The 95% confidence interval, tells us that we can be 95% confident that our point estimate, which could be a mean or a proportion, falls within our confidence interval. Or in other words, it tells us that if we would draw an infinite number of samples, similar to our extra sample and for every sample, we would compute a 95% confidence interval with a similar margin of error. In 95% of the cases, the population value would fall within this confidence interval. This, of course, also means that in 5% of the cases, this method will produce an interval that does not contain the actual population parameter. If you would like to reduce the chance of an incorrect inference, you could go for a larger confidence interval, such as, for instance, 99%. In this video, I will tell you how you can change your confidence level. And what the consequences are, of doing so. Imagine you asked a sample of 100 new parents, if the babies like to answer nature's call, during the diaper changing process. 17% reported that, this is the case. Our sample, proportion p, thus equals 0.17. The formula to compute a 95% confidence interval for a proportion is, p plus and minus the z score for the 95% confidence level, times the standard error, which equals the square root of p, multiplied with 1, minus p divided by n. You can look up a z score for a 95% confidence level, in the z table. Look at this standard normal distribution here. When you have a 0.95 probability, that your value falls within z standard errors from the mean, that means, that 0.025 probability falls in the two tails. If you look up the z scores, which are displayed here in the z table, we find values of plus or minus 1.96. You can see that here. We can now easily compute the interval, that's 0.17 plus and minus 1.96 times the standard error which is the square root of 0.17, times 0.83, divided by 100. This leads to a confidence interval with the end point 0.10 and 0.24. You can now imagine, that it is not so difficult to construct intervals with other confidence levels. Let's first look at the 99% confidence interval. This is the formula. p plus or minus the z score for the 99% confidence level, times the standard error. The only difference is, the different z score. Look at this standard normal distribution. When you have a 0.99 probability that your value falls within z standard errors from the mean, that means, that 0.005 probability falls in the two tails. If we look up the z scores, which are indicated here. We find values of plus and minus 2.58. You can see that here. We can now compute the interval. That is 0.17, plus and minus 2.58, times the standard error. Which was 0.038. This leads to a confidence interval with the endpoints 0.07 and 0.27. For the 90% confidence level, we find a z score of 1.645. This leads to a confidence interval of 0.17, plus and minus 1.645, times 0.038. That makes a confidence interval with the endpoints of 0.11 and 0.23. I have here displayed confidence intervals graphically. You can see that a higher confidence level leads to a wider confidence interval. In other words, the more confident we are that we draw a correct inference, the larger of margin of error. That means, that we have to compromise between confidence and precision. As one gets better, the other gets worse. We never settle for a 100% confidence interval, because the margin of error then is far too large, which means that our conclusions are not very informative. In most cases, the 95% confidence interval is used. We can also use other confidence intervals when we construct a confidence interval to estimate a population mean. Suppose, we've asked a sample of 30 new parents in Amsterdam, how much hours of sleep they've lost after the first child was born. The mean is 2.6 hours per night. And the standard deviation is 0.9 hours per night. This is the formula we use, to construct a 95% confidence interval. x-bar, plus and minus the t score for the 95% confidence level, times the standard error. Which equals the sample standard deviation, divided by the square root of the sample size. Now, what is the t score for 95% confidence level? That's dependent on the degrees of freedom, which equals n minus 1. That is, 30 minus 1 is 29. In the t table, we should look in the column of the 95% confidence level and in the row of 29 degrees of freedom. That gives a t score of 2.045. The confidence interval becomes 2.6 plus and minus 2.045, times 0.9, divided by the square root of 30. That gives an interval from 2.26 to 2.94. If we would want to construct an interval with a confidence level of 99%, we simply replace the t score for the 95% level with the t score for the 99% level. You can look it up in the table, it's 2.756. The confidence interval is 2.6 plus and minus 2.756, times 0.9, divided by the square root of 30. That leads to an interval from 2.15 to 3.05. You can also easily do that for other confidence levels. Let me conclude this video by giving you a step by step plan for constructing a confidence interval. First, decide which confidence level you want to use. For instance, do you settle for the regular 95% level? Or do you want to be more confident and less precise? Or more precise and less confident? Second, decide if you're dealing with a proportion or a mean. If you're interested in a proportion, you work with the z distribution, and if you're interested in a mean, you have to use the t distribution. So, in the case of a proportion, you look up the relevant z score and in the case of a mean, you look up the relevant t score. Don't forget, that if you're interested in a mean, you should also compute the degrees of freedom which is equal to n minus 1. Third, compute the endpoints of the confidence interval. And finally, four, interpret the results substantively. That's it. If you're not 95% confident now that you can construct a confidence interval yourself, rewatch the last couple of videos.
