 00:07:53.730 the homework, assignments, and also for a test. 122 00:07:53.730 --> 00:07:55.500 You will need to understand the process 123 00:07:55.500 --> 00:07:59.160 so you can perform well in that test 124 00:07:59.160 --> 00:08:02.913 and ultimately learn the material. 125 00:08:03.780 --> 00:08:05.580 So again, please feel free to reach out 126 00:08:05.580 --> 00:08:06.880 if you have any questions.