WEBVTT 1 00:00:03.900 --> 00:00:06.840 <v Instructor>Hello students, this is Problem 3.</v> 2 00:00:06.840 --> 00:00:09.513 And of course, we are gonna start with Problem 3a. 3 00:00:10.650 --> 00:00:11.550 Here we are going to go 4 00:00:11.550 --> 00:00:14.550 over an example calculating probability 5 00:00:14.550 --> 00:00:17.523 from a normal distribution using continuous data. 6 00:00:18.570 --> 00:00:20.920 So first what I'm gonna do is read the problem. 7 00:00:22.050 --> 00:00:22.890 Problem 3. 8 00:00:22.890 --> 00:00:26.430 Total cholesterol in children aged 10 to 15 years of age 9 00:00:26.430 --> 00:00:29.010 is assumed to follow a normal distribution 10 00:00:29.010 --> 00:00:32.690 with a mean of 191 and a standard deviation of 22.4. 11 00:00:34.860 --> 00:00:37.590 So part A is asking us what proportion 12 00:00:37.590 --> 00:00:41.430 of children 10 to 15 years of age have a total cholesterol 13 00:00:41.430 --> 00:00:44.883 between 180 and 190? 14 00:00:48.360 --> 00:00:50.280 So first thing I would suggest 15 00:00:50.280 --> 00:00:52.050 that you write down the information 16 00:00:52.050 --> 00:00:53.790 that has been provided to us 17 00:00:53.790 --> 00:00:58.790 that is mean equal to 191 and sigma equal to 22.4. 18 00:01:04.920 --> 00:01:08.880 And then, next I recommend that you draw the normal curve. 19 00:01:08.880 --> 00:01:11.470 So here, this is the normal curve 20 00:01:13.260 --> 00:01:15.363 trying to draw it as best as I can. 21 00:01:16.320 --> 00:01:20.520 And here is 191 22 00:01:20.520 --> 00:01:22.323 and this is X. 23 00:01:26.280 --> 00:01:30.570 So as per the question, now we will draw some lines 24 00:01:30.570 --> 00:01:34.290 so we can understand what the question is asking us. 25 00:01:34.290 --> 00:01:38.670 And basically, the question is asking us 26 00:01:38.670 --> 00:01:40.833 what is the area under the curve? 27 00:01:42.330 --> 00:01:45.550 So first, let's draw the lines 28 00:01:47.940 --> 00:01:50.070 and I will be using different colors 29 00:01:50.070 --> 00:01:53.283 to make this easier for us to see. 30 00:01:54.915 --> 00:01:58.812 Okay, so this is 190 31 00:01:58.812 --> 00:01:59.895 and this 180. 32 00:02:04.473 --> 00:02:05.306 Okay? 33 00:02:07.200 --> 00:02:09.670 And basically we are 34 00:02:14.070 --> 00:02:16.653 looking for this area, okay? 35 00:02:21.060 --> 00:02:24.210 And even though, we don't know the area under the curve 36 00:02:24.210 --> 00:02:26.070 for this distribution, 37 00:02:26.070 --> 00:02:28.470 we do know it for a normal distribution. 38 00:02:28.470 --> 00:02:29.940 So what we need to do here 39 00:02:29.940 --> 00:02:33.153 is transform our X values to our Z values. 40 00:02:34.110 --> 00:02:37.260 So after that, we can use the appendix from the back 41 00:02:37.260 --> 00:02:40.860 of our book to identify these probabilities. 42 00:02:40.860 --> 00:02:42.963 So let's start the process. 43 00:02:43.920 --> 00:02:46.260 I'm gonna go back to my blacking. 44 00:02:46.260 --> 00:02:49.920 So here's the Z, and again, we know that the Z distribution 45 00:02:49.920 --> 00:02:51.783 has a mean of zero. 46 00:02:54.330 --> 00:02:57.270 So basically now what we have to do is figure out 47 00:02:57.270 --> 00:02:59.793 what are the Z values for 180 and 190? 48 00:03:01.393 --> 00:03:02.880 And to figure that out, 49 00:03:02.880 --> 00:03:05.910 we have to be using the formula Z equal 50 00:03:05.910 --> 00:03:09.570 to X minus mu divided by sigma. 51 00:03:09.570 --> 00:03:11.613 So let's write down our formula here. 52 00:03:17.310 --> 00:03:22.310 Okay, so now this was the probability we were looking for 53 00:03:23.550 --> 00:03:24.453 180. 54 00:03:28.710 --> 00:03:29.543 Okay? 55 00:03:29.543 --> 00:03:33.153 So not as we try to convert this from X to Z, 56 00:03:34.440 --> 00:03:36.243 we have to replace 180 and 190. 57 00:04:05.046 --> 00:04:09.030 Okay? So now let's do the algebra and see what we get here. 58 00:04:09.030 --> 00:04:11.373 We get negative 0.49, 59 00:04:19.200 --> 00:04:21.398 negative 0.04. 60 00:04:21.398 --> 00:04:24.660 Okay, so now what we need to do is go to the back 61 00:04:24.660 --> 00:04:25.620 of our textbook, 62 00:04:25.620 --> 00:04:29.040 page 342 63 00:04:29.040 --> 00:04:30.900 and page 343. 64 00:04:30.900 --> 00:04:33.990 So first, the way we look down the Z values 65 00:04:33.990 --> 00:04:36.990 is we have to go down the left hand side 66 00:04:36.990 --> 00:04:40.980 and find negative 0.4. 67 00:04:40.980 --> 00:04:45.543 Then, we go across till we find 0.09. 68 00:04:47.670 --> 00:04:49.860 And the way we find this probability 69 00:04:49.860 --> 00:04:51.990 is the same way we'll find the probability 70 00:04:51.990 --> 00:04:53.853 for negative 0.04. 