In 2013, UNICEF rated Dutch children as the happiest in the world. Since I live in the Netherlands this will be great news for me. But do you believe it? If not, how could you prove them wrong? Would you for example ask every child in the country how happy they are? Of course not, that's unfeasible. There's a much more practical way. We could draw a representative sample. From that sample you could then learn something about all that children. Every research question defines a population of interest. In this lecture, we will discuss the basic principles to draw conclusions about an entire population using one single sample. This is called Statistical Inference. To do so we will first look at how sample and population relate to each other. Then you will learn what sampling variation is and we will discuss key concepts such as the standard error and the sampling distribution. We will then see how these concepts are used to draw inferences about a population parameter, as for example the mean. Finally, you will learn what a confidence interval is. Imagine we want to study the happiness of children in the country. Let us define the population more specifically. Girls and boys between the ages of eight and 12 living in the Netherlands. The next step is to draw a sample from our population. These are the children we are going to interviewing our study. It is very important that the sample of individuals is representative which means that we believe them to have similar characteristics to the population of interest. Finally, we will collect data from our sample. Some of these will be general characteristics like age and gender. But we will also collect specific data for answering our research question. In our case, to determine what is the average level of happiness of that children. With the same, we will measure the level of happiness of each child in our sample through a numerical score ranging from zero to 10. The mean happiness score in the total population is what we call parameter of interest. We use it to answer our research question and we will estimate it using the sample. We first describe the happiness score in the sample. Recall that we should describe two aspects; the average and the variation around that average. See for example here the results I got with 60 children. The sample mean, the mean happiness score in these 60 observed children is 7.9. The variation around the sample mean can be described with a standard deviation. In our case it is 1.3. The sample mean is the base case we can make with the collected data about the mean in the population. We call it the point estimate of the parameter of interest which here is the population mean. It is unlikely that the sample mean is exactly equal to the real mean score of the population. Moreover, if we will repeat exactly the same study with the same number of children of the same characteristics, it is highly unlikely that we would get exactly the same mean score of happiness. This phenomenon is called sampling variation. How can we say anything about the mean of the entire population if every time that we pick a different sample, the point estimate is going to be off. This is where we will apply statistics. To quantify how well our point estimate represents the population parameter, we first need to quantify how much variability our results can be expected by chance alone. The common approach to this problem is described by the central limit theorem. Let us go through this theory step-by-step. Imagine that you repeat your study and you draw many different samples from the same population. All of the same size and then calculate each time the mean. Many of these means will be close to the real mean of the population but some will be completely off. This means that there is a distribution of the sample means. It is called the sampling distribution. A nice property of the sampling distribution of the mean is that it is normal and that the mean of this distribution is the true population mean which we want to estimate well. This is true even if the distribution of the data is not normal as long as your sample is not too small. This is an important concept to grasp. Now, let us go a step further. The standard deviation of the sampling distribution is called a standard error and indicates the spread of your results if you would repeatedly select a sample with the same number of children and calculate the mean happiness score. The lower the standard error the better. A small standard error will tell us that every time you calculate the sample mean, you obtain a value which is close to the true population mean. In other words, a small standard error indicates that the sample mean is a very precise estimate of the population mean. Fortunately, we do not need to repeat our study many times to calculate the standard error. For the mean, the standard error can be calculated as the standard deviation of the data divided by the square root of the sample size. This means that the standard error depends on how much variation there is in the population we're studying and on the sample size. The larger the sample size, the smaller the standard error. For example, if you select a sample of 10 children you would get this sampling distribution for the mean happiness score. You see that it has a larger split. The standard error is large. This means that anytime I run a study with 10 children, it is likely that my sample mean happiness score is quite different from the real mean happiness in the population. The precision is low. If you now select a sample of 1000 children from the same population, you would get this sampling distribution. The standard error is smaller now. This means that repeating the exact same study will result in similar estimates of the sample mean close to the true population mean. With a large sample size, the sample mean is a precise estimator of the population mean. Let's apply all this toward that children. Here you see the sampling distribution obtained with our sample of 60 children. The sample mean is 7.9 and the standard deviation is 1.3. This results in a standard error of 0.17. Now, using the properties of the sampling distribution, we can go a step further and provide a range of plausible values of the level of happiness in the whole population of that children. This is called a confidence interval. In our case, the 95 percent confidence interval contains the scores between 7.6 and 8.2. The confidence interval is interpreted as follows. If we will repeat our study 20 times and every time we would calculate the confidence interval, the true mean happiness level will lie in the confidence interval 95 percent of the times. Unfortunately, one out of 20 times, the confidence interval will not contain the population mean. In this lecture, we have discussed how you can say something about the whole population by using just one sample. We have covered the basic concepts of statistical inference. Sampling variation, sampling distribution, standard error and confidence interval. You have learned that the precision of our sample estimate of a population parameter like for example the mean, can be summarized by the standard error and that the larger your representative sample is, the more precise your sample estimate will be. In the next reading activity, we will discuss how to apply the main ideas covered in this lecture to estimate other parameters like for example proportions. You will also learn how to calculate confidence intervals using R.