Complete Solutions Guide
This guide contains detailed solutions for all practice problems. Use this to check your work and understand problem-solving approaches.
tip
How to Use This Guide
- Attempt problem first - Don't peek at solutions
- Self-grade - Compare your answer
- Review explanation - Understand the methodology
- Read "Common Mistakes" - Learn what NOT to do
- Try similar problem - Reinforce learning
Module 1 Solutions
Linear Regression Solutions
Solution 1.1: Calculate Regression Line
Given Data:
X: [500, 800, 1200, 1500]
Y: [50, 70, 100, 130] (in thousands)
Step-by-step Solution:
Part a) Calculate means:
X̄ = (500 + 800 + 1200 + 1500) / 4 = 4000 / 4 = 1000
Ȳ = (50 + 70 + 100 + 130) / 4 = 350 / 4 = 87.5
Part b) Calculate slope (b₁):
b₁ = Σ(xᵢ - X̄)(yᵢ - Ȳ) / Σ(xᵢ - X̄)²
Numerator calculation:
(500-1000)(50-87.5) = (-500)(-37.5) = 18,750
(800-1000)(70-87.5) = (-200)(-17.5) = 3,500
(1200-1000)(100-87.5) = (200)(12.5) = 2,500
(1500-1000)(130-87.5) = (500)(42.5) = 21,250
Sum: 18,750 + 3,500 + 2,500 + 21,250 = 46,000
Denominator calculation:
(500-1000)² = 250,000
(800-1000)² = 40,000
(1200-1000)² = 40,000
(1500-1000)² = 250,000
Sum: 580,000
b₁ = 46,000 / 580,000 ≈ 0.0793 ≈ 0.08
Part c) Calculate intercept (b₀):
b₀ = Ȳ - b₁×X̄
b₀ = 87.5 - 0.0793×1000
b₀ = 87.5 - 79.3
b₀ ≈ 8.2
Part d) Fitted equation:
ŷ = 8.2 + 0.08×X
Part e) Prediction for 1000 sq ft:
ŷ = 8.2 + 0.08×1000 = 8.2 + 80 = 88.2 thousand = $88,200
Common Mistakes:
- Forgetting to square differences in denominator
- Arithmetic errors when summing
- Rounding too early (keep more decimal places)
- Confusing which variable is X vs Y
Solution 1.2: Calculate SSE, MSE, RMSE
Given:
Model: ŷ = 10 + 0.08×X
Actual vs Predicted:
X=500: ŷ=50k, Actual=48k (Residual: -2k)
X=800: ŷ=74k, Actual=75k (Residual: +1k)
X=1200: ŷ=106k, Actual=105k (Residual: -1k)
X=1500: ŷ=130k, Actual=132k (Residual: +2k)
Part a) Residuals:
e₁ = 48 - 50 = -2
e₂ = 75 - 74 = +1
e₃ = 105 - 106 = -1
e₄ = 132 - 130 = +2
Part b) SSE:
SSE = Σeᵢ² = (-2)² + (1)² + (-1)² + (2)²
= 4 + 1 + 1 + 4
= 10
Part c) MSE:
MSE = SSE / n = 10 / 4 = 2.5
Part d) RMSE:
RMSE = √MSE = √2.5 ≈ 1.58 thousand = $1,580
Part e) Interpretation:
On average, predictions are off by about $1,580.
For house prices in $50-130k range, this is ~1.2-3.2% error.
Reasonable for a simple model, but room for improvement.
Common Mistakes:
- Forgetting to square residuals (SSE is "squared" error)
- Using n vs (n-1) in denominator (depends on context)
- Not taking square root for RMSE
- Confusing error sign (residual can be ± but SSE always ≥ 0)
Bias-Variance Solutions
Solution 3.1: Bias-Variance Decomposition
Given:
Model A: Train 0.1, Test 0.11 (Gap: 0.01)
Model B: Train 0.05, Test 0.30 (Gap: 0.25)
Model C: Train 0.3, Test 0.32 (Gap: 0.02)
Part a) Which has high bias?
Answer: Model C
Reasoning:
- Training error = 0.30 (high!)
- Test error = 0.32 (also high, similar)
- Both train and test are bad → Model too simple
- Can't fit even training data well → HIGH BIAS
Part b) Which has high variance?
Answer: Model B
Reasoning:
- Training error = 0.05 (excellent!)
- Test error = 0.30 (terrible!)
- Huge gap (0.25) → Model fits train perfectly but fails on test
- Memorized training data → HIGH VARIANCE
Part c) Which has good tradeoff?
