Bootstrapping

For Quiz 12

Load the R packages we will use

library(tidyverse)
library(moderndive) 
library(infer) 
library(fivethirtyeight) 

• Set random seed generator to 123

• Take a sample of 100 from the dataset congress_age and assign it to congress_age_100

set.seed(4346)

congress_age_100 <- congress_age  %>% 
  rep_sample_n(size=100)

• congress_age is the population and congress_age_100 is the sample

• 18,635 is number of observations in the population and 100 is the number of observations in your sample

1. Use specify to indicate the variable from congress_age_100 that you are interested in

congress_age_100  %>% 
  specify(response = age)

Response: age (numeric)
# A tibble: 100 x 1
     age
   <dbl>
 1  58  
 2  27.3
 3  59.4
 4  47.8
 5  36.4
 6  62.3
 7  52.5
 8  55.5
 9  44  
10  48  
# ... with 90 more rows

---

2. Generate 1000 replicates of your sample of 100

congress_age_100  %>% 
  specify(response = age)  %>% 
  generate(reps = 1000, type= "bootstrap")

Response: age (numeric)
# A tibble: 100,000 x 2
# Groups:   replicate [1,000]
   replicate   age
       <int> <dbl>
 1         1  55.2
 2         1  40.8
 3         1  55.7
 4         1  52.5
 5         1  54.5
 6         1  35.8
 7         1  44.5
 8         1  47.9
 9         1  40.8
10         1  37.4
# ... with 99,990 more rows

• The output has 100,000 rows

---

3. Calculate the mean for each replicate

• Assgin to boostrap_distribution_mean_age

• Display bootstrap_distribution_mean_age

bootstrap_distribution_mean_age  <- congress_age_100  %>% 
  specify(response = age)  %>% 
  generate(reps = 1000, type = "bootstrap")  %>% 
  calculate(stat = "mean")

bootstrap_distribution_mean_age

# A tibble: 1,000 x 2
   replicate  stat
       <int> <dbl>
 1         1  51.3
 2         2  48.2
 3         3  49.7
 4         4  50.5
 5         5  51.6
 6         6  47.9
 7         7  49.5
 8         8  50.0
 9         9  51.0
10        10  51.0
# ... with 990 more rows

The bootstrap has 1000 rows

---

4. Visualize the bootstrap distribution

visualize(bootstrap_distribution_mean_age) 

calculate the 95% confidence interval using the percentile method

• Assign the output to congress_ci_percentile

• Display congress_ci_percentile

congress_ci_percentile  <- bootstrap_distribution_mean_age %>% 
  get_confidence_interval(type = "percentile", level = 0.95)

congress_ci_percentile

# A tibble: 1 x 2
  lower_ci upper_ci
     <dbl>    <dbl>
1     48.5     52.7

• Calculate the observed point estimate of th mean and assign it to obs_mean_age

• Display obs_mean_age

obs_mean_age  <-  congress_age_100  %>% 
  specify(response = age)  %>% 
  calculate(stat = "mean")  %>% 
  pull()

obs_mean_age

[1] 50.533

• Shade the confidence interval

• Add a line to the observed mean. obs_mean_age, to your visualization and color it “hot pink”

visualize(bootstrap_distribution_mean_age) +
  shade_confidence_interval(endpoints = congress_ci_percentile) + 
  geom_vline(xintercept = obs_mean_age, color = "hotpink", size = 1 )

• Calculate the population mean to see if it is in the 95% confidence interval

• Assign the output to pop_mean_age

• Display pop_mean_age

pop_mean_age  <- congress_age  %>% 
  summarize(pop_mean= mean(age))  %>% pull()

pop_mean_age

[1] 53.31373

• Add a line to the visualization aT the, populaTion mean, pop_mean_age to plot color in purple

visualize(bootstrap_distribution_mean_age) +
  shade_confidence_interval(endpoints = congress_ci_percentile) + 
   geom_vline(xintercept = obs_mean_age, color = "hotpink", size = 1) +
   geom_vline(xintercept = pop_mean_age, color = "purple", size = 3)

Save the previous plot to the preview.png and add to the yaml chunk at the top

ggsave(filename = "preview.png", 
       path = here::here("_posts", "2021-05-11-bootstrapping"))

• Is the population mean the 95% confidence interval constructed using the bootstrap distribution? yes

• Change set.seed(123) to set.seed(4346) rerun all the code

When you change the seed is the population mean in the 95% confidence interval constructed usin the boostrap? no

• If you construct 100 95% confidence intervals approxiamtely how many do we expect will contain the population? 95