Quiz 1:Regression Models (Data Science Specialization) Answers 2025 :

Question 1
Which value of μ minimizes $∑i=1nwi(xi−μ)2\sum_{i=1}^n w_i (x_i-\mu)^2$ ?

✅ 0.1471
❌ 0.300
❌ 1.077
❌ 0.0025

Explanation: The weighted least-squares minimizer is the weighted mean: $μ=∑wixi/∑wi=1.03/7≈0.1471.\mu = \sum w_i x_i / \sum w_i = 1.03/7 \approx 0.1471.$

Question 2
Regression through the origin (slope = sum(x*y)/sum(x^2)) — give the slope.

✅ 0.8263
❌ -0.04462
❌ 0.59915
❌ -1.713

Explanation: For regression through origin slope = $∑xiyi/∑xi2≈2.9227/3.5373≈0.8263.\sum x_i y_i / \sum x_i^2 \approx 2.9227 / 3.5373 \approx 0.8263.$

Question 3
From mtcars, slope of lm(mpg ~ wt).

❌ 0.5591
❌ 30.2851
✅ -5.344
❌ -9.559

Explanation: The familiar result for mpg ~ wt is a negative slope ≈ −5.344 (mpg decreases ~5.344 per 1000 lb increase in wt).

Question 4
Predictor SD is half of outcome SD; correlation = 0.5. Slope of Y ~ X ?

❌ 3
❌ 0.25
✅ 1
❌ 4

Explanation: Slope = Corr(Y,X) * (sd_Y / sd_X) = 0.5 * (1 / 0.5) = 1.

Question 5
Two normalized tests (mean 0, sd 1), correlation 0.4. Expected Quiz2 score if Quiz1 = 1.5?

✅ 0.6
❌ 0.16
❌ 0.4
❌ 1.0

Explanation: Conditional expectation (linear prediction) = r * z1 = 0.4 * 1.5 = 0.6.

Question 6
Normalize x; value of first measurement (z-score)?

❌ 8.86
❌ 8.58
✅ -0.9719
❌ 9.31

Explanation: mean ≈ 9.31, sd ≈ 0.7511, z = (8.58−9.31)/0.7511 ≈ −0.9719.

Question 7
Intercept for regression of y on x (same x,y as Q2)?

❌ 2.105
❌ -1.713
✅ 1.567
❌ 1.252

Explanation: slope ≈ −1.713, means: mean_x ≈ 0.573, mean_y ≈ 0.586. Intercept = mean_y − slope*mean_x ≈ 0.586 − (−1.713)(0.573) ≈ 1.567.

Question 8
If predictor and response both have mean 0, what about intercept?

❌ It must be exactly one.
❌ Nothing can be said.
❌ It is undefined.
✅ It must be identically 0.

Explanation: With both sample means 0, intercept = mean_y − b * mean_x = 0 − b*0 = 0.

Question 9
What value minimizes sum of squared distances for given x vector?

✅ 0.573
❌ 0.36
❌ 0.44
❌ 0.8

Explanation: The value minimizing squared distances is the sample mean: sum(x)/n = 5.73/10 = 0.573.

Question 10
Let β₁ be slope of Y~X and γ₁ slope of X~Y. What is β₁/γ₁ always equal to?

❌ 1
❌ Cor(Y,X)
✅ Var(Y)/Var(X)
❌ 2 SD(Y)/SD(X)

Explanation: β₁ = ρ·(σ_Y/σ_X), γ₁ = ρ·(σ_X/σ_Y) ⇒ β₁/γ₁ = (σ_Y²)/(σ_X²) = Var(Y)/Var(X).

🧾 Summary Table

Q#	✅ Correct Answer	Key Concept
1	0.1471	Weighted mean minimizes weighted squared error
2	0.8263	Slope through origin = Σ(xy)/Σ(x²)
3	-5.344	lm(mpg ~ wt) slope (mpg decreases with weight)
4	1	slope = corr · (sd_Y / sd_X)
5	0.6	Conditional expectation = r · z1 for standardized vars
6	-0.9719	z-score = (x − mean)/sd
7	1.567	Intercept = mean_y − b·mean_x
8	0	If means are 0, intercept = 0
9	0.573	Mean minimizes sum of squared deviations
10	Var(Y)/Var(X)	Ratio of slopes = variance ratio