71 00:04:55.980 --> 00:04:59.820 So these probabilities, the way our table is set up 72 00:04:59.820 --> 00:05:04.613 is giving us the value for less than the number, okay? 73 00:05:07.380 --> 00:05:11.280 And again, this can vary from Z-table to Z-table. 74 00:05:11.280 --> 00:05:14.100 And I know a lot of students when they're taking this class, 75 00:05:14.100 --> 00:05:16.413 they like to go to Z-tables on the internet. 76 00:05:18.240 --> 00:05:22.020 I have looked at some of them, yes, the Z-tables are all... 77 00:05:22.020 --> 00:05:24.000 At least the ones I've seen, they're all correct 78 00:05:24.000 --> 00:05:26.940 but it might be set up differently. 79 00:05:26.940 --> 00:05:30.390 And if you use a Z-table that is set up differently 80 00:05:30.390 --> 00:05:31.800 the way we are progressing 81 00:05:31.800 --> 00:05:33.330 with solving the problem will change. 82 00:05:33.330 --> 00:05:35.070 So for the purpose of our class, 83 00:05:35.070 --> 00:05:38.133 please, please use the Z-table from the back of our book. 84 00:05:39.300 --> 00:05:41.910 So again, we are looking for the area, 85 00:05:41.910 --> 00:05:43.470 the green-shaded area. 86 00:05:43.470 --> 00:05:46.020 And the way we are going to find out 87 00:05:46.020 --> 00:05:48.420 is of course, we have to take the difference 88 00:05:48.420 --> 00:05:49.983 between the two values. 89 00:05:51.030 --> 00:05:52.860 So now we have to do what? 90 00:05:52.860 --> 00:05:56.120 We have to look up the value and then we have to look at... 91 00:05:58.080 --> 00:06:03.080 Subtract and get the final answer we are looking for. 92 00:06:03.270 --> 00:06:07.650 So once we go to the back of our book, we see that the value 93 00:06:07.650 --> 00:06:09.743 for this is 0.3121 94 00:06:13.920 --> 00:06:16.823 and the value for this is 0.484. 95 00:06:18.870 --> 00:06:22.290 Again, if you are having a tough time 96 00:06:22.290 --> 00:06:25.980 figuring out these values from the Z-table, 97 00:06:25.980 --> 00:06:27.870 please, please let me know. 98 00:06:27.870 --> 00:06:30.690 I will be more than happy to have a Zoom meet with you 99 00:06:30.690 --> 00:06:32.820 go over this one-on-one. 100 00:06:32.820 --> 00:06:35.520 So now, the probability here is going to be... 101 00:06:35.520 --> 00:06:37.590 And I know we are kind of short of space here 102 00:06:37.590 --> 00:06:39.990 so I'm squeezing everything 103 00:06:39.990 --> 00:06:44.523 0.484 minus 0.3121. 104 00:06:45.570 --> 00:06:47.130 So when we do this, 105 00:06:47.130 --> 00:06:51.720 and again I like to put this parentheses around. 106 00:06:51.720 --> 00:06:54.507 So then, P is equal to 0.1719. 107 00:06:58.260 --> 00:07:01.530 Okay, so basically, we can say that based 108 00:07:01.530 --> 00:07:03.063 on the values provided, 109 00:07:04.440 --> 00:07:09.440 there is a 17.19% probability 110 00:07:09.960 --> 00:07:14.640 that the child will have a cholesterol between 180 and 190, 111 00:07:14.640 --> 00:07:15.660 Okay? 112 00:07:15.660 --> 00:07:17.730 And that is based on the values provided. 113 00:07:17.730 --> 00:07:21.900 Again, just to go over the key points here, 114 00:07:21.900 --> 00:07:26.140 I want to tell you that this part 115 00:07:27.930 --> 00:07:31.497 was given to us from the Z-table and that is 0.3121. 116 00:07:33.720 --> 00:07:37.080 And again, this part that was given to us 117 00:07:37.080 --> 00:07:38.343 from the Z-table, 118 00:07:40.320 --> 00:07:43.800 all the way here is 0.484. 119 00:07:43.800 --> 00:07:46.350 And we are looking for the difference 120 00:07:46.350 --> 00:07:49.620 and that's why we had to do the subtraction. 121 00:07:49.620 --> 00:07:52.350 Now, a couple of things to keep in mind. 122 00:07:52.350 --> 00:07:54.000 The area under the curve, 123 00:07:54.000 --> 00:07:57.543 the total area under the curve always adds up to one. 124 00:07:58.440 --> 00:08:00.990 And another thing to keep in mind is whenever you are 125 00:08:00.990 --> 00:08:04.380 in doubt, which one should I subtract from which one? 126 00:08:04.380 --> 00:08:06.690 Always keep in mind that the area under the curve 127 00:08:06.690 --> 00:08:09.090 is always a positive at, okay? 128 00:08:09.090 --> 00:08:11.640 So whichever number is bigger then the subtraction 129 00:08:11.640 --> 00:08:13.200 should be from that number. 130 00:08:13.200 --> 00:08:17.700 So here 0.484 is the bigger number than 0.3121. 131 00:08:17.700 --> 00:08:22.000 So therefore, we are going to subtract 0.3121 from 0.484 132 00:08:24.120 --> 00:08:26.853 because an area cannot be a negative value. 133 00:08:29.550 --> 00:08:32.370 So again, based on the values that were provided here, 134 00:08:32.370 --> 00:08:37.370 we can say that there is a 17.19% probability 135 00:08:37.740 --> 00:08:41.610 that a child's cholesterol given the values 136 00:08:41.610 --> 00:08:43.957 will be between 180 and 190.