Lab 5: Sampling distributions

install.packages("pscl", repos="https://cran.rstudio.com")

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library(openintro)
library(tidyverse)
install.packages("infer",repos = "http://cran.us.r-project.org")

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library(infer)

global_monitor <- tibble(
  scientist_work = c(rep("Benefits", 80000), rep("Doesn't benefit", 20000))
)

ggplot(global_monitor, aes(x = scientist_work)) +
  geom_bar() +
  labs(
    x = "", y = "",
    title = "Do you believe that the work scientists do benefit people like you?"
  ) +
  coord_flip() 

global_monitor %>%
  count(scientist_work) %>%
  mutate(p = n /sum(n))

## # A tibble: 2 x 3
##   scientist_work      n     p
## * <chr>           <int> <dbl>
## 1 Benefits        80000   0.8
## 2 Doesn't benefit 20000   0.2

samp1 <- global_monitor %>% sample_n(50)

Exercise 1

Describe the distribution of responses in this sample. How does it compare to the distribution of responses in the population. Hint: Although the sample_n function takes a random sample of observations (i.e. rows) from the dataset, you can still refer to the variables in the dataset with the same names. Code you presented earlier for visualizing and summarising the population data will still be useful for the sample, however be careful to not label your proportion p since you’re now calculating a sample statistic, not a population parameters. You can customize the label of the statistics to indicate that it comes from the sample.

samp1 %>%
  count(scientist_work) %>%
  mutate(p_hat = n /sum(n))

## # A tibble: 2 x 3
##   scientist_work      n p_hat
## * <chr>           <int> <dbl>
## 1 Benefits           36  0.72
## 2 Doesn't benefit    14  0.28

This sample is fairly similar to the population, 72% and 28% instead of 80% and 20%

Exercise 2

Would you expect the sample proportion to match the sample proportion of another student’s sample? Why, or why not? If the answer is no, would you expect the proportions to be somewhat different or very different? Ask a student team to confirm your answer.

It should be similar but not exactly the same. We are working with a fairly small sample size of 20 so there is going to be variation in the data.

Exercise 3

Take a second sample, also of size 50, and call it samp2. How does the sample proportion of samp2 compare with that of samp1? Suppose we took two more samples, one of size 100 and one of size 1000. Which would you think would provide a more accurate estimate of the population proportion?

samp2 <- global_monitor %>% sample_n(50)
samp3 <- global_monitor %>% sample_n(100)
samp4<- global_monitor %>% sample_n(1000)

samp2 %>%
  count(scientist_work) %>%
  mutate(p_hat = n /sum(n))

## # A tibble: 2 x 3
##   scientist_work      n p_hat
## * <chr>           <int> <dbl>
## 1 Benefits           43  0.86
## 2 Doesn't benefit     7  0.14

samp3 %>%
  count(scientist_work) %>%
  mutate(p_hat = n /sum(n))

## # A tibble: 2 x 3
##   scientist_work      n p_hat
## * <chr>           <int> <dbl>
## 1 Benefits           83  0.83
## 2 Doesn't benefit    17  0.17

samp4 %>%
  count(scientist_work) %>%
  mutate(p_hat = n /sum(n))

## # A tibble: 2 x 3
##   scientist_work      n p_hat
## * <chr>           <int> <dbl>
## 1 Benefits          804 0.804
## 2 Doesn't benefit   196 0.196

The data is pretty similar, it gets closer to an even 80%/20% as we get a larger sample size. The sample size of 1000 is the most accurate.

sample_props50 <- global_monitor %>%
                    rep_sample_n(size = 50, reps = 15000, replace = TRUE) %>%
                    count(scientist_work) %>%
                    mutate(p_hat = n /sum(n)) %>%
                    filter(scientist_work == "Doesn't benefit")

Exercise 4

How many elements are there in sample_props50? Describe the sampling distribution, and be sure to specifically note its center. Make sure to include a plot of the distribution in your answer.

ggplot(data = sample_props50, aes(x = p_hat)) +
  geom_histogram(binwidth = 0.02) +
  labs(
    x = "p_hat (Doesn't benefit)",
    title = "Sampling distribution of p_hat",
    subtitle = "Sample size = 50, Number of samples = 15000"
  )

There are 4 columns (0.1,0.2,0.3,0.4), and about 15000 observations.

global_monitor %>%
  sample_n(size = 50, replace = TRUE) %>%
  count(scientist_work) %>%
  mutate(p_hat = n /sum(n)) %>%
  filter(scientist_work == "Doesn't benefit")

## # A tibble: 1 x 3
##   scientist_work      n p_hat
##   <chr>           <int> <dbl>
## 1 Doesn't benefit    13  0.26

Exercise 5

To make sure you understand how sampling distributions are built, and exactly what the rep_sample_n function does, try modifying the code to create a sampling distribution of 25 sample proportions from samples of size 10, and put them in a data frame named sample_props_small. Print the output. How many observations are there in this object called sample_props_small? What does each observation represent?

