---
title: "Lab 5: Sampling and Uncertainty"
author: "Your name"
format: html
editor: visual
editor_options:
  chunk_output_type: console
---

## Policy question

Suppose the **Ames Sustainability Office** wants to estimate average home size for a residential energy-efficiency program but cannot inspect every home.

Today we ask:

> **How much should the city trust an estimate that comes from only one sample?**

## Setup

```{r}
library(dplyr)
library(ggplot2)

load(url("https://raw.githubusercontent.com/GarciaRios/govt_3990/gh-pages/Labs/lab4/data/ames.RData"))

ames <- ames |>
  rename(
    area = Gr.Liv.Area,
    price = SalePrice
  )
```

## Exercise 1: Start with the population

```{r}
population_stats <- ames |>
  summarise(
    N = n(),
    mu_area = mean(area),
    sigma_area = sd(area)
  )

population_stats
```

```{r}
mu_area <- mean(ames$area)
sigma_area <- sd(ames$area)
```

```{r}
ggplot(ames, aes(x = area)) +
  geom_histogram(binwidth = 250) +
  labs(
    x = "Living area (square feet)",
    y = "Number of homes",
    title = "Population distribution of living area"
  ) +
  theme_minimal()
```

1. What is the population mean living area, $\mu$?

> 

2. What is the population standard deviation, $\sigma$?

> 

3. Describe the shape of the population distribution. Why might that shape matter when we begin taking samples?

> 

## Exercise 2: Take one sample

Choose an integer seed for your group. Different groups should choose different seeds.

```{r}
set.seed(527)  # Replace 527 with an integer chosen by your group.

sample_50 <- ames |>
  slice_sample(n = 50)
```

```{r}
sample_50 |>
  summarise(
    x_bar = mean(area),
    s = sd(area)
  )
```

4. What is your sample mean, $\bar{x}$?

> 

5. How far is your $\bar{x}$ from the true population mean $\mu$? Report the difference in square feet.

> 

6. Compare your estimate with another group's estimate. Are the two sample means identical? Why or why not?

> 

## Exercise 3: Sample again and again

```{r}
sample_means_20 <- tibble(
  sample_id = 1:20,
  x_bar = replicate(
    20,
    mean(sample(ames$area, size = 50, replace = TRUE))
  )
)

sample_means_20
```

```{r}
ggplot(sample_means_20, aes(x = sample_id, y = x_bar)) +
  geom_point(size = 2.5) +
  geom_hline(
    yintercept = mu_area,
    linetype = "dashed"
  ) +
  labs(
    x = "Random sample",
    y = "Sample mean living area",
    title = "Twenty samples produce twenty estimates"
  ) +
  theme_minimal()
```

7. What does each point in this graph represent?

> 

8. Are all 20 estimates on the same side of the true population mean?

> 

9. What does this graph tell you about treating a single sample mean as if it were exact?

> 

## Exercise 4: Build a sampling distribution

```{r}
sample_means_50 <- tibble(
  x_bar = replicate(
    2000,
    mean(sample(ames$area, size = 50, replace = TRUE))
  )
)
```

```{r}
ggplot(sample_means_50, aes(x = x_bar)) +
  geom_histogram(binwidth = 25) +
  geom_vline(
    xintercept = mu_area,
    linetype = "dashed"
  ) +
  labs(
    x = "Sample mean living area",
    y = "Number of samples",
    title = "Sampling distribution of the mean, n = 50"
  ) +
  theme_minimal()
```

```{r}
sample_means_50 |>
  summarise(
    mean_of_sample_means = mean(x_bar),
    simulated_se = sd(x_bar)
  )
```

10. What does one observation in `sample_means_50` represent?

> 

11. Describe the **shape, center, and spread** of the sampling distribution.

> 

12. Compare the mean of the 2,000 sample means with $\mu$. What do you notice?

> 

## Exercise 5: Standard deviation versus standard error

```{r}
theoretical_se_50 <- sigma_area / sqrt(50)
theoretical_se_50
```

```{r}
sample_means_50 |>
  summarise(
    simulated_se = sd(x_bar),
    theoretical_se = theoretical_se_50
  )
```

