WEBVTT 1 00:00:05.280 --> 00:00:06.150 Hello, students. 2 00:00:06.150 --> 00:00:08.855 This is our second example for Chapter 6, 3 00:00:08.855 --> 00:00:13.079 and in this example we are going to go over 4 00:00:13.079 --> 00:00:16.000 how to calculate confidence interval 5 00:00:17.160 --> 00:00:20.640 for a single sample of dichotomous data. 6 00:00:20.640 --> 00:00:25.640 The example here states that we have 795 males 7 00:00:25.800 --> 00:00:29.037 and we have asked them if they have ever been told by a doctor 8 00:00:29.037 --> 00:00:32.460 that they have symptoms of prediabetes. 9 00:00:32.460 --> 00:00:34.500 The results we see here is 10 00:00:34.500 --> 00:00:37.560 that 374 have told us that, 11 00:00:37.560 --> 00:00:40.080 yes, they have been told 12 00:00:40.080 --> 00:00:43.846 and 421 responded negatively stating, 13 00:00:43.846 --> 00:00:46.260 they have not been told. 14 00:00:46.260 --> 00:00:49.230 What we want to do here is 15 00:00:49.230 --> 00:00:52.080 find out the true proportion of male 16 00:00:52.080 --> 00:00:54.900 who have been told that they have 17 00:00:54.900 --> 00:00:57.270 symptoms of pre-diabetes 18 00:00:57.270 --> 00:01:01.500 based on the population represented by the sample. 19 00:01:01.500 --> 00:01:04.530 So now the formula for this will come, of course, 20 00:01:04.530 --> 00:01:08.310 from our textbook, and that is from page 118. 21 00:01:08.310 --> 00:01:10.460 And I'm first gonna write the formula here. 22 00:01:32.520 --> 00:01:34.925 So here, we are using the Z distribution 23 00:01:34.925 --> 00:01:39.485 and the 95% is what was given in our question. 24 00:01:39.485 --> 00:01:42.976 So before we can start solving this equation, 25 00:01:42.976 --> 00:01:45.960 first, we need to find out what is P hat. 26 00:01:45.960 --> 00:01:48.450 So that's what we are going to find out here. 27 00:01:48.450 --> 00:01:49.653 P hat is, 28 00:02:02.430 --> 00:02:04.169 again, basically 374. 29 00:02:04.169 --> 00:02:09.169 That is the number of males who responded positively. 30 00:02:09.600 --> 00:02:12.130 And the denominator here is the 31 00:02:13.380 --> 00:02:16.740 addition of both 374 and 421, 32 00:02:16.740 --> 00:02:19.683 which is our sample size. 33 00:02:21.870 --> 00:02:24.510 So when we crunch the numbers here, 34 00:02:24.510 --> 00:02:28.800 what we get is 374 divided by, 35 00:02:28.800 --> 00:02:31.230 again, I prefer you do these calculations 36 00:02:31.230 --> 00:02:33.120 with me as I'm doing it. 37 00:02:33.120 --> 00:02:37.612 So let's add 374 and 421. 38 00:02:37.612 --> 00:02:40.680 We are supposed to get 795. 39 00:02:40.680 --> 00:02:43.257 And yes, that is exactly what we got. 40 00:02:43.257 --> 00:02:45.933 And then, when we do the division here, 41 00:02:53.460 --> 00:02:56.730 we get 0.47. 42 00:02:56.730 --> 00:03:00.690 So now what we need to find out is the Z value. 43 00:03:00.690 --> 00:03:03.810 So we go to the back of our book, Table 1B, 44 00:03:03.810 --> 00:03:08.520 and the critical Z` value here is 1.96. 45 00:03:08.520 --> 00:03:13.203 So now let's insert these numbers in our formula. 46 00:03:16.260 --> 00:03:20.040 P hat is exactly what we just calculated. 47 00:03:20.040 --> 00:03:23.070 The Z value here is 1.96, 48 00:03:23.070 --> 00:03:25.740 which we got from the back of our book. 49 00:03:25.740 --> 00:03:30.540 And then we have to calculate this part, which is 0.47, 50 00:03:30.540 --> 00:03:32.347 then 1 minus 0.47 51 00:03:39.480 --> 00:03:43.507 divided by the n, which is 795. 52 00:03:45.720 --> 00:03:50.163 So let's see what we get here as we crunch the numbers. 53 00:03:54.330 --> 00:03:56.620 So the numerator here will be 54 00:04:11.130 --> 00:04:11.963 2.49. 55 00:04:13.170 --> 00:04:16.953 So it's 0.249. 56 00:04:17.820 --> 00:04:21.573 And the denominator here is still 795. 57 00:04:22.590 --> 00:04:24.060 So let's see what happens here. 58 00:04:24.060 --> 00:04:25.103 Now, 0.249 59 00:04:29.310 --> 00:04:33.810 divided by 795. 60 00:04:33.810 --> 00:04:37.260 We get, wow, a very small number, 0.0003. 61 00:04:39.540 --> 00:04:42.093 So now we have to do the square root of that. 62 00:04:53.790 --> 00:04:56.007 So what we are going to get here is, 63 00:04:56.007 --> 00:04:57.600 I'm going to write it down. 64 00:04:57.600 --> 00:05:01.320 Again, this is still the same 0.47, 65 00:05:01.320 --> 00:05:05.913 and here is going to be 1.96 multiplied by, 66 00:05:13.500 --> 00:05:18.183 so if we multiply 1.96, 67 00:05:24.540 --> 00:05:25.373 we get, 68 00:05:27.317 --> 00:05:29.193 again, I'm writing it down, 69 00:05:32.830 --> 00:05:37.830 0.033. 70 00:05:38.490 --> 00:05:42.126 And I have already posted an announcement regarding 71 00:05:42.126 --> 00:05:44.970 decimal places and how far we want to go. 72 00:05:44.970 --> 00:05:47.550 So based on that, this is how far we are going. 73 00:05:47.550 --> 00:05:50.152 So now, when we do the subtraction 74 00:05:50.152 --> 00:05:53.643 as well as the addition here, this is what we get. 75 00:05:56.318 --> 00:05:57.452 Oops. 76 00:05:57.452 --> 00:05:58.285 0.47 77 00:06:04.950 --> 00:06:09.723 minus 0.033 is 0.437, 78 00:06:36.654 --> 00:06:37.487 0.503. 79 00:06:42.900 --> 00:06:47.080 So basically, what we can state is 80 00:06:48.150 --> 00:06:50.722 that our calculation tells us that 81 00:06:50.722 --> 00:06:55.230 the full population of men represented in the sample, 82 00:06:55.230 --> 00:06:58.080 the proportion of those who have been told 83 00:06:58.080 --> 00:07:01.110 that they have symptoms of prediabetes 84 00:07:01.110 --> 00:07:06.007 with 95% confidence is between 0.437 and 0.503 85 00:07:09.510 --> 00:07:14.510 or 43.7% to 50.3%. 86 00:07:19.020 --> 00:07:23.310 So basically, I converted the decimal to a percentage. 87 00:07:23.310 --> 00:07:25.713 So that's all for this problem.