WEBVTT 00:00:03.000 --> 00:00:15.000 so before we had talked about the different confidence levels, how we have 90%, 95 and 99% confidence levels. 00:00:15.000 --> 00:00:26.000 And we also saw that as our confidence levels increase, so did our margin of ignorance. 00:00:26.000 --> 00:00:35.000 Alright. So our Comp. As our confidence levels one from 90 to 95 to 99% As they increase, so did our margin of errors. 00:00:35.000 --> 00:00:43.000 So what we were left here to do, as we had to choose that between precision and confidence. 00:00:43.000 --> 00:00:49.000 And so this this part make sure you read through this, because we got we got. 00:00:49.000 --> 00:00:59.000 We go over that fact that going back and and talking about the difference between the confidence levels, and then what? 00:00:59.000 --> 00:01:02.000 That's what that does to our margin of errors. 00:01:02.000 --> 00:01:10.000 Okay. So now we're going to focus on 00:01:10.000 --> 00:01:17.000 What happens if we increase our sample size 00:01:17.000 --> 00:01:41.000 Alright, so ideally. We would like to have both precision and contact when we construct confidence intervals, so can we obtain that if we increase our sample size, alright ideally, we would like to have a narrow interval and when we have a narrow interval that means our margin 00:01:41.000 --> 00:01:49.000 of error is smaller, because our interval is smaller. So our margin of error smaller. 00:01:49.000 --> 00:02:09.000 So like a 90% confidence level would have the the narrower venturable when you compare it to 95 and 99 so would also have a smaller margin of air, so ideally would like to have a narrower interval and a high level of confidence right 00:02:09.000 --> 00:02:18.000 okay, So the 90% gives us the narrow interval with the smaller margin of air. 00:02:18.000 --> 00:02:24.000 So if we can help both of them, then that would be ideal. 00:02:24.000 --> 00:02:28.000 If so, we can achieve this by increasing the sample size. 00:02:28.000 --> 00:02:34.000 So that's what this example does for us It 00:02:34.000 --> 00:02:43.000 It just talks about how, if we do increase our sample size, we could see that 00:02:43.000 --> 00:02:47.000 We can. 00:02:47.000 --> 00:02:51.000 Decrease our margin of error. So that's that's what we see here. 00:02:51.000 --> 00:02:55.000 So a larger sample size gives a smaller margin of there. 00:02:55.000 --> 00:03:00.000 The larger sample size is, gonna give us a smaller margin of error. 00:03:00.000 --> 00:03:05.000 Alright, So I'm just gonna go ahead and write down some information here. 00:03:05.000 --> 00:03:10.000 So I'm gonna pause. The video and write down all the information so that you have it. 00:03:10.000 --> 00:03:29.000 And I want you to do the same thing. Pause the video, and then write down the formula for confidence interval and try to identify everything here in the problem. 00:03:31.000 --> 00:03:34.000 no. I went ahead, and I broke down our confidence. Interval formula. 00:03:34.000 --> 00:03:43.000 So you can follow along the question says that national surveys show that 43% of American adult support the legalization of marijuana. 00:03:43.000 --> 00:03:49.000 But we really want to find the proportion of students at this community college who support the legalization of marijuana. 00:03:49.000 --> 00:03:57.000 So we're interested in estimating the population proportion of students at this community college. 00:03:57.000 --> 00:04:06.000 So the 43% here doesn't really factor in any of our calculations, because this is extra information for us. 00:04:06.000 --> 00:04:16.000 The population proportion. We want to estimate is the proportion of community college students who support legalization. 00:04:16.000 --> 00:04:19.000 Okay. So there's 2 surveys that were conducted. 