How do the confidence intervals vary with 1) sample size (i.e., n = 30 vs n = 100) and 2) confidence level (i.e., 95% vs 99%)?
3 Summary
How do the confidence intervals vary with sample size?
How do the confidence intervals vary with confidence level?
How do the confidence intervals vary with population distribution?
Did you notice anything else?
Source Code
---title: "BTS 510 Lab 7"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)set.seed(12345)theme_set(theme_classic(base_size =16))```## Learning objectives* **Compare** and **contrast** *point estimates* and *interval estimates** **Construct** *confidence intervals** **Interpret** *confidence intervals*## Vary population distribution, sample size, confidence interval level### Normally distributed: $population \sim \mathcal{N}(\mu = 10, \sigma^2 = 25)$* Two sample sizes: 30 and 100* Note that `rnorm()` takes standard deviation as an argument, not variance```{r}norm30 <-rnorm(n =30, mean =10, sd =5)norm100 <-rnorm(n =100, mean =10, sd =5)```* Calculate the means, variances, and standard deviations for each sample```{r}mean(norm30)mean(norm100)var(norm30)var(norm100)sd(norm30)sd(norm100)```* Construct the 95% and 99% confidence intervals for each mean$[\bar{X} - z_{1-\alpha/2}\ \sqrt{\frac{\sigma^2}{n}}, \bar{X} + z_{1-\alpha/2}\ \sqrt{\frac{\sigma^2}{n}}]$* I'm just having R make the calculations for me, rather than doing by hand * I'm also doing it in several steps so you can see where each piece comes from, but you can do it all in one step if you're confident```{r}norm30_95moe <-1.96*sqrt(var(norm30)/30)norm30_95lcl <-mean(norm30) - norm30_95moenorm30_95ucl <-mean(norm30) + norm30_95moenorm30_99moe <-2.326*sqrt(var(norm30)/30)norm30_99lcl <-mean(norm30) - norm30_99moenorm30_99ucl <-mean(norm30) + norm30_99moe norm100_95moe <-1.96*sqrt(var(norm100)/100)norm100_95lcl <-mean(norm100) - norm100_95moenorm100_95ucl <-mean(norm100) + norm100_95moenorm100_99moe <-2.326*sqrt(var(norm100)/100)norm100_99lcl <-mean(norm100) - norm100_99moenorm100_99ucl <-mean(norm100) + norm100_99moe ```* You can look at them in the output, or you can have R print them in your Quarto document * (Look at the code to see how this was done) * n = 30, 95%: [`r round(norm30_95lcl, 3)`, `r round(norm30_95ucl, 3)`] * n = 30, 99%: [`r round(norm30_99lcl, 3)`, `r round(norm30_99ucl, 3)`] * n = 100, 95%: [`r round(norm100_95lcl, 3)`, `r round(norm100_95ucl, 3)`] * n = 100, 99%: [`r round(norm100_99lcl, 3)`, `r round(norm100_99ucl, 3)`]* How do the confidence intervals vary with 1) sample size (i.e., n = 30 vs n = 100) and 2) confidence level (i.e., 95% vs 99%)?### Bernoulli distributed: $population \sim B(0.3)$* Two sample sizes: 30 and 100```{r}bern30 <-rbinom(30, 1, 0.3)bern100 <-rbinom(100, 1, 0.3)``````{r}ggplot(data =data.frame(bern30), aes(x = bern30)) +geom_histogram()```* Calculate the means, variances, and standard deviations for each sample```{r}mean(bern30)sd(bern30)var(bern30)mean(bern100)sd(bern100)var(bern100)```* Construct the 95% and 99% confidence intervals for each mean$[\bar{X} - z_{1-\alpha/2}\ \sqrt{\frac{\sigma^2}{n}}, \bar{X} + z_{1-\alpha/2}\ \sqrt{\frac{\sigma^2}{n}}]$```{r}bern30_95moe <-1.96*sqrt(var(bern30)/30)bern30_95lcl <-mean(bern30) - bern30_95moebern30_95ucl <-mean(bern30) + bern30_95moebern30_99moe <-2.326*sqrt(var(bern30)/30)bern30_99lcl <-mean(bern30) - bern30_99moebern30_99ucl <-mean(bern30) + bern30_99moe bern100_95moe <-1.96*sqrt(var(bern100)/100)bern100_95lcl <-mean(bern100) - bern100_95moebern100_95ucl <-mean(bern100) + bern100_95moebern100_99moe <-2.326*sqrt(var(bern100)/100)bern100_99lcl <-mean(bern100) - bern100_99moebern100_99ucl <-mean(bern100) + bern100_99moe ```* You can look at them in the output, or you can have R print them in your Quarto document * (Look at the code to see how this was done) * n = 30, 95%: [`r round(bern30_95lcl, 3)`, `r round(bern30_95ucl, 3)`] * n = 30, 99%: [`r round(bern30_99lcl, 3)`, `r round(bern30_99ucl, 3)`] * n = 100, 95%: [`r round(bern100_95lcl, 3)`, `r round(bern100_95ucl, 3)`] * n = 100, 99%: [`r round(bern100_99lcl, 3)`, `r round(bern100_99ucl, 3)`]* How do the confidence intervals vary with 1) sample size (i.e., n = 30 vs n = 100) and 2) confidence level (i.e., 95% vs 99%)?### Binomial distributed: $population \sim Bin(50, 0.3)$* Two sample sizes: 30 and 100```{r}binom30 <-rbinom(30, 50, 0.3)binom100 <-rbinom(100, 50, 0.3)```* Calculate the means, variances, and standard deviations for each sample```{r}```* Construct the 95% and 99% confidence intervals for each mean$[\bar{X} - z_{1-\alpha/2}\ \sqrt{\frac{\sigma^2}{n}}, \bar{X} + z_{1-\alpha/2}\ \sqrt{\frac{\sigma^2}{n}}]$```{r}```* How do the confidence intervals vary with 1) sample size (i.e., n = 30 vs n = 100) and 2) confidence level (i.e., 95% vs 99%)?### Poisson distributed: $population \sim Poisson(1.5)$* Two sample sizes: 30 and 100```{r}pois30 <-rpois(n =30, lambda =1.5)pois100 <-rpois(n =30, lambda =1.5)```* Calculate the means, variances, and standard deviations for each sample```{r}```* Construct the 95% and 99% confidence intervals for each mean$[\bar{X} - z_{1-\alpha/2}\ \sqrt{\frac{\sigma^2}{n}}, \bar{X} + z_{1-\alpha/2}\ \sqrt{\frac{\sigma^2}{n}}]$```{r}```* How do the confidence intervals vary with 1) sample size (i.e., n = 30 vs n = 100) and 2) confidence level (i.e., 95% vs 99%)?## Summary* How do the confidence intervals vary with sample size?* How do the confidence intervals vary with confidence level?* How do the confidence intervals vary with population distribution?* Did you notice anything else?