sample_props_small <- global_monitor %>%
  sample_n(size = 10, reps = 25, replace = TRUE) %>%
  count(scientist_work) %>%
  mutate(p_hat = n /sum(n)) %>%
  filter(scientist_work == "Doesn't benefit")

sample_props_small

## # A tibble: 1 x 3
##   scientist_work      n p_hat
##   <chr>           <int> <dbl>
## 1 Doesn't benefit     1   0.1

There are 25 observations, each representing a sample size of 10.

ggplot(data = sample_props50, aes(x = p_hat)) +
  geom_histogram(binwidth = 0.02)

Exercise 6

Use the app below to create sampling distributions of proportions of Doesn’t benefit from samples of size 10, 50, and 100. Use 5,000 simulations. What does each observation in the sampling distribution represent? How does the mean, standar error, and shape of the sampling distribution change as the sample size increases? How (if at all) do these values change if you increase the number of simulations? (You do not need to include plots in your answer.)

Each observation is a random sample of the size selected, in this case it is 10,50,and 100. The more observations added the more the graph looked like a normal bell distribution, at 10 it is very difficult to make out any trend at all. The mean grew closer to 0.2 with an increasing sample size, and the standard deviation got closer to 0.04. The more simulations added just makes the graph look more and more normal, with the mean and standard deviation also growing closer.

Exercise 7

Take a sample of size 15 from the population and calculate the proportion of people in this sample who think the work scientists do enchances their lives. Using this sample, what is your best point estimate of the population proportion of people who think the work scientists do enchances their lives?

set.seed(1)

samp5 <- global_monitor %>%
  sample_n(15)

samp5 %>%
  count(scientist_work) %>%
  mutate(p_hat5 = n /sum(n))

## # A tibble: 2 x 3
##   scientist_work      n p_hat5
## * <chr>           <int>  <dbl>
## 1 Benefits           14 0.933 
## 2 Doesn't benefit     1 0.0667

93.3% is the best point estimate.

Exercise 8

Since you have access to the population, simulate the sampling distribution of proportion of those who think the work scientists do enchances their lives for samples of size 15 by taking 2000 samples from the population of size 15 and computing 2000 sample proportions. Store these proportions in as sample_props15. Plot the data, then describe the shape of this sampling distribution. Based on this sampling distribution, what would you guess the true proportion of those who think the work scientists do enchances their lives to be? Finally, calculate and report the population proportion.

sample_props15 <- global_monitor %>%
  rep_sample_n(size = 15, reps = 2000, replace = TRUE) %>%
  count(scientist_work) %>%
  mutate(p_hat = n /sum(n)) %>%
  filter(scientist_work == "Benefits")
glimpse(sample_props15)

## Rows: 2,000
## Columns: 4
## Groups: replicate [2,000]
## $ replicate      <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16,…
## $ scientist_work <chr> "Benefits", "Benefits", "Benefits", "Benefits", "Benef…
## $ n              <int> 7, 14, 12, 13, 11, 12, 13, 12, 14, 11, 11, 13, 15, 11,…
## $ p_hat          <dbl> 0.4666667, 0.9333333, 0.8000000, 0.8666667, 0.7333333,…

ggplot(data = sample_props15, aes(x = p_hat)) +
  geom_histogram(binwidth = 0.075) +
  labs(x = "p_hat (Benefits)",
    title = "Sampling distribution of p_hat",
    subtitle = "Sample size = 15, Number of samples = 2000")

The histogram is very left skewed, maybe 75% of scientists think their work enhances their lives. Looks like the actual percent is 80%.

Exercise 9

Change your sample size from 15 to 150, then compute the sampling distribution using the same method as above, and store these proportions in a new object called sample_props150. Describe the shape of this sampling distribution and compare it to the sampling distribution for a sample size of 15. Based on this sampling distribution, what would you guess to be the true proportion of those who think the work scientists do enchances their lives?

sample_props150 <- global_monitor %>%
  rep_sample_n(size = 150, reps = 2000, replace = TRUE) %>%
  count(scientist_work) %>%
  mutate(p_hat = n /sum(n)) %>%
  filter(scientist_work == "Benefits")
glimpse(sample_props150)

## Rows: 2,000
## Columns: 4
## Groups: replicate [2,000]
## $ replicate      <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16,…
## $ scientist_work <chr> "Benefits", "Benefits", "Benefits", "Benefits", "Benef…
## $ n              <int> 119, 124, 122, 120, 122, 121, 117, 121, 116, 114, 117,…
## $ p_hat          <dbl> 0.7933333, 0.8266667, 0.8133333, 0.8000000, 0.8133333,…

ggplot(data = sample_props150, aes(x = p_hat)) +
  geom_histogram(binwidth = 0.02) +
  labs(x = "p_hat (Benefits)",
    title = "Sampling distribution of p_hat",
    subtitle = "Sample size = 150, Number of samples = 2000")

This graph is much more bell shaped and uniform, although still has a slight left skew. Looks like about 80% of scientists enhance lives.

Exercise 10

Of the sampling distributions from 2 and 3, which has a smaller spread? If you’re concerned with making estimates that are more often close to the true value, would you prefer a sampling distribution with a large or small spread?

Since the smaller spread has the larger sample size, I would prefer this one to find the sampling distribution.