00:04:19.000 --> 00:04:31.000 One was size 100, So I'm gonna call that N one equals 100 and then the second sample was 400 so and 2 would be 400. 00:04:31.000 --> 00:04:35.000 Little N. Always represents a random sample, because we have 2. 00:04:35.000 --> 00:04:48.000 I call one n one and the other one and 2. Now the results from both of the surveys gave us a sample proportion of 55% who are in favor. 00:04:48.000 --> 00:04:55.000 So the P. Hat is point 5 5. Now it happened in both of the samples. 00:04:55.000 --> 00:04:59.000 And that it's that surprisingly so we wouldn't. 00:04:59.000 --> 00:05:03.000 We don't expect our p-hats to be the same in the different samples. 00:05:03.000 --> 00:05:08.000 It just happens to be the same for this particular scenario. 00:05:08.000 --> 00:05:16.000 So we have point 5, 5 for the first sample and point 5 5 for the second sample. 00:05:16.000 --> 00:05:22.000 It just happened to give us the same P. Hat for both samples, which were different samples. 00:05:22.000 --> 00:05:29.000 One was the size 100 and the other one was the size 400, so I make sure that I put P. 00:05:29.000 --> 00:05:33.000 Hat one for the first sample, and p hat, 2 for the second sample. 00:05:33.000 --> 00:05:38.000 So we're able to know which sample we're talking about. 00:05:38.000 --> 00:05:43.000 Now look at our formula. So this part here is P. 00:05:43.000 --> 00:05:48.000 Hat. So I just wanna make sure. You see where I'm getting this from. 00:05:48.000 --> 00:05:56.000 The is P. Hat. This is our C. And then this right here is our standard error. 00:05:56.000 --> 00:06:01.000 Okay. So this part of the formula, with the square root, that's our standard error. 00:06:01.000 --> 00:06:21.000 So when we plug everything in the P. Hat, and the one minus p hat, and if you notice, if you do one minus P hat on the side, which is one minus point, 1 5, you can do This on the side so you already know what to plug in for the one minus p so point 5, 5, if you do 00:06:21.000 --> 00:06:27.000 one minus point 5 5 you get. You get the point 4, 5 is what I meant to write. 00:06:27.000 --> 00:06:32.000 Point, 4, 5, 00:06:32.000 --> 00:06:35.000 Okay, And so that's what you see here. The P. 00:06:35.000 --> 00:06:42.000 Had times of one minus p hat, and that gets divided by the 100, which is our sample size. 00:06:42.000 --> 00:07:00.000 Okay, So because you might be using a different calculator than I am, I want you to go ahead and just pause the video and try to see if you can find what the standard error is. 00:07:03.000 --> 00:07:19.000 so on the side. This is what you should have done if you're if you can't put this in your calculator all at once, then I'm just helping you with the calculation on the side here just for the standard error part, and then we're gonna take it and multiply, it 00:07:19.000 --> 00:07:26.000 by the 1.9 6. Now we're using 1 point, 9, 6, because this was a 95% confidence. 00:07:26.000 --> 00:07:30.000 So remember that we're asked to construct a 95% confidence. 00:07:30.000 --> 00:07:37.000 So the Z. That is, representing the 95% confidence is 1.9, 6. 00:07:37.000 --> 00:07:44.000 So we're constructing 2 confidence intervals to 95% confidence intervals, one with size, 100. 00:07:44.000 --> 00:07:52.000 And the other one was size 400. Okay? So the 1.9, 6 doesn't change because they're both 95% confidence. 00:07:52.000 --> 00:07:55.000 The P. Hats do not change, because both samples have. P. 00:07:55.000 --> 00:07:59.000 Had a point 5 5. So really, the only thing that's changing is your standard error. 00:07:59.000 --> 00:08:03.000 So that's why I'm doing the calculation on the side. 00:08:03.000 --> 00:08:09.000 Okay, So it's the square root of p hat times, one minus P hat divided by N. 00:08:09.000 --> 00:08:14.000 So if you just do the numerator part point 5 5 times point 4, 5. 