-Load the R package we will use.

install.packages('fivethirtyeightdata', repos =
'https://fivethirtyeightdata.github.io/drat/', type = 'source')

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

Replace all the instances of ???. These are answers on your moodle quiz. Run all the individual code chunks to make sure the answers in this file correspond with your quiz answers After you check all your code chunks run then you can knit it. It won’t knit until the ??? are replaced Save a plot to be your preview plot Look at the variable definitions in congress_age What is the average age of members that have served in congress? 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

# A tibble: 18,635 x 13
   congress chamber bioguide firstname middlename lastname suffix
      <int> <chr>   <chr>    <chr>     <chr>      <chr>    <chr> 
 1       80 house   M000112  Joseph    Jefferson  Mansfie… <NA>  
 2       80 house   D000448  Robert    Lee        Doughton <NA>  
 3       80 house   S000001  Adolph    Joachim    Sabath   <NA>  
 4       80 house   E000023  Charles   Aubrey     Eaton    <NA>  
 5       80 house   L000296  William   <NA>       Lewis    <NA>  
 6       80 house   G000017  James     A.         Gallagh… <NA>  
 7       80 house   W000265  Richard   Joseph     Welch    <NA>  
 8       80 house   B000565  Sol       <NA>       Bloom    <NA>  
 9       80 house   H000943  Merlin    <NA>       Hull     <NA>  
10       80 house   G000169  Charles   Laceille   Gifford  <NA>  
# … with 18,625 more rows, and 6 more variables: birthday <date>,
#   state <chr>, party <chr>, incumbent <lgl>, termstart <date>,
#   age <dbl>

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

congress_age is the population and congress_age_100 is the sample 18635 is number of observations in the the population and 100 is the number of observations in your sample

Construct the confidence interval

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

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 10,000 rows 3. calculate the mean for each replicate -Assign to bootstrap_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_distribution_mean_age has 1000 means 4. visualize the bootstrap distribution

visualize(bootstrap_distribution_mean_age)

ggsave(filename = "preview.png", 
       path = here::here("_posts", "2021-05-04-bootstrapping-and-confidence-intervals"))

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 = .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 the 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 at the observed mean, obs_mean_age, to your visualization and color it “hotpink”

visualize(bootstrap_distribution_mean_age) +
  shade_confidence_interval(endpoints = congress_ci_percentile) + 
  geom_vline(xintercept = 53.597, 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 the plot color it “purple”

visualize(bootstrap_distribution_mean_age) +
  shade_confidence_interval(endpoints = congress_ci_percentile) + 
   geom_vline(xintercept = 53.313, color = "hotpink", size = 1) +
   geom_vline(xintercept = 53.597 , color = "purple", size = 3)

Is population mean in 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 using the bootstrap distribution? NO -If you construct 100 95% confidence intervals approximately how many do you expect will contain the population mean? 0

Bootstrapping and Confidence Intervals

Construct the confidence interval

Calculate the 95% confidence interval using the percentile method

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