A 95% confidence interval to estimate a population mean tells us that we have 95% confidence that this interval contains the actual population mean. With this formula, you can compute the endpoints of the interval. There is one big problem with this formula, however. To compute the confidence interval you need to know the population standard deviation, and usually we don't know this value. After all, we use the sample to draw inferences about population parameters. In this video, I show you how you can solve this problem. The solution is that we estimate the population's standard deviation, and therefore have to employ another distribution than the standard normal distribution. It is the T distribution. Let me tell you how that works. Imagine we've asked a sample of 60 new parents in Amsterdam how much sleeping hours they have lost after their first children were born. The mean is 2.6 hours and the standard deviation is 0.9 hours. To construct the 95% confidence interval, we need this formula, x-bar plus and minus 1.96 times the standard deviation of the sampling distribution of the sample mean, which equals the population standard deviation divided by the square root of the sample size. We can also write that as x-bar plus and minus the z score for 95% confidence level, which is 1.96, times the standard deviation of the sampling distribution. As you can see, it is impossible to compute the confidence interval because we don't know the value of the population standard deviation. Therefore, we cannot compute the standard deviation of the sampling distribution of the sample mean. To solve that problem, we estimate the population's standard deviation with the sample standard deviation. This leads to the following formula, x-bar plus and minus the z score for the 95% confidence level times the estimated standard deviation of the sampling distribution of the sample mean, which equals the sample standard deviation divided by the square root of the sample size. We call this estimated standard deviation of the sampling distribution the standard error. But because we're now estimating the standard deviation, we add extra error in the computation. For that reason, we employ another distribution than the standard normal distribution. Also called the z distribution we employed previously. Because of the extra error, we now use the T-distribution. That leads to this formula. x-bar plus and minus the t score for the 95% confidence level, times the estimated standard deviation of the sampling distribution of the sample mean. Let me now tell you a little more about t distributions and t scores. The t distribution strongly resembles the standard normal distribution. It is bell-shaped, symmetric, and has a mean of zero. Still, however, it is slightly different. Because we're now estimating the standard deviation of the sampling distribution, we introduce extra error. That error can be substantial when we have a small sample. The t distribution takes into account that extra error for small samples. The t distribution therefore has slightly thicker tails than a normal distribution and has a larger standard deviation. You can see that here. The black distribution is a standard normal distribution. The blue one is a t distribution. The exact shape of the t distribution depends on the size of the sample. The larger the sample, the more the t distribution looks like the normal distribution. More precisely, the shape of the t distribution is dependent on a single parameter, the so-called degrees of freedom symbolized by df. The degrees of freedom parameter in the t distribution equals the sample size n minus 1. This means that we actually have many different t distributions. One separate distribution for every df. The blue t distribution you see here has two degrees of freedom. A t distribution with five degrees of freedom looks like this. And a t distribution with 30 degrees of freedom looks like this. This means that when we have 30 or more degrees of freedom, the t distribution is almost identical to the standard normal distribution. More precisely the standard normal distribution is the t distribution with df equals infinity. Just like with the standard normal distribution and Z-scores, we can find cumulative probabilities pertaining to particular t scores. An important difference with the standard normal distribution however, is that these probabilities depend on the degrees of freedom. When you're computing a 95% confidence interval, you can find the t scores pertaining to the 95% confidence level for all the different possible degrees of freedom in a so called t table. This table is similar to the z table. Let me show you how that works by means of our new parent study. The mean lost sleeping hours in our sample was 2.6. And the standard deviation was 0.9. The sample size was 60. This is the formula to compute endpoints of the 95% confidence interval. Let's start with computing the standard error. It equals a sample standard deviation s divided by the square root of n. That is 0.9 divided by the square root of 60. That makes 0.116. So our standard error, or in other words, the estimated standard deviation of the sampling distribution of the sample mean equals 0.116. To compute a margin of error, we have to multiply this value with the t score for the 95% confidence level. As you know, the t score depends on the degrees of freedom. df equals n minus 1. We have a sample of 60, so 60 minus 1 equals 59. In our t table, we look in the column of the 95% confidence level and in the row of 59 degrees of freedom. Because the table does not report 59 degrees of freedom, we settle for the closest lower value, that's 50 degrees of freedom. The relevant t score is 2.009. So we multiply 0.116 with 2.009. That's about 0.23. So we subtract this value from our sample mean 2.6 and also add it to the mean. That gives us the interval endpoints 2.37 and 2.83. The 95% confidence interval goes from 2.37 to 2.83. We have 95% confidence that this interval contains the actual population mean. To compute a confidence interval for a population mean, two assumptions need to be satisfied. First, your data should be obtained by randomization. In other words, the sample should be a random sample otherwise your findings will not be valid. Second, your population should be approximately normally distributed. This might seem to be problematic because many variables are not normally distributed in the population. However, things are not as bad as they seem. Using the t distribution to construct a confidence interval for a mean is robust against violations of the second assumption. That a statistical method is robust means that it performs well even if the assumption is violated. Finally, you should also be wary of extreme outliers when constructing a confidence interval based on the t distribution. When your data have extreme outliers the method doesn't work well. So always check your data for outliers before you start.
