Lab 4: Normal and Binomial Models with Ames Housing

From observed data to probability models and policy planning

Download the student practice template

The question

How can a probability model help us say something meaningful about real housing data?

Today we will use residential home sales from Ames, Iowa. Rather than treating the normal and binomial distributions as isolated formulas, we will use them to answer substantive questions about the Ames housing market.

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.

Before implementing the program, analysts want to know:

  • How common are homes that would require an extended audit?
  • If the office used a normal model instead of the complete housing data, how accurate would its estimate be?
  • If 10 homes are scheduled in a day, how likely is it that 4 or more require extended audits?
  • How should those probabilities inform staffing and scheduling?

This is a simplified policy scenario, but it gives the quantities we calculate a concrete decision context.

Our workflow is:

  1. See the actual data.
  2. Calculate a useful quantity.
  3. Build a probability model.
  4. Compare the model with reality.
  5. Interpret what the number means.

The goal is not just to get a probability.

See it. Model it. Calculate it. Interpret it.

Setup

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)

The data contain residential home sales in Ames, Iowa, from 2006 through 2010. Each row is one home sale.

The original dataset stores above-ground living area as Gr.Liv.Area. For readability, we rename it area and use that shorter name throughout the lab.

glimpse(ames)
nrow(ames)

1. Start with the actual distribution

Before using a probability model, look at the data.

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()

Now calculate several summaries:

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

Your turn

  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.

A model does not have to reproduce every feature of the data to be useful. But we should understand where it fits well and where it does not.

2. Give a z-score substantive meaning

Suppose we want to know how unusual a 2,000-square-foot home is relative to homes sold in Ames.

Store the mean and standard deviation:

mu_area <- mean(ames$area)
sigma_area <- sd(ames$area)

Now calculate the z-score:

z_2000 <- (2000 - mu_area) / sigma_area
z_2000

Recall:

\[ z = \frac{x-\mu}{\sigma}. \]

Your turn

  1. How many standard deviations above or below the Ames mean is a 2,000-square-foot home?
  2. Write one sentence interpreting the z-score in the context of Ames housing.
  3. Based only on the z-score, would you describe 2,000 square feet as extremely unusual? Why or why not?

The z-score changes the units from square feet to standard deviations from the mean.

3. Ask the same question two ways

Now ask:

What proportion of Ames homes have at least 2,000 square feet of living area?

Because we have the full Ames dataset, we can answer this directly from the observed data.

Empirical probability

p_large_empirical <- ames |>
  summarise(
    p_large = mean(area >= 2000)
  ) |>
  pull(p_large)

p_large_empirical

This is the observed proportion of Ames home sales meeting our definition of a large home.

Normal-model probability

Now suppose we model living area with a normal distribution having the same mean and standard deviation as the Ames data:

\[ X \sim N(\mu_{area},\sigma_{area}). \]

The model-based probability of at least 2,000 square feet is:

p_large_normal <- 1 - pnorm(
  2000,
  mean = mu_area,
  sd = sigma_area
)

p_large_normal

Your turn

  1. What proportion of Ames homes are actually at least 2,000 square feet?
  2. What proportion does the fitted normal model predict?
  3. Are the two probabilities identical? Which is larger?
  4. 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?

This is an important distinction:

The empirical probability describes what we observed. The model probability describes what the fitted probability model predicts.

4. Visualize the model against reality

Put the observed distribution and the fitted normal curve on the same graph.

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()

Your turn

  1. Where does the normal model fit the observed distribution reasonably well?
  2. Where does it fit less well?
  3. Why is it useful to look at the graph before trusting a probability calculated from a model?

5. Turn an observed proportion into a binomial model

Now define a success as:

\[ \text{Success} = \text{a home has at least 2,000 square feet of living area}. \]

We already estimated the probability of success from the Ames population:

p_large_empirical

Suppose we consider a group of 10 home sales and define

\[ Y = \text{number of the 10 homes with area at least 2,000 square feet}. \]

If the BINS conditions are reasonable, then

\[ Y \sim \operatorname{Binom}(10,p), \]

where \(p\) is the observed Ames proportion.

Check BINS

  • Binary: each home is either at least 2,000 square feet or it is not.
  • Independent: knowing the size category of one selected home should not determine another.
  • Number fixed: we consider 10 home sales.
  • Same probability: we use the same Ames probability for each trial.

Your turn

  1. What is \(n\) in this model?
  2. What is \(p\)? Report the value calculated from the data.
  3. What exactly counts as a success?
  4. Which BINS condition would you worry about most if these 10 homes all came from the same neighborhood or development? Explain.

6. Make the binomial probabilities mean something

What is the probability that exactly 4 of 10 selected Ames homes are at least 2,000 square feet?

First calculate it using the binomial formula:

\[ P(Y=4) = {10\choose4} p^4(1-p)^6. \]

choose(10, 4) *
  p_large_empirical^4 *
  (1 - p_large_empirical)^6

Now verify with R:

dbinom(
  4,
  size = 10,
  prob = p_large_empirical
)

Now calculate the probability that at least 4 of 10 are large:

1 - pbinom(
  3,
  size = 10,
  prob = p_large_empirical
)

Your turn

  1. What is \(P(Y=4)\)? Interpret it in words.
  2. What is \(P(Y\geq4)\)? Interpret it in words.
  3. Why is \(P(Y\geq4)\) larger than \(P(Y=4)\)?
  4. 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?

7. See the entire binomial distribution

Calculate the probability for every possible number of large homes in a group of 10.

binom_ames <- tibble(
  large_homes = 0:10
) |>
  mutate(
    probability = dbinom(
      large_homes,
      size = 10,
      prob = p_large_empirical
    )
  )

binom_ames

Visualize it:

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()

Calculate the expected value and standard deviation:

n <- 10
p <- p_large_empirical

expected_large <- n * p
sd_large <- sqrt(n * p * (1 - p))

expected_large
sd_large

Your turn

  1. Which number of large homes is most probable according to the graph?
  2. What is the expected number of large homes among 10 sales?
  3. Explain what the expected value means. Does it mean every group of 10 will contain exactly that many large homes?
  4. 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

  1. State the threshold you selected and briefly explain whether it represents the broader or narrower rule.

  2. Calculate the empirical proportion of Ames homes meeting your definition.

B. Compare reality with the normal model

  1. Calculate the z-score for your threshold.
  1. Use the fitted normal model to calculate the probability that a home is at least as large as your threshold.
  1. 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.

  1. State the binomial model using your estimated value of \(p\).

  2. Calculate \(P(W=4)\).

  1. Calculate \(P(W\geq4)\).
  1. Calculate the expected value and standard deviation of \(W\).
  1. In 2–3 sentences, explain what your binomial results mean for staffing a day with 12 scheduled homes.

D. Policy memo sentence

  1. 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.

The strongest answers today connect the number back to the data:

What does this probability tell us about actual homes?