---
title: "Lab 6: Confidence Intervals"
author: "Your name"
format: html
editor: visual
editor_options:
  chunk_output_type: console
---

## Policy question

The **Ames Sustainability Office** needs an estimate of average home size for a residential energy-efficiency program.

Today we ask:

> **How precise is the estimate, and is it precise enough to make a policy decision?**

## 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
  )

mu_area <- mean(ames$area)
```

## Exercise 1: One policy sample

Choose an integer seed for your group.

```{r}
set.seed(614)  # Replace 614 with your group's seed.

n <- 60

sample_60 <- ames |>
  slice_sample(n = n)

sample_stats <- sample_60 |>
  summarise(
    x_bar = mean(area),
    s = sd(area)
  )

sample_stats
```

1. What is your point estimate of the population mean living area?

> 

2. What does the sample standard deviation tell you about the homes in your sample?

> 

3. Compare your sample mean with another group. Why should the city expect different analysts to get slightly different estimates even when both use good random samples?

> 

## Exercise 2: Standard error and margin of error

```{r}
x_bar <- mean(sample_60$area)
s <- sd(sample_60$area)

se <- s / sqrt(n)
se

t_star <- qt(
  0.975,
  df = n - 1
)

t_star

me <- t_star * se
me
```

4. What is the standard error of your sample mean? Interpret it in context.

> 

5. What is the 95% critical value $t^*$ for your sample?

> 

6. What is your margin of error? What units is it measured in?

> 

## Exercise 3: Construct the 95% confidence interval

```{r}
lower_95 <- x_bar - me
upper_95 <- x_bar + me

c(
  lower = lower_95,
  estimate = x_bar,
  upper = upper_95
)
```

7. Report your 95% confidence interval.

> 

8. Write a one-sentence interpretation for the Ames Sustainability Office. Your sentence should refer to the **population mean living area**, not individual homes.

> 

9. Does your interval contain the actual Ames population mean, $\mu$? Remember that in a real application, the city would not know the true value.

> 

## Exercise 4: What does 95% confidence mean?

Choose the best interpretation:

A. There is a 95% probability that $\mu$ is inside the interval we just calculated.

B. 95% of individual Ames homes fall inside the interval.

C. If we repeatedly took random samples and built intervals the same way, about 95% of those intervals would contain the true population mean.

D. The sample mean has a 95% chance of being correct.

10. Which answer did you choose? Explain why the other interpretations are wrong.

> 

## Exercise 5: Compare intervals across groups

Write your group's lower bound, sample mean, and upper bound where everyone can see them.

11. Are the intervals identical across groups?

> 

12. Do they all have exactly the same width? Why might widths differ slightly even though every group used $n=60$?

> 

13. Did every group capture the true population mean? If one did not, does that mean the group made a mistake?

> 

## Exercise 6: Simulate 100 confidence intervals

```{r}
draw_ci <- function() {
  x <- sample(
    ames$area,
    size = n,
    replace = FALSE
  )

  x_bar_i <- mean(x)
  s_i <- sd(x)
  se_i <- s_i / sqrt(n)
  t_star_i <- qt(0.975, df = n - 1)

  c(
    x_bar = x_bar_i,
    lower = x_bar_i - t_star_i * se_i,
    upper = x_bar_i + t_star_i * se_i
  )
}

ci_100 <- replicate(
  100,
  draw_ci()
) |>
  t() |>
  as.data.frame() |>
  tibble::as_tibble() |>
  mutate(
    interval_id = row_number(),
    captures_mu = lower <= mu_area & upper >= mu_area
  )
```

```{r}
ci_100 |>
  summarise(
    capture_rate = mean(captures_mu)
  )
```

14. What proportion of your 100 intervals captured $\mu$?

> 

15. Is it exactly 95%? Should it have to be exactly 95% in a set of only 100 intervals?

> 

16. If we simulated 100,000 intervals instead, what capture rate would you expect to see approximately?

> 

## Exercise 7: See confidence in action

```{r}
ci_100 |>
  filter(interval_id <= 30) |>
  ggplot(
    aes(
      y = interval_id,
      x = lower,
      xend = upper,
      yend = interval_id,
      color = captures_mu
    )
  ) +
  geom_segment(linewidth = 1) +
  geom_point(
    aes(x = x_bar),
    size = 2
  ) +
  geom_vline(
    xintercept = mu_area,
    linetype = "dashed"
  ) +
  labs(
    x = "Living area (square feet)",
    y = "Confidence interval",
    color = "Captures true mean?",
    title = "Thirty 95% confidence intervals"
  ) +
  theme_minimal()
```

17. What distinguishes the intervals that fail to capture $\mu$?

> 

18. Does a failed interval look obviously “bad” before we reveal the population mean?

> 

19. Why is this graph a better explanation of 95% confidence than saying “there is a 95% probability that $\mu$ is in this interval”?

> 

## Exercise 8: Confidence versus precision

```{r}
ci_levels <- tibble(
  confidence = c(0.90, 0.95, 0.99),
  t_star = c(
    qt(0.95, df = n - 1),
    qt(0.975, df = n - 1),
    qt(0.995, df = n - 1)
  )
) |>
  mutate(
    margin_of_error = t_star * se,
    lower = x_bar - margin_of_error,
    upper = x_bar + margin_of_error,
    width = upper - lower
  )

ci_levels
```

20. What happens to interval width as confidence increases?

> 

21. Why does greater confidence require a wider interval when the data and sample size stay fixed?

> 

22. A program manager says, “I want 99% confidence **and** a narrower interval, but I do not want to collect more data.” Explain the tradeoff.

> 

## Exercise 9: Is the estimate precise enough?

The Sustainability Office says it can plan confidently only if the margin of error for average living area is **no more than 75 square feet**.

23. Does your current sample of 60 homes meet the city's precision requirement?

> 

24. If it does not, what can the city change to make the interval narrower without lowering its confidence level?

> 

25. Why would “just report the sample mean and ignore the interval” be a poor response to the city's concern?

> 

## Exercise 10: Plan a larger sample

```{r}
target_me <- 75

n_needed <- ceiling(
  (1.96 * s / target_me)^2
)

n_needed
```

26. How many homes should the city plan to sample to target a margin of error of about 75 square feet?

> 

27. Why do we round the required sample size **up** rather than down?

> 

28. If the city wanted to cut the margin of error roughly in half, about how much larger would the sample need to be? Connect your answer to the square-root relationship from last week.

> 

# On Your Own: Estimate mean home price

Use a new random sample of **80 homes**.

## A. Build a 95% confidence interval

29. Draw a random sample of 80 homes and calculate $\bar{x}$, $s$, $SE$, $t^*$, margin of error, and the 95% confidence interval for mean sale price.

```{r}

```

30. Write the interval in dollars and interpret it in one sentence for the housing program manager.

> 

## B. Evaluate the interval

31. Calculate the true population mean sale price and determine whether your interval captured it.

```{r}

```

32. Construct 90% and 99% intervals using the **same sample**. Compare their widths with your 95% interval.

```{r}

```

> 

## C. Policy recommendation

The program manager wants a margin of error no larger than **$15,000** at 95% confidence.

33. Does your current sample of 80 homes meet that precision target?

> 

34. Using your sample standard deviation as the planning value, estimate how many homes the city would need to sample to target a $15,000 margin of error.

```{r}

```

35. Write **2--3 sentences to the program manager** explaining whether the current sample is precise enough and, if not, what you recommend.

>