Answer: Model A
Reasoning:
- Training error: 0.1 (not perfect, acceptable)
- Test error: 0.11 (slightly higher, expected)
- Gap: 0.01 (tiny! Generalizes well)
- Balanced performance → GOOD BIAS-VARIANCE TRADEOFF
Definitions:
Bias: Error from oversimplifying (underfitting)
- Model too simple to capture patterns
- High training AND test error
Variance: Error from overfitting
- Model too complex, captures noise
- Low training, high test error
Common Mistakes:
- Thinking high training error means high variance (it's high BIAS)
- Confusing gap with variance (gap ≠ variance exactly)
- Saying Model A has overfitting (gap is too small)
Module 2 Solutions
Cross-Validation Solutions
Solution 1.1: 5-Fold CV Calculation
Given:
Fold 1: 0.25
Fold 2: 0.28
Fold 3: 0.22
Fold 4: 0.26
Fold 5: 0.29
Part a) CV error (mean):
CV Error = (0.25 + 0.28 + 0.22 + 0.26 + 0.29) / 5
= 1.30 / 5
= 0.26
Part b) CV standard deviation:
Mean = 0.26
Deviations from mean:
0.25 - 0.26 = -0.01 → (-0.01)² = 0.0001
0.28 - 0.26 = +0.02 → (+0.02)² = 0.0004
0.22 - 0.26 = -0.04 → (-0.04)² = 0.0016
0.26 - 0.26 = 0.00 → (0.00)² = 0.0000
0.29 - 0.26 = +0.03 → (+0.03)² = 0.0009
Sum of squared deviations: 0.0030
Variance = 0.0030 / 5 = 0.0006
Std Dev = √0.0006 ≈ 0.024
(Note: Could use n-1 for sample std dev = 0.027)
Part c) What does std dev tell you?
Low std dev (0.024) means:
- Fold errors are consistent
- Each fold gives similar estimate
- Model generalizes consistently
- Results are reproducible
Part d) Report result:
CV MSE = 0.26 ± 0.024
Part e) Is model stable?
Answer: YES
Reasoning:
- Low std dev relative to mean (0.024 / 0.26 ≈ 9%)
- Small variation across folds
- Reliable error estimate
Common Mistakes:
- Dividing by (n-1) when shouldn't
- Forgetting to square deviations
- Calculating standard error instead of standard deviation
- Misinterpreting what std dev means
Logistic Regression Solutions
Solution 2.1: Sigmoid Function
Sigmoid: σ(z) = 1 / (1 + e^(-z))
Part a) σ(z=0):
σ(0) = 1 / (1 + e^0) = 1 / (1 + 1) = 1/2 = 0.5
Part b) σ(z=1):
σ(1) = 1 / (1 + e^(-1))
= 1 / (1 + 0.368)
= 1 / 1.368
≈ 0.73
Part c) σ(z=-1):
σ(-1) = 1 / (1 + e^1)
= 1 / (1 + 2.718)
= 1 / 3.718
≈ 0.27
Part d) σ(z=2):
σ(2) = 1 / (1 + e^(-2))
= 1 / (1 + 0.135)
= 1 / 1.135
≈ 0.88
Part e) σ(z=-2):
σ(-2) = 1 / (1 + e^2)
= 1 / (1 + 7.389)
= 1 / 8.389
≈ 0.12
Part b) Range of sigmoid:
Answer: [0, 1]
As z → ∞: e^(-z) → 0, so σ(z) → 1/(1+0) = 1
As z → -∞: e^(-z) → ∞, so σ(z) → 1/∞ = 0
Part c) When σ(z) = 0.5:
0.5 = 1 / (1 + e^(-z))
1 + e^(-z) = 2
e^(-z) = 1
-z = 0
z = 0
Part d & e) Limits:
As z → ∞: σ → 1.0
As z → -∞: σ → 0.0
Common Mistakes:
- Computing e^z instead of e^(-z)
- Forgetting order of operations
- Rounding too early in intermediate steps
- Thinking sigmoid can go outside [0,1]
Classification Metrics Solutions
Solution 3.2: Precision & Recall
Given:
Confusion matrix:
Predicted Disease
Yes No
Actual Yes 45 5
No 10 440
Total positives: 50, Total negatives: 450
Part a) Count TP, TN, FP, FN:
TP = 45 (correctly predicted disease)
TN = 440 (correctly predicted no disease)
FP = 10 (predicted disease but didn't have)
FN = 5 (predicted no disease but had)
Part b) Precision:
Precision = TP / (TP + FP)
= 45 / (45 + 10)
= 45 / 55
≈ 0.82 = 82%
Interpretation: "Of positive predictions, 82% correct"
Part c) Recall:
Recall = TP / (TP + FN)
= 45 / (45 + 5)
= 45 / 50
= 0.90 = 90%
Interpretation: "Of actual positives, 90% detected"
Part d) Precision interpretation:
"Of all people we predicted as having disease,
82% actually had it (18% false alarms)"
