# Foundations of Inference

Learn how to draw conclusions about a population from a sample of data via a process known as statistical inference.

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## Course Description

One of the foundational aspects of statistical analysis is inference, or the process of drawing conclusions about a larger population from a sample of data. Although counter intuitive, the standard practice is to attempt to disprove a research claim that is not of interest. For example, to show that one medical treatment is better than another, we can assume that the two treatments lead to equal survival rates only to then be disproved by the data. Additionally, we introduce the idea of a p-value, or the degree of disagreement between the data and the hypothesis. We also dive into confidence intervals, which measure the magnitude of the effect of interest (e.g. how much better one treatment is than another).

- 1
### Introduction to ideas of inference

**Free**In this chapter, you will investigate how repeated samples taken from a population can vary. It is the variability in samples that allow you to make claims about the population of interest. It is important to remember that the research claims of interest focus on the population while the information available comes only from the sample data.

Welcome to the course!50 xpHypotheses (1)50 xpHypotheses (2)50 xpRandomized distributions50 xpWorking with the NHANES data100 xpCalculating statistic of interest100 xpRandomized data under null model of independence100 xpRandomized statistics and dotplot100 xpRandomization density100 xpUsing the randomization distribution50 xpDo the data come from the population?100 xpWhat can you conclude?50 xpStudy conclusions50 xp - 2
### Completing a randomization test: gender discrimination

In this chapter, you will gain the tools and knowledge to complete a full hypothesis test. That is, given a dataset, you will know whether or not is appropriate to reject the null hypothesis in favor of the research claim of interest.

Example: gender discrimination50 xpGender discrimination hypotheses50 xpSummarizing gender discrimination100 xpStep-by-step through the permutation100 xpRandomizing gender discrimination100 xpDistribution of statistics50 xpReflecting on analysis50 xpCritical region100 xpTwo-sided critical region100 xpWhy 0.05?50 xpHow does sample size affect results?50 xpSample size in randomization distribution100 xpSample size for critical region100 xpWhat is a p-value?50 xpCalculating the p-values100 xpPractice calculating p-values100 xpCalculating two-sided p-values100 xpSummary of gender discrimination50 xp - 3
### Hypothesis testing errors: opportunity cost

You will continue learning about hypothesis testing with a new example and the same structure of randomization tests. In this chapter, however, the focus will be on different errors (type I and type II), how they are made, when one is worse than another, and how things like sample size and effect size impact the error rates.

Example: opportunity cost50 xpSummarizing opportunity cost (1)100 xpPlotting opportunity cost100 xpRandomizing opportunity cost100 xpSummarizing opportunity cost (2)100 xpOpportunity cost conclusion50 xpErrors and their consequences50 xpDifferent choice of error rate50 xpErrors for two-sided hypotheses50 xpp-value for two-sided hypotheses: opportunity costs100 xpSummary of opportunity costs50 xp - 4
### Confidence intervals

As a complement to hypothesis testing, confidence intervals allow you to estimate a population parameter. Recall that your interest is always in some characteristic of the population, but you only have incomplete information to estimate the parameter using sample data. Here, the parameter is the true proportion of successes in a population. Bootstrapping is used to estimate the variability needed to form the confidence interval.

Parameters and confidence intervals50 xpWhat is the parameter?50 xpHypothesis test or confidence interval?50 xpBootstrapping50 xpResampling from a sample100 xpVisualizing the variability of p-hat100 xpAlways resample the original number of observations50 xpVariability in p-hat50 xpEmpirical Rule100 xpBootstrap t-confidence interval100 xpBootstrap percentile interval100 xpInterpreting CIs and technical conditions50 xpSample size effects on bootstrap CIs100 xpSample proportion value effects on bootstrap CIs100 xpPercentile effects on bootstrap CIs100 xpSummary of statistical inference50 xp

Datasets

All pollsPolling dataBig discrimination datasetNew discrimination datasetSmall discrimination datasetCollaborators

#### Jo Hardin

Professor at Pomona College

Jo Hardin is a professor of mathematics and statistics at Pomona College. Her statistical research focuses on developing new robust methods for high throughput data. Recently, she has also worked closely with the statistics education community on ways to integrate data science early into a statistics curriculum. When not working with students or on her research, she loves to put on a pair of running shoes and hit the road.

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