00:08:14.000 --> 00:08:28.000 It's point 2, 4, 7, 5, Make sure you divide that by 100, and when you do, that's point 0, 0 2, 4, 7, 5. 00:08:28.000 --> 00:08:32.000 Now remember, we still have to take the square root of all that right. 00:08:32.000 --> 00:08:43.000 So don't forget to take the square root. If you take the square root you're going to end up with a standard error Point 0. 4, 9. 00:08:43.000 --> 00:08:56.000 So the question is, how do I? Round this point? 0 4, 9, 7, 4 points are all 4, 9, 7, 5, 9, 3, 00:08:56.000 --> 00:09:02.000 7, and a continuous alright. So you want to be careful not to round to, not to round too early. 00:09:02.000 --> 00:09:07.000 And also I can't write too many places, either, cause you know they don't really, fit. 00:09:07.000 --> 00:09:21.000 But you need to be careful not to round to to fewer decimal places, so points are 497 point, 4, 4, 9, 7, 4, 9. 00:09:21.000 --> 00:09:27.000 I'm gonna leave it. Points are 4, 9, 7, 4, 9. 00:09:27.000 --> 00:09:29.000 So I can take it to 1, 2, 3, 4, 5, 6 places. 00:09:29.000 --> 00:09:37.000 That should be pretty safe. Right? Alright. So we still have to multiply by that 1 point. 00:09:37.000 --> 00:09:55.000 9, 6, so you're going to do the 1.9, 6 times our standard error, which is point 0, 4, 9, 7, 4, 9, right? 00:09:55.000 --> 00:10:04.000 So this purple, my purple marking here That's the margin affair, That's what this point 0: 9, 8 does. Okay. 00:10:04.000 --> 00:10:15.000 But remember, I was very careful for my standard error. I kept it to like 6 decimal places, just to be safe just to make sure I can get that point through a nine-eight margin of error right here. 00:10:15.000 --> 00:10:18.000 Alright so now I'm going to do 1.9, 6. 00:10:18.000 --> 00:10:31.000 So the writing and purple. It's 1.9 6 times points point 0, 4 9.0, 4, 9, 7, 4, 9 alright. 00:10:31.000 --> 00:10:36.000 So when I do this calculation, I get point 0 9, 7, 5. 00:10:36.000 --> 00:10:41.000 So now we're okay rounding. Think we're going to round it? 00:10:41.000 --> 00:10:48.000 3 to 3 places here, so that 5 Well, it's a 5 or bigger, so that takes that 7. 00:10:48.000 --> 00:10:53.000 And it's exactly what we see here. It's points around 9, 8, so it rounds to points around. 00:10:53.000 --> 00:11:05.000 I need. I wanted just to take a minute to do this to help you guys through some of these calculations, But of course, if you have a scientific calculator or something where you can just enter it as you see it. 00:11:05.000 --> 00:11:13.000 Well, then you'll get points around 9, 8, right away, If you don't, then you're going to have to do the calculation inside the square root first. 00:11:13.000 --> 00:11:24.000 So notice how I did. The numerator, and then I did that, and then I put in the denominator, and I do it step by step. 00:11:24.000 --> 00:11:33.000 Okay. So you'll notice how we ended up coming up with this margin of error. 00:11:33.000 --> 00:11:37.000 So the Z Times. The standard error is the margin of error. 00:11:37.000 --> 00:11:39.000 Okay, Now, let's go ahead and pause the video. 00:11:39.000 --> 00:11:51.000 And see if you can come up with the standard air. Here, see if you can come up with the value and site the square root here, and then take the square root of whatever the insight is. 00:11:51.000 --> 00:12:01.000 So whatever we did here, go ahead, and do again for the bottom, one just for practice. 00:12:03.000 --> 00:12:20.000 alright. So the only thing that would have changed in our calculation for our standard error was the 400 right, because right this one had a larger sample size, 400 so let's see what the standard error is you multiply the numerator that's still the same divided by 400 00:12:20.000 --> 00:12:24.000 icon. This number my calculator gave me scientific notation. 