Part e) Recall interpretation:
"Of all people who actually had disease,
we detected 90% (missed 10%)"
Common Mistakes:
- Swapping numerator and denominator
- Confusing recall with specificity
- Not understanding what each measures
- Saying precision = TP/all predictions (should be TP/(TP+FP))
Real-World Solutions
Solution 1.1: Customer Churn Prediction
Setup:
100,000 customers, 5% churn (5,000 churners)
Intervention cost: $50/customer
Retained value: $5,000
Net benefit if saved: $5,000 - $50 = $4,950
Model A:
Precision: 0.60, Recall: 0.80
Predicted positives: 0.80 × 5,000 = 4,000 churners
Expected TP: 0.60 × 4,000 = 2,400 correct
Expected FP: 0.40 × 4,000 = 1,600 false alarms
Expected FN: 0.20 × 5,000 = 1,000 missed
Cost calculation:
Interventions: 4,000 × $50 = $200,000
Benefit (saved): 2,400 × $4,950 = $11,880,000
Cost of missed: 1,000 × $5,000 = $5,000,000 loss
Net: $11,880,000 - $200,000 - $5,000,000 = $6,680,000
Model B:
Precision: 0.75, Recall: 0.85
Predicted positives: 0.85 × 5,000 = 4,250 churners
Expected TP: 0.75 × 4,250 = 3,187 correct
Expected FP: 0.25 × 4,250 = 1,063 false alarms
Expected FN: 0.15 × 5,000 = 750 missed
Cost calculation:
Interventions: 4,250 × $50 = $212,500
Benefit (saved): 3,187 × $4,950 = $15,776,150
Cost of missed: 750 × $5,000 = $3,750,000 loss
Net: $15,776,150 - $212,500 - $3,750,000 = $11,813,650
Comparison:
Model A: $6,680,000 net benefit
Model B: $11,813,650 net benefit
Model B is better by $5.1M despite higher deployment cost!
ROI for Model B: ($11,813,650 - $50,000) / $50,000 = 23,527%
Common Mistakes:
- Using accuracy instead of precision/recall
- Forgetting deployment costs
- Not accounting for false negatives
- Calculating ROI wrong
Quick Reference Index
Formula Sheet
Linear Regression:
b₁ = Σ(xᵢ - X̄)(yᵢ - Ȳ) / Σ(xᵢ - X̄)²
b₀ = Ȳ - b₁×X̄
SSE = Σ(yᵢ - ŷᵢ)²
MSE = SSE / n
RMSE = √MSE
R² = 1 - (SSE / TSS)
Cross-Validation:
CV Error = mean of fold errors
CV Std Dev = std dev of fold errors
Logistic Regression:
σ(z) = 1 / (1 + e^(-z))
P(y=1|x) = σ(b₀ + b₁×x)
log-odds = b₀ + b₁×x
odds = e^(log-odds) = e^(b₀ + b₁×x)
Classification Metrics:
Accuracy = (TP + TN) / Total
Precision = TP / (TP + FP)
Recall = TP / (TP + FN)
F1 = 2 × (Precision × Recall) / (Precision + Recall)
Specificity = TN / (TN + FP)
Common Mistakes by Topic
Regression Mistakes
- Forgetting absolute value in residuals
- Using n vs n-1 inconsistently
- Rounding too early
- Misinterpreting R²
Classification Mistakes
- Confusing precision with recall
- Using accuracy on imbalanced data
- Picking wrong threshold
- Misreading confusion matrix
CV Mistakes
- Not using stratified CV for imbalanced data
- Ignoring standard deviation
- Using wrong number of folds
- Not fixing random seed
Study Tips for Solutions
✓ Work backwards: Given answer, trace back to understand methodology
✓ Variation practice: Change numbers, redo calculation
✓ Teach someone: Explain solution to peer
✓ Time yourself: Practice speed calculation
✓ Check reasonableness: Does answer make sense?
When to Use This Guide
✓ After attempting problem
✓ When you're stuck (but try first!)
✓ To verify your answer
✓ To learn alternative approaches
✓ Right before exam (quick reference)
✗ Don't: Use before attempting problem (defeats learning)
Next Steps
- Review solutions for problems you got wrong
- Identify patterns in your mistakes
- Return to concept pages for weak areas
- Try similar problems with different numbers
- Time yourself to build speed
Ready for exam? → Good luck!