WEBVTT 00:00:01.030 --> 00:00:03.150 In this segment we introduce the concept of 00:00:03.150 --> 00:00:05.120 a confidence interval. 00:00:05.120 --> 00:00:08.109 The starting point is that an estimate, the value of an 00:00:08.109 --> 00:00:11.110 estimator, does not tell the whole story. 00:00:11.110 --> 00:00:13.910 One option is to also provide the standard error of the 00:00:13.910 --> 00:00:17.290 estimator, but a more common practice is to report a 00:00:17.290 --> 00:00:18.910 confidence interval. 00:00:18.910 --> 00:00:20.040 What is that? 00:00:20.040 --> 00:00:22.060 We'll introduce the notion of a confidence 00:00:22.060 --> 00:00:23.740 interval through a story. 00:00:23.740 --> 00:00:25.910 You're working for a polling company. 00:00:25.910 --> 00:00:31.520 You carry out a poll, and then you go and report to your boss 00:00:31.520 --> 00:00:38.000 that my estimate is this particular number. 00:00:38.000 --> 00:00:41.620 And then your boss says, I appreciate the five digit 00:00:41.620 --> 00:00:45.510 accuracy, but are your conclusions that accurate? 00:00:45.510 --> 00:00:49.240 You go back to your desk, you do some more calculations, and 00:00:49.240 --> 00:00:55.880 then you tell your boss, here is a 95% confidence interval. 00:00:58.480 --> 00:01:01.930 Your boss tells you, that looks great, but what does 00:01:01.930 --> 00:01:03.570 that exactly mean? 00:01:03.570 --> 00:01:06.980 You go back to your textbook, you pull out the definition, 00:01:06.980 --> 00:01:09.420 and you reply as follows. 00:01:09.420 --> 00:01:13.130 Well, a 95% confidence interval-- 00:01:13.130 --> 00:01:17.550 so here I am letting alpha to be 5%-- 00:01:17.550 --> 00:01:22.800 a 95% confidence interval is an interval that has the 00:01:22.800 --> 00:01:24.360 following property. 00:01:24.360 --> 00:01:27.710 That the unknown value of the parameter that we're trying to 00:01:27.710 --> 00:01:32.060 estimate falls inside this interval with 00:01:32.060 --> 00:01:35.320 probability at least 95%. 00:01:35.320 --> 00:01:39.390 And if you wish, I could also let alpha be equal to 1%, in 00:01:39.390 --> 00:01:43.630 which case I could give you a 99% confidence interval. 00:01:43.630 --> 00:01:46.220 Your boss replies, no, that sounds good. 00:01:46.220 --> 00:01:49.920 A 95% interval sounds fine. 00:01:49.920 --> 00:01:53.310 And your boss goes out to the press, holds a press 00:01:53.310 --> 00:01:58.740 conference, and reports that the true value of the 00:01:58.740 --> 00:02:04.490 parameter lies inside this range, inside the reported 00:02:04.490 --> 00:02:08.627 confidence interval with probability at least 95%. 00:02:11.680 --> 00:02:14.610 Does this statement make sense? 00:02:14.610 --> 00:02:16.220 Actually, no. 00:02:16.220 --> 00:02:20.300 This statement is the most common misconception of what a 00:02:20.300 --> 00:02:22.740 confidence interval is. 00:02:22.740 --> 00:02:24.790 To see why this statement does not make 00:02:24.790 --> 00:02:27.030 sense, look at it carefully. 00:02:27.030 --> 00:02:29.880 We're talking about the probability of something. 00:02:29.880 --> 00:02:33.460 But that something does not involve anything random. 00:02:33.460 --> 00:02:36.300 0.3 and 0.52 are just numbers. 00:02:36.300 --> 00:02:41.079 And theta is also a number which we do not know what it 00:02:41.079 --> 00:02:42.816 is, but it is not random. 00:02:42.816 --> 00:02:44.650 It is a constant. 00:02:44.650 --> 00:02:49.350 So this statement is incorrect on a purely syntactic basis. 00:02:49.350 --> 00:02:52.370 I mean the true parameter theta either is inside this 00:02:52.370 --> 00:02:54.190 interval, or it is not. 00:02:54.190 --> 00:02:57.640 But there's nothing random here, and so this statement 00:02:57.640 --> 00:02:59.980 does not make sense. 00:02:59.980 --> 00:03:04.070 Instead, let us look carefully at this definition. 00:03:04.070 --> 00:03:06.810 This statement does make sense because it 00:03:06.810 --> 00:03:09.230 involves random variables. 00:03:09.230 --> 00:03:13.150 The lower and the upper end of the confidence interval are 00:03:13.150 --> 00:03:17.140 quantities that are determined by the data, and therefore 00:03:17.140 --> 00:03:18.530 they are random. 00:03:18.530 --> 00:03:21.400 So we do have random variables in here. 00:03:21.400 --> 00:03:25.160 And so it makes sense to talk about probabilities. 00:03:25.160 --> 00:03:29.590 To really understand what's going on, think as follows. 00:03:29.590 --> 00:03:33.860 We're dealing with a poll that is trying to estimate some 00:03:33.860 --> 00:03:36.860 unknown value, theta. 00:03:36.860 --> 00:03:40.860 You carry out the poll, and you come up with a confidence 00:03:40.860 --> 00:03:42.990 interval based on the data. 00:03:42.990 --> 00:03:46.820 You might be lucky, and your confidence interval happens to 00:03:46.820 --> 00:03:49.100 capture the true parameter. 00:03:49.100 --> 00:03:52.579 You carry the poll one more time, maybe on another day. 00:03:52.579 --> 00:03:55.170 You come up with another confidence interval. 00:03:55.170 --> 00:03:58.360 And you're again lucky, and it captures the true parameter. 00:03:58.360 --> 00:04:01.260 You carry it on another day, and you come up with a 00:04:01.260 --> 00:04:02.550 confidence interval. 00:04:02.550 --> 00:04:06.770 And maybe the data that you got were kind of skewed. 00:04:06.770 --> 00:04:10.010 You were unlucky, and your confidence interval does not 00:04:10.010 --> 00:04:12.750 capture the true parameter. 00:04:12.750 --> 00:04:18.870 Having a 95% confidence interval means that 95% of the 00:04:18.870 --> 00:04:24.980 time, 95% of the polls that you carry out will capture the 00:04:24.980 --> 00:04:26.380 true parameter. 00:04:26.380 --> 00:04:30.190 So the word 95% really talks about your method of 00:04:30.190 --> 00:04:32.490 constructing confidence intervals. 00:04:32.490 --> 00:04:35.850 It's a method that 95% of the time will 00:04:35.850 --> 00:04:37.590 capture the true parameter. 00:04:37.590 --> 00:04:42.380 It is not a statement about the actual numbers that you 00:04:42.380 --> 00:04:46.040 are reporting on one specific poll. 00:04:46.040 --> 00:04:49.190 So it is important to keep this in mind, and to always 00:04:49.190 --> 00:04:53.700 interpret confidence intervals the correct way. 00:04:53.700 --> 00:04:56.909 So how does one come up with confidence intervals? 00:04:56.909 --> 00:04:59.240 The most common method is based on normal 00:04:59.240 --> 00:05:02.140 approximations, as we will be seeing next.