Topic Review — Statistical Inference

← Statistical Inference

This review covers all four lessons: sampling distributions and the Central Limit Theorem, confidence intervals for population mean, hypothesis testing for population mean, and Type I and Type II errors.

Review Questions

1. A random sample of size n = 36 is drawn from a population with mean μ = 120 and standard deviation σ = 18. State the distribution of the sample mean &x̄, giving its mean and standard deviation.
2. Using the distribution from Q1, find P(&x̄ > 124).
3. A population has σ = 15 (known). A sample of n = 25 gives &x̄ = 84. Construct a 95% confidence interval for μ.
4. Interpret the confidence interval (78.12, 89.88) in the context of Q3.
5. A researcher wants a 99% confidence interval for μ with σ = 20 and margin of error no larger than 4. What is the minimum sample size required?
6. A hypothesis test has H₀: μ = 50 vs H₁: μ ≠ 50, with σ = 10 known, n = 100. A sample gives &x̄ = 52. Perform the test at α = 0.05 and state your conclusion.
7. For the test in Q6, calculate the p-value and use it to confirm your conclusion.
8. A one-tailed test has H₀: μ = 200 vs H₁: μ < 200, σ = 25, n = 49, α = 0.01. The critical value is z = −2.326. Find the rejection region (critical value of &x̄).
9. A factory claims its bolts have mean diameter μ = 12 mm. Quality control suspects the bolts are undersize. With σ = 0.8 mm and a sample of 64 bolts giving &x̄ = 11.85 mm, test at α = 0.05.
10. In Q9, describe the Type II error that could have been made. What would need to happen to reduce the risk of this error?
11. Define the power of a hypothesis test. A test has α = 0.05 and detects a particular alternative with power 0.75. What is the probability of a Type II error?
12. A 95% CI for μ (with known σ = 12, n = 36) gives the interval (58.08, 61.92). A researcher claims μ = 63. Is this claim consistent with the interval? What would the result of a two-tailed test at α = 0.05 be?
13. Increasing the confidence level from 95% to 99% (same n and σ) makes the confidence interval wider. Explain why, and state the trade-off involved.
14. A random variable X ~ N(μ, 64). A sample of 16 gives &x̄ = 42. Assuming H₀: μ = 45 vs H₁: μ < 45 at α = 0.05:
    • (a) Carry out the test and state your conclusion.
    • (b) Suppose the true mean is μ = 40. Calculate β (P(Type II error)).
15. A manufacturing company tests a new process with n = 100, σ = 5. The test is H₀: μ = 200 vs H₁: μ ≠ 200 at α = 0.02.
    • (a) Find the critical values for &x̄.
    • (b) A sample gives &x̄ = 198.9. State the conclusion.
    • (c) Comment on the possibility of each type of error given your conclusion.