Confidence Intervals: Margin of Error

00:01 Welcome back.

00:02 In the last few lessons, we've been concentrating on confidence intervals.

00:07 We learned about two situations when the population variance is known, which happens very rarely in practice, and when the variance is unknown and follows a student's T distribution.

00:18 The goal of this lesson is less about learning and more about clarifying.

00:23 All right. These are the formulas that allow us to find confidence intervals.

00:29 As we noted in our previous lecture, these parts are the ones that determine the span of the confidence interval.

00:36 There is a special name for these expressions.

00:38 Margin of error.

00:41 When the population variance is known, the margin of error is equal to this expression.

00:47 And in the case of population variance unknown, the margin of error is equal to this expression instead.

00:54 So basically the confidence intervals could be summarized as follows.

00:59 The true population mean falls in the interval defined by the sample mean plus minus the margin of error.

01:08 Now we get to the core of this lesson.

01:10 Getting a smaller margin of error means that the confidence interval would be narrower, as we want a better prediction.

01:16 It is in our interest to have the narrowest possible confidence interval.

01:21 The best part is that we can control the margin of error.

01:25 Let's see its parts.

01:27 There is a statistic, a standard deviation and the sample size.

01:33 The statistic and the standard deviation are in the numerator, so smaller statistics and smaller standard deviations will reduce the margin of error.

01:42 How do we do that? A higher level of confidence increases the statistic.

01:48 A higher statistic means a higher margin of error.

01:51 This leads to a wider confidence interval.

01:55 Think about this example.

01:56 You took an exam, and you want to make a prediction about the mean grade obtained by all exam takers.

02:03 As A, B, C, D and F are all the possible grades.

02:08 We are 100% confident that the mean grade will be in the confidence interval from F to A. Now, if we lower the confidence level to 99%, we may end up with a confidence interval from F plus to a minus. Remember the interpretation.

02:26 In 99% of the cases, the population mean falls in the interval.

02:31 So with a 50% confidence level in 50% of the cases, the true mean will fall in the specified interval.

02:40 The only scenario under which this is possible is if the interval is narrower.

02:46 Therefore, if the standard deviation and the sample size are kept constant, a lower confidence level result in a narrower interval.

02:56 What about the standard deviation? That's easy. A lower standard deviation means that the data set is more concentrated around the mean. So we have a better chance to get it right.

03:07 For instance, the mean grade in your class is B, and you know that there were people with A's, B's, C's, D's, and a few F's.

03:16 How likely is it that you got to be? Now compare this to a situation when the teacher said the mean of the class is around B and the lowest grade is C.

03:27 In this case, you are much more likely to get a B right.

03:31 In the first case, the grades are dispersed, while in the second they are concentrated. Lastly, we have the sample size in the denominator.

03:43 Higher sample sizes will decrease the margin of error.

03:46 This is also quite intuitive.

03:48 The more observations you have in your sample, the more certain you are in the prediction. This time.

03:54 You have a B-plus.

03:57 There are 30 people in the class, and you want to know if you performed above the average. You ask three of your friends, and they all got A's.

04:06 Your sample of three people leads you to think you underperformed.

04:11 You get frustrated and start asking around some more.

04:14 The next five people, you ask, got Ds.

04:18 Now you have a sample of eight people, and it seems you did quite well.

04:24 After asking everyone in class, the whole population that is, you find out that the mean grade is B, you are above average by a small margin.

04:35 The conclusion is that the more observations there are in the sample, the higher the chances of getting a good idea about the true mean of the entire population.

04:46 All right. Hopefully this clears up some doubts you may have had.

04:51 Thanks for watching.