##Libraries

library(tidyverse)
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## âś” lubridate 1.9.2     âś” tidyr     1.3.0
## âś” purrr     1.0.1     
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## â„ą Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errors
library(openintro)
## Loading required package: airports
## Loading required package: cherryblossom
## Loading required package: usdata
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))
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 summarizing 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.

The

ggplot(samp1, aes(x = scientist_work)) +
  geom_bar() +
  labs(
    x = "", y = "",
    title = "(Sample 1) Do you believe that the work scientists do benefit people like you?"
  ) +
  coord_flip() 

samp1%>%
  count(scientist_work) %>%
  mutate(samp1_p = n /sum(n))

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.

No I don’t think that another students sample would match mine. I would expect it to be slightly different since the sample is chosen at random

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

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?

The samp1 is .02 more in the benefits and 0.02 less in the doesn’t benefit category than samp2. I think that the larger the sample size the more accurate we can be to estimate the population proportion. So I think that the sample size of 1000 would provide the most accurate estimation.

samp2 <- global_monitor %>%
  sample_n(50)

samp2 %>%
  count(scientist_work) %>%
  mutate(samp2_p = n /sum(n))
samp3 <- global_monitor %>%
  sample_n(100)

samp3 %>%
  count(scientist_work) %>%
  mutate(samp2_p = n /sum(n))
samp4 <- global_monitor %>%
  sample_n(1000)

samp4 %>%
  count(scientist_work) %>%
  mutate(samp2_p = n /sum(n))
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")
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"
  )

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.

There are about 2000 elements in the graph. The distribution seems to follow a normal distribution and the center of this distribution seems to be closer to 0.2 which is the proportion that represents the gloabal proportion.

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?

There are 24 observations. Each observation represents a simulation

sample_props_small <- global_monitor %>%
                    rep_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
ggplot(data = sample_props_small, 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 = 10"
  )

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, standard 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.)

The standard deviation seems to decrease as the sample size increases. The mean also decreases as it seems to go closer and closer to the true mean The shape of the distrubution will most likely get closer and closer to the the actual distribution.

sample_props_10 <- global_monitor %>%
                    rep_sample_n(size = 10, reps = 5000, replace = TRUE) %>%
                    count(scientist_work) %>%
                    mutate(p_hat = n /sum(n)) %>%
                    filter(scientist_work == "Doesn't benefit")

sample_props_50 <- global_monitor %>%
                    rep_sample_n(size = 50, reps = 5000, replace = TRUE) %>%
                    count(scientist_work) %>%
                    mutate(p_hat = n /sum(n)) %>%
                    filter(scientist_work == "Doesn't benefit")

sample_props_100 <- global_monitor %>%
                    rep_sample_n(size = 100, reps = 5000, replace = TRUE) %>%
                    count(scientist_work) %>%
                    mutate(p_hat = n /sum(n)) %>%
                    filter(scientist_work == "Doesn't benefit")
mean(sample_props_10$p_hat)
## [1] 0.2233895
mean(sample_props_50$p_hat)
## [1] 0.198736
mean(sample_props_100$p_hat)
## [1] 0.200072
sd(sample_props_10$p_hat)
## [1] 0.1102339
sd(sample_props_50$p_hat)
## [1] 0.05717216
sd(sample_props_100$p_hat)
## [1] 0.04035989

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 enhances 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?

I think that 80% of people think that it does benefit them.

sample_props_benefits <- global_monitor %>%
                    rep_sample_n(size = 15, reps = 25, replace = TRUE) %>%
                    count(scientist_work) %>%
                    mutate(p_hat = n /sum(n)) %>%
                    filter(scientist_work == "Benefits")

sample_props_benefits
ggplot(data = sample_props_benefits, aes(x = p_hat)) +
  geom_histogram(binwidth = 0.02) +
  labs(
    x = "p_hat (Benefits)",
    title = "Sampling distribution of p_hat",
    subtitle = "Sample size = 10"
  )

mean(sample_props_benefits$p_hat)
## [1] 0.816

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")

sample_props15
ggplot(data = sample_props15, aes(x = p_hat)) +
  geom_histogram() +
  labs(
    x = "p_hat (Benefits)",
    title = "Sampling distribution of p_hat",
    subtitle = "Sample size = 15, reps = 2000"
  )
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

mean(sample_props15$p_hat)
## [1] 0.8008333

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?

It looks like the proportion is still around 80%

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")

sample_props150
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"
  )

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?

The spread is much more uniform in the sample with the size of 150