Lesson 30 - Course Review

Lesson Administration

Calendar

Day 1

Day 2

Army Math Basketball

Math vs USMAPS

4-1

Math vs USMAPS

5-1

Math vs Garrison

6-1

Math vs Garrison

7-1

Army Math Volleyball

Math vs EECS

3-0

Math vs EECS

4-0

Reese

TEE Schedule

Date Start End
Wed, 17 Dec 2025 0730 1100
Wed, 17 Dec 2025 1300 1630
Thu, 18 Dec 2025 0730 1100

TEE Rooms

Wed, 17 Dec (1300-1630): TH324

  1. Afari-Aikins, John
  2. Coldren, Nathan
  3. Conti, Annabella
  4. Freeman, Brandon
  5. George, Joshua
  6. Goetz, Charles

Wed, 17 Dec (1300-1630): TH322

  1. Lavery, Harrison
  2. McDaniel, Jack
  3. McDonnell, Hunter
  4. McKillop, John

Wed, 17 Dec (1300-1630): TH323

  1. Meers, Tehya
  2. Midberry, James

Wed, 17 Dec (1300-1630): TH321

  1. Minicozzi, John
  2. Noack, Macoy

Thu, 18 Dec (0730-1100): TH321

  1. Din, Jenna
  2. Gupta, Aarav
  3. Lawrence, Karina

Wed, 17 Dec (0730-1100): BH 171A

  1. Zagame, Samuel

Wed, 17 Dec (1300-1630): TH322

  1. Arengo, Mary
  2. Chambers, Cherokee
  3. Dohl, Chad
  4. Hudson-Odoi, Vanessa
  5. Kinkead, Lucas
  6. Records, Benjamin
  7. Shelton, Sawyer
  8. Stockbower, Tatiana
  9. Thanepohn, Trevor
  10. Walter, Benjamin
  11. Wills, Liam
  12. Wint, Logan

Thu, 18 Dec (0730-1100): TH321

  1. Chau, Paul
  2. Johnson, Joseph
  3. Rubio, Andrew
  4. Sager, Campbell
  5. Vann, Nehemiah

Wed, 17 Dec (1300-1630): TH323

  1. Aguilar Winchell, Benjamin
  2. Ahn, David
  3. Andrade, Elena
  4. Bettencourt, Jacob
  5. Bhutani, Dillon
  6. Campbell, Evan
  7. Forgues, Barbara
  8. Jo, Alex
  9. Lanham, Logan
  10. Maan, Bahawal
  11. Schwartz, Joseph
  12. Sindler, Allan
  13. Tahmazian, Isabela
  14. Wamre, Gabrielle

Thu, 18 Dec (0730-1100): TH321

  1. Helmkamp, Braeden
  2. Ogordi, Daniel
  3. Park, Sangwoo
  4. Smith, Gennaro

Wed, 17 Dec (1300-1630): TH321

  1. Ardisana, James
  2. Bachmann, Christian
  3. Barksdale, Jordon
  4. Barvitskie, Mason
  5. Corbett, Chas
  6. Davidson, Justin
  7. Groebner, Samuel
  8. Harris, Parker
  9. Kim, Danny
  10. Mantell, Jack
  11. McKane, Angelina
  12. Nguyen, Ta
  13. Oxendine, Jake
  14. Patterson, Alyssa
  15. Speaks, Brennan

Thu, 18 Dec (0730-1100): TH321

  1. Arterberry, Myles
  2. McPherson, Paige
  3. Williams, Caleb

TEE Overview

TEE Cover Sheet

TEE Admin

Authorized Resources:

  • Your computer with access to a blank RStudio document (.R, .rmd, or .qmd tab)
  • Course Guide
  • Tidyverse Tutorial
  • Two pages (front and back) of personally handwritten notes
  • Issued calculator

Unauthorized Resources:

  • Internet
  • Generative AI
  • Textbook
  • Course applets
  • Email or other electronic communications
  • Music devices
  • Friends / Instructor

Important Reminders:

  • Show enough work to logically present your thought process (R code or equations) - demonstrate mastery of course material, not simply a correct answer
  • Prepare RStudio by clicking the broom button in “History,” “Environment,” and “Plots” panes; use CTRL+L to clear the Console
  • Academic security: Do not discuss content until released on 19 December @ 1630

Advice:

  • Don’t just put down an answer - show how you got to that answer even if you used R/calculator
  • You’ve been taught everything on this exam!

