Module 6 Practice Problems
20 Comprehensive Problems • Covers All Module 6 Topics
How to Use These Practice Problems
- Work through problems on paper before checking solutions
- Use the hints if you're stuck
- Problems are organized by topic and difficulty
- Show all your work - understanding the process is key!
Problems 1-5: CI Interpretation & Concepts
Problem 1Interpreting Confidence IntervalsEasy
A 95% CI for mean commute time is (28.3, 35.7) minutes. What is the margin of error?
Solution:
Margin of error = (35.7 - 28.3) / 2 = 7.4 / 2 = 3.7 minutes
Alternatively: Center = (28.3 + 35.7) / 2 = 32, so E = 35.7 - 32 = 3.7
Problem 2CI Correct InterpretationEasy
A poll reports: "We are 95% confident that between 52% and 58% of voters support the measure." Which interpretation is correct?
A: There's a 95% chance the true proportion is between 0.52 and 0.58.
B: If we repeated this process many times, 95% of intervals would contain the true proportion.
Solution:
Answer: B is correct.
Explanation: The parameter (true proportion) is fixed. The 95% refers to confidence in the method: 95% of intervals constructed this way will capture the true parameter. Interpretation A incorrectly treats the parameter as random.
Problems 6-10: CIs for Means
Problem 6CI for Mean CalculationMedium
A sample of n = 20 students has x̄ = 85 and s = 10. Construct a 95% CI for μ. (Use t* = 2.093 for df = 19)
Solution:
Given: x̄ = 85, s = 10, n = 20, df = 19, t* = 2.093
Standard error: SE = s/√n = 10/√20 = 10/4.472 = 2.236
Margin of error: E = t* × SE = 2.093 × 2.236 = 4.68
CI: 85 ± 4.68 = (80.32, 89.68)
Interpretation: We are 95% confident that the true mean score is between 80.32 and 89.68.
Problems 11-15: CIs for Proportions
Problem 11CI for ProportionMedium
In a survey of 400 voters, 228 support Candidate A. Construct a 95% CI for the proportion of all voters who support A. (Use z* = 1.96)
Solution:
Calculate p̂: p̂ = 228/400 = 0.57
Check conditions: np̂ = 400(0.57) = 228 ≥ 10 , n(1-p̂) = 400(0.43) = 172 ≥ 10
SE: √(0.57 × 0.43 / 400) = √(0.0006128) = 0.02475
E: 1.96 × 0.02475 = 0.0485
CI: 0.57 ± 0.049 = (0.521, 0.619) or (52.1%, 61.9%)
Problems 16-20: Sample Size
Problem 16Sample Size for MeanHard
How many students must be sampled to estimate mean GPA within ±0.1 with 95% confidence if σ ≈ 0.5? (Use z* = 1.96)
Solution:
Given: E = 0.1, z* = 1.96, σ = 0.5
Calculate: n = (1.96 × 0.5 / 0.1)² = (9.8)² = 96.04
Round up: n = 97 students
Problem 20Sample Size for Proportion (Conservative)Hard
A company wants to estimate customer satisfaction proportion within ±4% with 95% confidence. They have no prior estimate. How many customers should they survey?
Solution:
Conservative approach: p̂ = 0.5
Given: E = 0.04, z* = 1.96
Calculate: n = 0.5(0.5) × (1.96/0.04)² = 0.25 × (49)² = 0.25 × 2401 = 600.25
Round up: n = 601 customers