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1 Learning objectives
Understand probability as a measure of uncertainty
Summarize probabilities with probability distributions
Describe the difference between population and sample
Think about uncertainty in sampling
Summarize uncertainty in sampling: Sampling distribution
2 Mathematically-derived sampling distribution
Many combinations of population distribution and test statistic have mathematically-derived sampling distributions
For example, with a normally distributed population (\mathcal{N}(\mu, \sigma^2)), the sampling distribution of the mean is \bar{X} \sim \mathcal{N}(\mu, \frac{\sigma^2}{n})
In this case, the population distribution and sampling distribution are the same distribution: Normal
The variance (or standard deviation) of the sampling distribution decreases as n increases: A larger sample leads to a smaller, more precise standard error
Note
There is not a mathematically-derived sampling distribution for every population distribution and test statistic
If you’re dealing with unknown or unusual distributions or test statistics, your only option may be computational methods
3 Computationally-derived sampling distribution
3.1 Getting started
Set a random number seed
If you don’t do this, every time you run the code, you’ll get different numbers
Setting a random seed makes your simulation replicable
(I normally set a seed at the start of each document, but I moved it down here because it’s very important in this lab.)
---title: "BTS 510 Lab 6"format: html: embed-resources: true self-contained-math: true html-math-method: katex number-sections: true toc: true code-tools: true code-block-bg: true code-block-border-left: "#31BAE9"---```{r}#| label: setuplibrary(tidyverse)theme_set(theme_classic(base_size =16))```## Learning objectives* Understand **probability** as a measure of **uncertainty*** **Summarize** probabilities with *probability distributions* * Describe the difference between **population** and **sample*** Think about **uncertainty** in sampling* **Summarize** uncertainty in sampling: **Sampling distribution**## Mathematically-derived sampling distribution* Many combinations of **population distribution** and **test statistic** have *mathematically-derived* sampling distributions * For example, with a **normally distributed** population ($\mathcal{N}(\mu, \sigma^2)$), the sampling distribution of the **mean** is $\bar{X} \sim \mathcal{N}(\mu, \frac{\sigma^2}{n})$ * In this case, the *population* distribution and *sampling* distribution are the **same distribution**: Normal * The variance (or standard deviation) of the sampling distribution **decreases** as $n$ increases: A *larger sample* leads to a smaller, *more precise standard error*::: {.callout-note}* There is not a mathematically-derived sampling distribution for every population distribution and test statistic * If you're dealing with *unknown or unusual distributions or test statistics*, your only option may be *computational methods*:::## Computationally-derived sampling distribution### Getting started* Set a **random number seed** * If you don't do this, every time you run the code, you'll get different numbers * Setting a random seed makes your simulation *replicable* * (I normally set a seed at the start of each document, but I moved it down here because it's very important in this lab.)* Load (and install, if needed) the [**infer** package](https://infer.tidymodels.org/) * **infer** has a function that we'll use to *repeatedly sample* * There are other ways to do this, but this is very simple and works well* Only have to do this once per document```{r}set.seed(12345)library(infer)```### Standard Normal population: $population \sim \mathcal{N}(0, 1)$#### Create a population* `rnorm()` creates a random variable that follows a **normal** distribution * First argument = Number of observations (population size) * Second argument = $\mu$ * Third argument = $\sigma$ (not $\sigma^2$)```{r}norm_pop <-rnorm(1000000, 0, 1)mean(norm_pop)var(norm_pop)```#### Sample from the population* Small sample: $n$ = 10 * `rep_sample_n()` repeatedly samples from the dataset * Piped (`%>%`) from the `norm_pop` dataset, so it resamples from that * `size = 10`: Sample size for each sample * `reps = 10000`: Number of repeated samples (could be larger but this is good enough to show how things work and doesn't take too long to run) * `replace = TRUE`: Sample with replacement * `summarise(x_bar = mean(norm_pop))`: Create a data frame with each of the means from each sample```{r}norm_means10 <-data.frame(norm_pop) %>%rep_sample_n(size =10, reps =10000, replace =TRUE) %>%summarise(x_bar =mean(norm_pop))ggplot(data = norm_means10, aes(x = x_bar)) +geom_histogram(bins =50) +xlim(-1.5, 1.5)```* Moderate sample: $n$ = 50```{r}norm_means50 <-data.frame(norm_pop) %>%rep_sample_n(size =50, reps =10000, replace =TRUE) %>%summarise(x_bar =mean(norm_pop))ggplot(data = norm_means50, aes(x = x_bar)) +geom_histogram(bins =50) +xlim(-1.5, 1.5) ```* Large sample: $n$ = 100```{r}norm_means100 <-data.frame(norm_pop) %>%rep_sample_n(size =100, reps =10000, replace =TRUE) %>%summarise(x_bar =mean(norm_pop))ggplot(data = norm_means100, aes(x = x_bar)) +geom_histogram(bins =50) +xlim(-1.5, 1.5)```::: {.callout-note}* Note that I adjusted the X axis to be the same for all three plots using `xlim()` so that you can easily compare the distributions * You'll want to do the same in your later plots too:::#### Compare estimates* Compare means and variances for each sample size```{r}mean(norm_means10$x_bar)mean(norm_means50$x_bar)mean(norm_means100$x_bar)``````{r}var(norm_means10$x_bar)var(norm_means50$x_bar)var(norm_means100$x_bar)```* How does each value correspond to what you'd expect based on the mathematically-derived sampling distribution? * $\bar{X} \sim \mathcal{N}(\mu, \frac{\sigma^2}{n})$### Bernoulli population: $population \sim B(0.3)$#### Create a population* `rbinom()` creates a random variable that follows a **binomial** distribution * First argument = Number of observations (population size) * Second argument = number of trials (here, just 1 for a **Bernoulli** trial) * Third argument = $p$ (probability of success)```{r}bernoulli_pop <-rbinom(1000000, 1, 0.3)mean(bernoulli_pop)var(bernoulli_pop)```#### Sample from the population* Small sample: $n$ = 10```{r}```* Moderate sample: $n$ = 50```{r}```* Large sample: $n$ = 100```{r}```#### Compare estimates* Compare means and variances for each sample size```{r}``````{r}```### Binomial population: $population \sim Bin(50, 0.3)$#### Create a population* `rbinom()` creates a random variable that follows a **binomial** distribution * First argument = Number of observations (population size) * Second argument = $m$ (number of trials) * Third argument = $p$ (probability of success)```{r}binomial_pop <-rbinom(1000000, 50, 0.3)mean(binomial_pop)var(binomial_pop)```#### Sample from the population* Small sample: $n$ = 10```{r}```* Moderate sample: $n$ = 50```{r}```* Large sample: $n$ = 100```{r}```#### Compare estimates* Compare means and variances for each sample size```{r}``````{r}```### Poisson population: $population \sim Poisson(1.5)$#### Create a population* `rpois()` creates a random variable that follows a **Poisson** distribution * First argument = Number of observations (population size) * Second argument = $\lambda$ (mean and variance)```{r}poisson_pop <-rpois(1000000, 1.5)mean(poisson_pop)var(poisson_pop)```#### Sample from the population* Small sample: $n$ = 10```{r}```* Moderate sample: $n$ = 50```{r}```* Large sample: $n$ = 100```{r}```#### Compare estimates* Compare means and variances for each sample size```{r}``````{r}```
