Sampling Distributions and Inference for Means and Proportions — HS Statistics practice questions

A study surveys 30 people from a city of 100,000. What is the sample size?

1. 30
2. 100,000
3. 3,000
4. It depends on the response rate

Answer: A — 30

A) Correct — The sample is the group actually measured. B) Is the population size. C) Multiplies the two numbers in some way. D) The sample size is the number surveyed.

A p-value is:

1. the probability that H₀ is true
2. the probability of results at least this extreme, if H₀ is true
3. the probability that Hₐ is true
4. the probability of making any error

Answer: B — the probability of results at least this extreme, if H₀ is true

B) Correct — It measures how surprising the data would be under the null. A) A common misreading; the p-value assumes H₀ is true. C) The p-value is not a probability about a hypothesis. D) Error rates are α and β.

A Type II error is:

1. rejecting H₀ when it is true
2. rejecting H₀ when it is false
3. choosing α too small
4. failing to reject H₀ when it is actually false

Answer: D — failing to reject H₀ when it is actually false

D) Correct — A missed detection; its probability is β. A) Is a Type I error. B) Is a correct decision. C) Can make Type II errors more likely, but is not one.

A 95% confidence interval for a proportion is (0.42, 0.58). What are the point estimate and margin of error?

1. p̂ = 0.50, margin of error 0.16
2. p̂ = 0.50, margin of error 0.08
3. p̂ = 0.42, margin of error 0.58
4. p̂ = 0.16, margin of error 0.50

Answer: B — p̂ = 0.50, margin of error 0.08

B) Correct — The center is the estimate, and half the width is the margin of error. A) Uses the full width as the margin of error. C) Reads the endpoints as the estimate and margin. D) Swaps the width and the center.

With n = 100,000, a 0.1-point difference in mean test scores is statistically significant. What does this mean?

1. The difference is unlikely to be due to chance, but it may be too small to matter
2. The difference is large and important
3. The study must be flawed
4. The null hypothesis is proven false

Answer: A — The difference is unlikely to be due to chance, but it may be too small to matter

A) Correct — Huge samples can detect tiny effects; practical importance is a separate question. B) Statistical significance does not measure size. C) Large samples are valuable; they just detect small effects. D) Tests give evidence, not proof.

The sample size is doubled. How does the standard error σ/√n change?

1. It is divided by √2
2. It is halved
3. It doubles
4. It does not change

Answer: A — It is divided by √2

A) Correct — n doubles, so √n grows by √2. B) Halving would need 4 times the sample. C) More data makes the estimate less variable, not more. D) The standard error depends on n.

p̂ = 0.5 and n = 100. What is the margin of error for a 95% confidence interval?

1. 0.05
2. About 0.098
3. 0.98
4. 0.0098

Answer: B — About 0.098

B) Correct — 1.96 × √(0.5 × 0.5/100) = 1.96 × 0.05. A) Is the standard error without the critical value. C) Forgets to divide by n inside the square root. D) Divides by n instead of √n.

p-value definition?

Probability of observing data (or more extreme) if H₀ true.

Confidence interval?

Range likely containing population parameter. E.g., 95% CI: 95% confident μ in interval.

How can you increase the power of a test?

Increase the sample size, raise α, reduce variability, or look for a larger effect. A larger n is the usual choice.

Sampling distribution of p̂?

Mean p, SD √(p(1 − p)/n). Approximately normal when np ≥ 10 and n(1 − p) ≥ 10.