Course Review - Non Exhaustive List of Topics

Probability Rules

  • Addition Rule: \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)
  • Multiplication Rule: \(P(A \cap B) = P(A) \cdot P(B|A)\)
  • Complement Rule: \(P(A^c) = 1 - P(A)\)
  • Conditional Probability: \(P(A|B) = \frac{P(A \cap B)}{P(B)}\)
  • Bayes’ Theorem: \(P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}\)
  • Mutually Exclusive: \(P(A \cap B) = 0\), so \(P(A \cup B) = P(A) + P(B)\)
  • Independence: \(P(A|B) = P(A)\), so \(P(A \cap B) = P(A) \cdot P(B)\)

Study Design & Data

  • Identify observational units in a study
  • Recognize variable types (categorical vs quantitative)
  • Distinguish between observational study and experimental study
  • Identify confounding variables
  • Know sampling methods (convenience sample, random sample, etc.)
  • Choose appropriate plot types based on variable type:
    • Categorical: bar graph
    • Quantitative: histogram, dot plot, box plot

Parameters vs Statistics

  • Know the difference between parameter (population) and statistic (sample)
  • Know when we can claim causality (random assignment)
  • Know when we can claim generalization (random sampling)

Random Variables & Distributions

  • Know when to use PMF, PDF, and CDF for discrete and continuous variables
  • Calculate probabilities using PMF (discrete) and PDF (continuous)
  • Go from PDF to CDF for continuous random variables to include piecewise PDFs (integration)
  • Calculate expected value \(E(X)\), variance \(Var(X)\), and standard deviation \(SD(X)\)
  • Apply linear transformation rules: \(E(aX + b) = aE(X) + b\) and \(Var(aX + b) = a^2 Var(X)\)

Hypothesis Testing

Know when to use each test, validity conditions, and how to execute:

Test When to Use
One-proportion z-test One categorical variable, testing proportion against a value
One-sample t-test One quantitative variable, testing mean against a value
Two-proportion z-test Comparing proportions between two groups
Two-sample t-test Comparing means between two independent groups
Paired t-test Comparing means for paired/matched data

For each test, know how to:

  • Write null and alternative hypotheses (symbols and words)
  • Check validity conditions
  • Calculate statistic, standardized statistic, and p-value
  • Interpret p-value (probability of observing data at least as extreme, given null is true)
  • Draw conclusions comparing p-value to significance level
  • Know when to use z-distribution vs t-distribution:
    • z: proportions (known population standard deviation)
    • t: means (unknown population standard deviation, using sample SD)

Confidence Intervals

  • Interpret confidence intervals correctly
  • Calculate margin of error (half the width of CI)
  • Determine if a value is plausible based on CI

Linear Regression

  • Describe scatterplot characteristics: direction, form, strength of association
  • Execute simple and multiple linear regression with data
  • Interpret coefficients for continuous variables (“for each 1-unit increase…”)
  • Interpret coefficients for categorical variables (“compared to the reference group…”)
  • Interpret coefficients “after controlling for” or “after accounting for” other variables
  • Interpret interaction terms
  • Compare models using p-values and \(R^2\)
  • Assess statistical significance of coefficients

In-Class Examples

Example 1

Researchers studied where people stop when approaching a stop sign. Do people have a preference for stopping position, or do they choose randomly between stopping as a single car, following another car, or leading a group?

Stop <- ma206data::chap8_Stop
head(Stop)
# A tibble: 6 × 1
  position_stop
  <chr>        
1 single       
2 single       
3 single       
4 single       
5 single       
6 single       
table(Stop)
position_stop
follow   lead single 
    98     42    176

Step 1: Set up hypotheses

If people had no preference, they would choose each position with equal probability (1/3).

\(H_0: \pi = 1/3\) (no preference - people choose “single” at the same rate as random chance)

\(H_a: \pi > 1/3\) (people prefer to stop as a single car)

Step 2: Calculate the statistic

# Count successes (single) and total
n <- nrow(Stop)
x <- sum(Stop$position_stop == "single")
p_hat <- x / n

cat("n =", n, "\n")
n = 316 
cat("x (single) =", x, "\n")
x (single) = 176 
cat("p-hat =", round(p_hat, 3), "\n")
p-hat = 0.557

Step 3: Conduct the test

\[z = \frac{\hat{p} - \pi_0}{\sqrt{\frac{\pi_0(1-\pi_0)}{n}}}\]

# Null hypothesis value
pi_0 <- 1/3

# Standard error under the null
SE <- sqrt(pi_0 * (1 - pi_0) / n)

# Standardized statistic (z-score)
z <- (p_hat - pi_0) / SE

cat("Standard Error =", round(SE, 4), "\n")
Standard Error = 0.0265 
cat("z =", round(z, 3), "\n")
z = 8.433 
# p-value (one-sided, greater than)
p_value <- 1 - pnorm(z)
cat("p-value =", round(p_value, 4), "\n")
p-value = 0

Step 4: Confidence Interval

\[\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\]

# For CI, use standard error based on p-hat (not pi_0)
SE_ci <- sqrt(p_hat * (1 - p_hat) / n)

# 90% CI (for alpha = 0.10)
z_star <- qnorm(0.95)  # one-sided, so 0.95
CI_lower <- p_hat - z_star * SE_ci
CI_upper <- p_hat + z_star * SE_ci

cat("90% CI: (", round(CI_lower, 3), ",", round(CI_upper, 3), ")\n")
90% CI: ( 0.511 , 0.603 )

Step 5: Draw conclusion

Compare p-value to significance level and state conclusion in context.

