Simulating a null hypothesis concerning correlation

The observed correlation between female illiteracy and fertility in the data set of 162 countries may just be by chance; the fertility of a given country may actually be totally independent of its illiteracy. You will test this null hypothesis in the next exercise.

To do the test, you need to simulate the data assuming the null hypothesis is true. Of the following choices, which is the best way to to do it?

This exercise is part of the course

Statistical Thinking in Python (Part 2)

Exercise instructions

Choose 162 random numbers to represent the illiteracy rate and 162 random numbers to represent the corresponding fertility rate.,Do a pairs bootstrap: Sample pairs of observed data with replacement to generate a new set of (illiteracy, fertility) data.,Do a bootstrap sampling in which you sample 162 illiteracy values with replacement and then 162 fertility values with replacement.,Do a permutation test: Permute the illiteracy values but leave the fertility values fixed to generate a new set of (illiteracy, fertility) data.,Do a permutation test: Permute both the illiteracy and fertility values to generate a new set of (illiteracy, fertility data).

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This exercise is part of the course

Statistical Thinking in Python (Part 2)

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When doing statistical inference, we speak the language of probability. A probability distribution that describes your data has parameters. So, a major goal of statistical inference is to estimate the values of these parameters, which allows us to concisely and unambiguously describe our data and draw conclusions from it. In this chapter, you will learn how to find the optimal parameters, those that best describe your data.

Exercise 1: Welcome to Introduction to Statistical Thinking with Python II Exercise 2: Optimal parameters Exercise 3: How often do we get no-hitters?Exercise 4: Do the data follow our story?Exercise 5: How is this parameter optimal?Exercise 6: Linear regression by least squares Exercise 7: EDA of literacy/fertility data Exercise 8: Linear regression Exercise 9: How is it optimal?Exercise 10: The importance of EDA: Anscombe's quartet Exercise 11: The importance of EDA Exercise 12: Linear regression on appropriate Anscombe data Exercise 13: Linear regression on all Anscombe data

To "pull yourself up by your bootstraps" is a classic idiom meaning that you achieve a difficult task by yourself with no help at all. In statistical inference, you want to know what would happen if you could repeat your data acquisition an infinite number of times. This task is impossible, but can we use only the data we actually have to get close to the same result as an infinitude of experiments? The answer is yes! The technique to do it is aptly called bootstrapping. This chapter will introduce you to this extraordinarily powerful tool.

Exercise 1: Generating bootstrap replicates Exercise 2: Getting the terminology down Exercise 3: Bootstrapping by hand Exercise 4: Visualizing bootstrap samples Exercise 5: Bootstrap confidence intervals Exercise 6: Generating many bootstrap replicates Exercise 7: Bootstrap replicates of the mean and the SEM Exercise 8: Confidence intervals of rainfall data Exercise 9: Bootstrap replicates of other statistics Exercise 10: Confidence interval on the rate of no-hitters Exercise 11: Pairs bootstrap Exercise 12: A function to do pairs bootstrap Exercise 13: Pairs bootstrap of literacy/fertility data Exercise 14: Plotting bootstrap regressions

You now know how to define and estimate parameters given a model. But the question remains: how reasonable is it to observe your data if a model is true? This question is addressed by hypothesis tests. They are the icing on the inference cake. After completing this chapter, you will be able to carefully construct and test hypotheses using hacker statistics.

Exercise 1: Formulating and simulating a hypothesis Exercise 2: Generating a permutation sample Exercise 3: Visualizing permutation sampling Exercise 4: Test statistics and p-values Exercise 5: Test statistics Exercise 6: What is a p-value?Exercise 7: Generating permutation replicates Exercise 8: Look before you leap: EDA before hypothesis testing Exercise 9: Permutation test on frog data Exercise 10: Bootstrap hypothesis tests Exercise 11: A one-sample bootstrap hypothesis test Exercise 12: A bootstrap test for identical distributions Exercise 13: A two-sample bootstrap hypothesis test for difference of means.

As you saw from the last chapter, hypothesis testing can be a bit tricky. You need to define the null hypothesis, figure out how to simulate it, and define clearly what it means to be "more extreme" in order to compute the p-value. Like any skill, practice makes perfect, and this chapter gives you some good practice with hypothesis tests.

Exercise 1: A/B testing Exercise 2: The vote for the Civil Rights Act in 1964 Exercise 3: What is equivalent?Exercise 4: A time-on-website analog Exercise 5: What should you have done first?Exercise 6: Test of correlation Exercise 7: Simulating a null hypothesis concerning correlation

Current Exercise

Exercise 8: Hypothesis test on Pearson correlation Exercise 9: Do neonicotinoid insecticides have unintended consequences?Exercise 10: Bootstrap hypothesis test on bee sperm counts

Every year for the past 40-plus years, Peter and Rosemary Grant have gone to the Galápagos island of Daphne Major and collected data on Darwin's finches. Using your skills in statistical inference, you will spend this chapter with their data, and witness first hand, through data, evolution in action. It's an exhilarating way to end the course!

Exercise 1: Finch beaks and the need for statistics Exercise 2: EDA of beak depths of Darwin's finches Exercise 3: ECDFs of beak depths Exercise 4: Parameter estimates of beak depths Exercise 5: Hypothesis test: Are beaks deeper in 2012?Exercise 6: Variation of beak shapes Exercise 7: EDA of beak length and depth Exercise 8: Linear regressions Exercise 9: Displaying the linear regression results Exercise 10: Beak length to depth ratio Exercise 11: How different is the ratio?Exercise 12: Calculation of heritability Exercise 13: EDA of heritability Exercise 14: Correlation of offspring and parental data Exercise 15: Pearson correlation of offspring and parental data Exercise 16: Measuring heritability Exercise 17: Is beak depth heritable at all in *G. scandens*?Exercise 18: Final thoughts