Bootstrap, Confidence Intervals and Hypothesis Testing with Python

Bootstrap is a statistical method that allows you to make inferences about a population from a sample. The basic idea is to repeatedly sample from the original sample with replacement to create many new samples, called bootstrap samples. These samples are then used to estimate the population parameters and construct confidence intervals.

Confidence intervals are a measure of the uncertainty of an estimate. They provide a range of plausible values for a population parameter based on a sample. The most common method for constructing confidence intervals is the percentile method, where the lower and upper bounds of the interval are defined by the percentiles of the distribution of the bootstrap samples.

Hypothesis testing is a statistical method that allows you to make inferences about a population based on a sample. The basic idea is to formulate a null hypothesis (e.g. the population mean is equal to a certain value) and an alternative hypothesis (e.g. the population mean is different from the certain value), and then use the sample data to test whether the null hypothesis can be rejected in favor of the alternative hypothesis.

In Python, the scikit-learn library provides a simple and easy-to-use implementation of bootstrap through the resample function from the sklearn.utils module. Here is an example of how to use it to create bootstrap samples and estimate the mean of a population:

In addition, the scipy.stats module provides the norm.interval function to calculate the confidence intervals. Here is an example of how to use it to calculate a 95% confidence interval for the mean of a population:

In this example, I've used the norm.interval function to calculate the 95% confidence interval for the mean of the population. The first parameter is the level of confidence (0.95 for a 95% interval), the loc parameter is the mean of the bootstrap sample, and the scale parameter is the margin of error.

As for hypothesis testing, the scipy.stats module provides various statistical test functions such as ttest_1samp, ttest_ind and ttest_rel for one sample, two independent samples, and two dependent samples respectively. Here is an example of how to use the ttest_1samp function to test the null hypothesis that the population mean is equal to 10:

In this example, I've used the ttest_1samp function to test the null hypothesis that the population mean is equal to 10. The first parameter is the sample data, and the second parameter is the hypothesized population mean. The function returns the test statistic and the p-value.

In summary, Bootstrap is a statistical method that allows you to make inferences about a population from a sample, Confidence intervals are a measure of the uncertainty of an estimate, and Hypothesis testing is a statistical method that allows you to make inferences about a population based on a sample. Python provides multiple libraries to implement those concepts, the scikit-learn library provides the resample function for bootstrap, the scipy.stats module provides the norm.interval function for confidence intervals and various statistical test functions for hypothesis testing.