00:12:24.000 --> 00:12:38.000 But that what that really means is that there are 3 zeros in front of the 6, so that Decimal point got moved over to the left 4 times. 00:12:38.000 --> 00:12:42.000 Okay, So if you were not sure what that number meant That's what it meant. 00:12:42.000 --> 00:12:46.000 Now, remember, you start to take the square root of this to get your standard air. 00:12:46.000 --> 00:12:52.000 So my standard error for the second confidence interval, with the larger sample size. 00:12:52.000 --> 00:13:01.000 That was the square root of point 0 0, 0, 6 1, 8, 7, 5. 00:13:01.000 --> 00:13:10.000 So if you take the square root of that, we got point 0 2, 4, 8, 7 point, 0, 2, 4, 8, 7. 00:13:10.000 --> 00:13:17.000 I'm just gonna move that one more place. So I rounded it to 1, 2, 3, 4 to what is that? 00:13:17.000 --> 00:13:25.000 6 places again. I'm just I'm just being conservative here, and trying to make sure I don't round too early, because that will change my margin of error. 00:13:25.000 --> 00:13:33.000 It will change it So 6 places is pretty safe. Okay? So remember that I saw the multiply by 1 point, 9, 6. 00:13:33.000 --> 00:13:45.000 So 1.9, 6, which is the Z. Whenever we have a 95% confidence that gets multiplied to my point, 0, 2, 4, 8, 7, 5. 00:13:45.000 --> 00:13:58.000 That was my standard error. Okay. So when you multiply these 2 numbers, so 1.9, 6 times point 0, 2, 4, 8, 00:13:58.000 --> 00:14:05.000 7, 5. That gives you your margin of error. Now we're rounding it to 3 places. 00:14:05.000 --> 00:14:11.000 So point 0 4, 9 is appropriate. Okay, So this is what I wanted to. 00:14:11.000 --> 00:14:18.000 This is what I wanted to make sure that use that you're able to understand. 00:14:18.000 --> 00:14:26.000 So as the sample size increases and I'll use red here as your sample size increases. 00:14:26.000 --> 00:14:38.000 So we had here. We had 100 sample sizable, 100, and here we had a sample size of 400, so as a sample size increase. 00:14:38.000 --> 00:14:44.000 Look what happened to your margin of error. It went from Point 0 9, 8 to point 0, 4, 9. 00:14:44.000 --> 00:14:53.000 So, as a sample size increased, the margin of error decreased 00:14:53.000 --> 00:14:57.000 Alright. Now we kept the same 95% confidence. 00:14:57.000 --> 00:15:10.000 So we were able to reduce our margin of error with this 95% confidence by having a larger sample size 00:15:10.000 --> 00:15:15.000 Alright. So the larger sample size really does give a smaller margin of air. 00:15:15.000 --> 00:15:25.000 So that's good news. Now, before we knew that larger samples 00:15:25.000 --> 00:15:29.000 Before we knew the larger samples do give us better estimates for the population. 00:15:29.000 --> 00:15:38.000 Proportion; and we also know that sample size affects variability and sample proportions. 00:15:38.000 --> 00:15:44.000 Proportions from larger samples, very less right proportions from larger samples. 00:15:44.000 --> 00:15:54.000 Very lost. So proportions from larger samples give us a smaller standard deviation 00:15:54.000 --> 00:15:58.000 If there's less variability in a sampling distribution, the standard error is smaller. 00:15:58.000 --> 00:16:07.000 So if your standard error is smaller, then your margin of error is smaller, because we use our standard error to find the margin of error Right? 00:16:07.000 --> 00:16:15.000 So that's what this says. Since we use the standard error to find the margin of error, larger samples will produce a smaller margin of error. 00:16:15.000 --> 00:16:22.000 So, if you're standard error is smaller. Then your margin of error is going to be smaller. 00:16:22.000 --> 00:16:44.000 Okay, So that's the main point from this example is that we can reduce our margin of error if we are able to increase our sample size.