Example 2: Multiple Linear Regression with Interaction

Researchers want to understand factors that affect lung capacity (FEV - Forced Expiratory Volume). They collected data on age, height, gender, and smoking status.

FEV <- ma206data::fev
head(FEV)
# A tibble: 6 × 5
    Age   FEV Height Gender Smoker
  <dbl> <dbl>  <dbl> <chr>  <chr> 
1    11  3.90   67   Female no    
2    11  3.98   68.5 Male   no    
3     8  2.17   57   Male   no    
4    11  3.74   68   Male   no    
5    11  2.94   63   Female no    
6    15  2.73   63   Female no

Part A: Scatterplot

ggplot(FEV, aes(x = Height, y = FEV)) +
  geom_point() +
  labs(x = "Height (inches)", y = "FEV (liters)",
       title = "Height vs. Lung Capacity")

Lets talk association direction, form, and strength:

Direction:

Form:

Strength:

Part B: Multiple Linear Regression (No Interaction)

model1 <- lm(FEV ~ Height + Age + Smoker, data = FEV)
summary(model1)

Call:
lm(formula = FEV ~ Height + Age + Smoker, data = FEV)

Residuals:
    Min      1Q  Median      3Q     Max 
-1.5349 -0.2903 -0.0146  0.2812  1.9197 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) -4.77287    0.24718 -19.309  < 2e-16 ***
Height       0.11288    0.00520  21.707  < 2e-16 ***
Age          0.05142    0.01049   4.900 1.24e-06 ***
Smokeryes   -0.13031    0.06685  -1.949   0.0517 .  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.4422 on 589 degrees of freedom
Multiple R-squared:  0.7466,    Adjusted R-squared:  0.7454 
F-statistic: 578.6 on 3 and 589 DF,  p-value: < 2.2e-16

Interpretation of Height coefficient:

After controlling for Age and Smoker status, for each additional inch of height, FEV increases by approximately ___ liters, on average.

Interpretation of Smoker coefficient:

After controlling for Height and Age, smokers have an FEV that is approximately ___ liters [higher/lower] than non-smokers, on average.

Is Height statistically significant at \(\alpha = 0.10\)?

Look at p-value for Height coefficient.

Part C: Multiple Linear Regression (With Interaction)

model2 <- lm(FEV ~ Height * Smoker + Age, data = FEV)
summary(model2)

Call:
lm(formula = FEV ~ Height * Smoker + Age, data = FEV)

Residuals:
     Min       1Q   Median       3Q      Max 
-1.51896 -0.28187 -0.01598  0.27434  1.92389 

Coefficients:
                  Estimate Std. Error t value Pr(>|t|)    
(Intercept)      -4.603951   0.252466 -18.236  < 2e-16 ***
Height            0.109355   0.005309  20.599  < 2e-16 ***
Smokeryes        -3.781015   1.261569  -2.997  0.00284 ** 
Age               0.056087   0.010552   5.315 1.51e-07 ***
Height:Smokeryes  0.055393   0.019115   2.898  0.00390 ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.4395 on 588 degrees of freedom
Multiple R-squared:  0.7502,    Adjusted R-squared:  0.7485 
F-statistic: 441.5 on 4 and 588 DF,  p-value: < 2.2e-16

Does smoking status change the association between Height and FEV?

Look at the p-value for the interaction term (Height:Smokeryes). If p-value < \(\alpha\), then yes, the relationship between Height and FEV differs for smokers vs non-smokers.

Part D: Model Comparison

Which model would you recommend?

Model R-squared Key p-values
Model 1 (no interaction) ___
Model 2 (with interaction) ___ Interaction p-value: ___

Recommendation: If the interaction term is not significant, prefer the simpler model (Model 1). If significant, the interaction model (Model 2) better captures the relationship.

Exam Preparation

  • Review Course Related Reviews with Solutions for WPR1 and WPR2
  • TEE Review
  • Review WPR 1 and WPR 2
  • Review all Exploration Exercises
  • Review Board Problems

Work these problems as if you’ve never seen them before - don’t just skip to the answer and assume you know it.

Thank You

Thank you for a great semester in MA206! Good luck on your TEE and future endeavors.

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