13. How close is the simulated standard error to $\sigma/\sqrt{50}$?

> 

14. In one sentence, explain the difference between **standard deviation** and **standard error**.

> 

15. Why is the standard error much smaller than the population standard deviation?

> 

## Exercise 6: What changes when the sample gets larger?

```{r}
sample_means_10 <- tibble(
  n = "n = 10",
  x_bar = replicate(
    2000,
    mean(sample(ames$area, size = 10, replace = TRUE))
  )
)

sample_means_100 <- tibble(
  n = "n = 100",
  x_bar = replicate(
    2000,
    mean(sample(ames$area, size = 100, replace = TRUE))
  )
)

sampling_sizes <- bind_rows(
  sample_means_10,
  sample_means_50 |> mutate(n = "n = 50"),
  sample_means_100
) |>
  mutate(
    n = factor(n, levels = c("n = 10", "n = 50", "n = 100"))
  )
```

```{r}
ggplot(sampling_sizes, aes(x = x_bar)) +
  geom_histogram(binwidth = 25) +
  facet_wrap(~ n, ncol = 1) +
  geom_vline(
    xintercept = mu_area,
    linetype = "dashed"
  ) +
  labs(
    x = "Sample mean living area",
    y = "Number of samples",
    title = "Larger samples produce more precise estimates"
  ) +
  theme_minimal()
```

```{r}
sampling_sizes |>
  group_by(n) |>
  summarise(
    center = mean(x_bar),
    simulated_se = sd(x_bar)
  )
```

```{r}
tibble(
  n = c(10, 50, 100),
  theoretical_se = sigma_area / sqrt(n)
)
```

16. What happens to the **center** of the sampling distribution as $n$ increases?

> 

17. What happens to the **spread** as $n$ increases?

> 

18. What happens to the **shape** as $n$ increases, especially given that the population itself is somewhat right-skewed?

> 

19. Which sample size gives the most precise estimate of the population mean? How can you tell from both the graph and the standard error?

> 

20. Suppose the city wants to cut the standard error in half. By what factor would it need to increase the sample size? Explain using the $\sqrt{n}$ in the denominator.

> 

## Exercise 7: Turn uncertainty into a policy statement

```{r}
prob_over_by_100 <- 1 - pnorm(
  mu_area + 100,
  mean = mu_area,
  sd = theoretical_se_50
)

prob_over_by_100
```

```{r}
sample_means_50 |>
  summarise(
    simulated_probability = mean(x_bar >= mu_area + 100)
  )
```

21. Interpret `prob_over_by_100` in words for the Sustainability Office.

> 

22. How close is the simulated probability to the probability predicted by the CLT?

> 

23. If overestimating average home size leads the city to over-budget staff time or retrofit resources, what does this probability tell the program manager?

> 

24. What could the city do to reduce this sampling uncertainty?

> 

# On Your Own: Estimate average home price

The Sustainability Office is now considering a housing program whose budget depends on **home prices** rather than home size.

## A. One sample

25. Take a random sample of **25 homes** and calculate the sample mean sale price. Treat this as your point estimate of the population mean.

```{r}

```

26. Calculate the actual population mean sale price so you can evaluate your estimate. How far away was your sample estimate?

```{r}

```

## B. Build a sampling distribution

27. Simulate **2,000 sample means** using samples of size 25. Store them in a tibble called `price_means_25`.

```{r}

```

28. Plot the sampling distribution. Describe its shape and center.

```{r}

```

> 

29. Calculate the simulated standard error and compare it with

$$
\frac{\sigma_{price}}{\sqrt{25}}.
$$

```{r}

```

> 

## C. Increase the sample size

30. Repeat the simulation using samples of **100 homes** and store the results as `price_means_100`.

```{r}

```

31. Compare the sampling distributions for $n=25$ and $n=100$. Which gives the city a more precise estimate, and why?

> 

## D. Policy recommendation

32. Write **2--3 sentences to the program manager** explaining why a larger sample can improve a policy estimate even though no single sample is guaranteed to equal the population truth.

>