--- title: "Lab 4: Normal and Binomial Models with Ames Housing" author: "Your name" format: html editor: visual editor_options: chunk_output_type: console --- ## Policy scenario Suppose the **Ames Sustainability Office** is planning a hypothetical residential energy-audit program. Homes with at least **2,000 square feet** of living area require a longer inspection slot. A field team can comfortably handle at most **3 extended audits in a day** before the office needs additional staff time or overtime. The analysis will help answer how common extended-audit homes are, whether a normal model describes that share well, and how often a day's workload could exceed planned capacity. ## 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) ``` ## Exercise 1: Start with the actual distribution ```{r} ggplot(ames, aes(x = area)) + geom_histogram(binwidth = 250) + labs( x = "Living area (square feet)", y = "Number of home sales", title = "Living area of homes sold in Ames" ) + theme_minimal() ``` ```{r} area_stats <- ames |> summarise( mean_area = mean(area), median_area = median(area), sd_area = sd(area), q1 = quantile(area, 0.25), q3 = quantile(area, 0.75), min_area = min(area), max_area = max(area) ) area_stats ``` 1. Describe the shape of the distribution of living area. > 2. What would you call a "typical" Ames home based on this distribution? Explain which measure of center you chose. > 3. Is the mean larger than the median? What does that tell you about the shape? > 4. Based on the histogram, does a normal model look perfect, reasonable, or poor? Explain. > ## Exercise 2: Give a z-score substantive meaning ```{r} mu_area <- mean(ames$area) sigma_area <- sd(ames$area) z_2000 <- (2000 - mu_area) / sigma_area z_2000 ``` 5. How many standard deviations above or below the Ames mean is a 2,000-square-foot home? > 6. Write one sentence interpreting the z-score in the context of Ames housing. > 7. Based only on the z-score, would you describe 2,000 square feet as extremely unusual? Why or why not? > ## Exercise 3: Ask the same question two ways Calculate the observed proportion of Ames homes with at least 2,000 square feet of living area. ```{r} p_large_empirical <- ames |> summarise( p_large = mean(area >= 2000) ) |> pull(p_large) p_large_empirical ``` Now calculate the probability predicted by a fitted normal model. ```{r} p_large_normal <- 1 - pnorm( 2000, mean = mu_area, sd = sigma_area ) p_large_normal ``` 8. What proportion of Ames homes are actually at least 2,000 square feet? > 9. What proportion does the fitted normal model predict? > 10. Are the two probabilities identical? Which is larger? > 11. Look back at the histogram. What feature of the actual distribution might help explain the difference? > ### Policy interpretation Suppose the Sustainability Office estimated the share of extended-audit homes using the normal model rather than the observed Ames data. Would the normal model lead the office to plan for too many or too few extended audits? How could that affect staffing or scheduling? > ## Exercise 4: Visualize the model against reality ```{r} ggplot(ames, aes(x = area)) + geom_histogram( aes(y = after_stat(density)), binwidth = 250 ) + stat_function( fun = dnorm, args = list(mean = mu_area, sd = sigma_area), linewidth = 1 ) + geom_vline( xintercept = 2000, linetype = "dashed" ) + labs( x = "Living area (square feet)", y = "Density", title = "Ames living area and a fitted normal model" ) + theme_minimal() ``` 12. Where does the normal model fit the observed distribution reasonably well? > 13. Where does it fit less well? > 14. Why is it useful to look at the graph before trusting a probability calculated from a model? > ## Exercise 5: Turn an observed proportion into a binomial model Suppose a success is a home with at least 2,000 square feet of living area. For a group of 10 home sales, let $$ Y = \text{number of the 10 homes with area at least 2,000 square feet}. $$ 15. What is $n$ in this model? > 16. What is $p$? Report the value calculated from the data. > 17. What exactly counts as a success? > 18. Which BINS condition would you worry about most if these 10 homes all came from the same neighborhood or development? Explain. > ## Exercise 6: Make the binomial probabilities mean something Calculate $P(Y=4)$ using the binomial formula. ```{r} choose(10, 4) * p_large_empirical^4 * (1 - p_large_empirical)^6 ``` Verify with `dbinom()`. ```{r} dbinom( 4, size = 10, prob = p_large_empirical ) ``` Calculate $P(Y \geq 4)$. ```{r} 1 - pbinom( 3, size = 10, prob = p_large_empirical ) ``` 19. What is $P(Y=4)$? Interpret it in words. > 20. What is $P(Y\geq4)$? Interpret it in words. > 21. Why is $P(Y\geq4)$ larger than $P(Y=4)$? > 22. Explain why the value of $p$ here is more meaningful than simply being handed an arbitrary number such as $p=0.30$. > ### Policy interpretation Recall that the field team can handle at most **3 extended audits per day** without additional staffing. What does $P(Y\geq4)$ tell the program manager about the risk that a 10-home day exceeds planned capacity? > ## Exercise 7: See the entire binomial distribution ```{r} binom_ames <- tibble( large_homes = 0:10 ) |> mutate( probability = dbinom( large_homes, size = 10, prob = p_large_empirical ) ) binom_ames ``` ```{r} ggplot( binom_ames, aes(x = large_homes, y = probability) ) + geom_col() + labs( x = "Large homes among 10 sales", y = "Probability", title = "Number of Ames homes at least 2,000 square feet" ) + theme_minimal() ``` ```{r} n <- 10 p <- p_large_empirical expected_large <- n * p sd_large <- sqrt(n * p * (1 - p)) expected_large sd_large ``` 23. Which number of large homes is most probable according to the graph? > 24. What is the expected number of large homes among 10 sales? > 25. Explain what the expected value means. Does it mean every group of 10 will contain exactly that many large homes? > 26. What does the standard deviation tell us about how much the count tends to vary across repeated groups of 10? > # On Your Own: Evaluate an alternative policy rule The Sustainability Office is considering changing the threshold for an extended audit. Choose one proposed rule: - **Broader rule:** extended audit for homes at or above **1,800 square feet** - **Narrower rule:** extended audit for homes at or above **2,500 square feet** Treat your choice as a policy alternative and evaluate what it would imply for program operations. ## A. Define the policy 27. State the threshold you selected and briefly explain whether it represents the broader or narrower rule. > 28. Calculate the empirical proportion of Ames homes meeting your definition. ```{r} ``` ## B. Compare reality with the normal model 29. Calculate the z-score for your threshold. ```{r} ``` 30. Use the fitted normal model to calculate the probability that a home is at least as large as your threshold. ```{r} ``` 31. Compare the empirical probability with the normal-model probability in 2--3 sentences. Does the normal model overestimate or underestimate the observed upper-tail probability? > ## C. Build a binomial model For a group of **12 home sales**, let $W$ be the number meeting your new definition of a large home. 32. State the binomial model using your estimated value of $p$. > 33. Calculate $P(W=4)$. ```{r} ``` 34. Calculate $P(W\geq4)$. ```{r} ``` 35. Calculate the expected value and standard deviation of $W$. ```{r} ``` 36. In 2--3 sentences, explain what your binomial results mean for staffing a day with 12 scheduled homes. > ## D. Policy memo sentence 37. Write **one sentence to the program manager** answering this question: Under your proposed threshold, should the office view 4 or more extended audits in a 12-home day as a routine planning scenario or a relatively uncommon one? Support your answer with the probability you calculated. >