Statistical Inference via Data Science A ModernDive into R and the Tidyverse (Second Edition)

We studied sampling in Chapter 7. Recall, for example, getting many random samples of red and white balls from a bowl, finding the sample proportions of red balls from each of those samples, and studying the distribution of the sample proportions. We can summarize our findings as follows:

Similarly, when sampling chocolate-covered almonds and getting the sample mean weight from each sample, the characteristics described above are also encountered in the sampling distribution of the sample mean; namely,

Moreover, these characteristics also apply to sampling distributions for the difference in sample means, the difference in sample proportions, and others. Recall that the sampling distribution is not restricted by the distribution of the population. As long as the samples taken are fairly large and we use the appropriate standard error, we can generalize these results appropriately.

The study of the sampling distribution is motivated by another question we have not yet answered: how can we determine the average weight of all the almonds if we do not have access to the entire bowl? We have seen by using simulations in Chapter 7 that the average of the sample means, derived from many random samples, will be fairly close to the expected value of the sample mean, which is precisely the population mean weight.

However, in real-life situations, we do not have access to many random samples, only to a single random sample. This chapter introduces methods and techniques that can help us approximate the information of the entire population, such as the population mean weight, by using a single random sample from this population. This undertaking is called estimation, and it is central to Statistics and Data Science.

We introduce some statistical jargon about estimation. If we are using a sample statistic to estimate a population parameter, e.g., using the sample mean from a random sample to estimate the population mean, we call this statistic a point estimate to make emphasis that it is a single value that is used to estimate the parameter of interest. Now, you may recall that, due to sampling variation, the sample mean typically does not match the population mean exactly, even if the sample is large. To account for this variation, we use an interval to estimate the parameter instead of a single value, and appropriately call it an interval estimate or, if given some level of accuracy, a confidence interval of the population parameter. In this chapter, we explain how to find confidence intervals, the advantages of using them, and the different methods that can be used to determine them.

In Section 8.1 we introduce a method to build a confidence interval for the population mean that uses the random sample taken and theoretical characteristics of the sampling distribution discussed in Chapter 7. We call this the theory-based approach for constructing intervals. In Section 8.2 we introduce another method, called the bootstrap, that produces confidence intervals by resampling a large number of times from the original sample. Since resampling is done via simulations, we call this the simulation-based approach for constructing confidence intervals. Finally, in Section 8.5 we summarize and present extensions of these methods.

If needed, read Section 1.3 for information on how to install and load R packages.

library(tidyverse)
library(moderndive)
library(infer)

Recall that loading the tidyverse package loads many packages that we have encountered earlier. For details refer to Section 1.3. The packages moderndive and infer contain functions and data frames that will be used in this chapter.