Quiz 4:Statistical Inference(Data Science Specialization):Answers2025

Question 1

Paired data (n=5). Two-sided paired t-test P-value for mean reduction?

✅ 0.087
❌ 0.043
❌ 0.10
❌ 0.05

Explanation: Differences = [8,3,−1,2,5], mean = 3.4, s ≈ 3.363, t = 3.4 / (3.363/√5) ≈ 2.26 with df=4. Two-sided p ≈ 0.087 (between 0.10 and 0.05).

Question 2

For n=9, mean=1100, sd=30. Values of μ₀ for which two-sided 5% t-test fails to reject?

✅ 1077 to 1123
❌ 1031 to 1169
❌ 1081 to 1119
❌ 1080 to 1120

Explanation: 95% t-CI = 1100 ± t_{0.975,8}·(30/3) ≈ 1100 ± 23 → [1077,1123]. Any μ₀ inside this CI would not be rejected at α=0.05.

Question 3

4 people, 3 prefer Coke. One-sided exact test (H₀: p=0.5 vs H₁: p>0.5). P-value?

❌ 0.62
✅ 0.31
❌ 0.10
❌ 0.005

Explanation: X~Binomial(4,0.5). P(X≥3)=P(3)+P(4)=(4+1)/16=5/16=0.3125 ≈ 0.31.

Question 4

Benchmark 1 infection /100 person-days. Observed 10 infections over 1787 person-days. One-sided P-value for test hospital is below the standard?

✅ 0.03
❌ 0.52
❌ 0.22
❌ 0.11

Explanation: Under H₀ mean = 1787*(1/100)=17.87. P(X ≤ 10) for Poisson(17.87) ≈ 0.03 (Poisson→normal approx gives similar ≈0.03–0.04). So one-sided p ≈ 0.03.

Question 5

Two groups (n=9 each). Treated mean change −3 (sd=1.5), Placebo mean change 1 (sd=1.8). Two-sided pooled t-test p-value?

❌ Less than 0.10 but larger than 0.05
✅ Less than 0.01
❌ Less than 0.05, but larger than 0.01
❌ Larger than 0.10

Explanation: Difference = −4, pooled sd ≈1.6565, SE ≈0.782, t ≈ −5.12 with df=16 → p ≪ 0.01.

Question 6

Given a 90% CI [1077,1123] for μ, do you reject H₀: μ = 1078 at two-sided 5%?

❌ It’s impossible to tell.
✅ No you wouldn’t reject.
❌ Where does Brian come up with these questions?
❌ Yes you would reject.

Explanation: 1078 is inside the 90% CI; a 90% CI is narrower than a 95% CI, so 1078 will also lie inside the 95% CI — thus fail to reject at α=0.05.

Question 7

n=100, detect mean loss 0.01 mm³, σ=0.04, one-sided α=0.05. Power ≈ ?

❌ 0.70
❌ 0.50
✅ 0.80
❌ 0.60

Explanation: se = 0.04/√100 = 0.004. Noncentral z = 0.01/0.004 = 2.5. Power = P(Z > z_{0.95} − 2.5) = P(Z > 1.645−2.5)=P(Z > −0.855) ≈ 0.80.

Question 8

Same setup, required n for 90% power (one-sided α=0.05)?

❌ 180
❌ 120
❌ 160
✅ 140

Explanation: n ≈ [ (z_{α} + z_{power})·σ / δ ]² = [(1.645+1.282)·0.04/0.01]² ≈ (2.927·4)² ≈137 → ~140 (closest choice).

Question 9

As α increases, what happens to power?

❌ You will get smaller power.
✅ You will get larger power.
❌ No, for real, where does Brian come up with these problems?
❌ It’s impossible to tell given the information in the problem.

Explanation: Larger α increases the rejection region, so the probability of rejecting under the alternative (power) increases.

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