Confidence Intervals: Further Explanations

00:01 After introducing confidence intervals and showing how to calculate them, there were several exercises.

00:07 You were asked to use the same data to find other confidence intervals.

00:13 Let's take a step back and try to understand confidence intervals a bit better.

00:18 Here is a graph of a normal distribution.

00:22 You know where the sample mean is in the middle of the graph.

00:26 Now, if we know that a variable is normally distributed, we are basically making the statement that the majority of observations will be around the mean and the rest far away from it. Let's draw a confidence interval.

00:41 There is the lower limit and the upper limit.

00:45 A 95% confidence interval would imply that we are 95% confident that the true population mean falls within this interval.

00:55 There is 2.5% chance that it will be on the left of the lower limit and 2.5% chance it will be on the right.

01:04 Overall, there is 5% chance that our confidence interval does not contain the true population mean.

01:11 So when alpha is 0.05 or 5%, we have alpha divided by two or 2.5% chance that the true mean is on the left of the interval and 2.5% on the right.

01:27 Okay, great.

01:29 Using the Z score in the formula, we are implicitly starting from a standard normal distribution. Therefore, the mean is zero.

01:38 The lower limit is minus C, while the upper one z.

01:43 For a 95% confidence interval using the Z table, we can find that these limits are -1.96 and 1.96.

01:53 That's exactly what we did in the previous lecture.

01:58 Finally, the formula makes sure we go back to the original range of values, and we get the interval for our particular data set.

02:06 Okay. What if we are looking at a 90% confidence interval? In that case, the interval looks like this and there is a 10% chance that the true mean is outside the interval.

02:20 Actually 5% on each side.

02:22 This causes the confidence interval to shrink.

02:25 So when our confidence is lower, the confidence interval itself is smaller. Similarly, for a 99% confidence interval, we would have a higher confidence, but a much larger confidence interval.

02:40 Let's see an example, just to make sure we have solidified this knowledge.

02:45 I don't know your age to your student, but I am 95% confident that you are between 18 and 55 years old.

02:52 Based on the fact that you are taking an online statistics course.

02:56 That's not much information to begin with.

02:58 Plus, I don't have any information about the age of any of the students.

03:02 Hence, the wide interval.

03:05 Okay. So, I am 95% confident you are between 18 and 55 years old. Also, I am 99% confident that you are between ten and 70 years old.

03:20 I am 100% confident that you are between zero and 118 years old, which is the age of the oldest person alive at the time of recording.

03:31 Finally, I am 5% confident that you are 25 years old.

03:37 Obviously, this is a completely arbitrary number.

03:40 As you can see, there is a trade-off between the level of confidence and the range of the interval. 100% confidence interval is completely useless, as I must include all ages possible in order to gain 100% confidence.

03:58 99% confidence gives me a much narrower range, but it's still not insightful enough for this particular problem.

04:05 25 years old, on the other hand, is a pretty useful estimate as we have an exact number.

04:10 But the level of confidence of 5% is too small for us to make use of in any meaningful analysis.

04:17 There is always a trade-off, which depends on the problem at hand.

04:22 95% is the accepted norm, as we don't compromise with accuracy too much but still get a relatively narrow interval.

04:31 Great. After this short clarification, let's carry on with more statistical problems.

04:37 Thanks for watching.