Statistics 371: Midterm 2

What is a population parameter?

A fixed, usually unknown number describing the population (e.g., μ, σ, p).

What is a statistic?

A value computed from sample data (e.g., X̄, s, P̂) used to estimate a parameter.

Estimator vs estimate?

Estimator = formula (X̄); Estimate = computed value (x̄ = 27.4).

Meaning of iid?

Independent and identically distributed observations.

Define a sampling distribution.

Distribution of a statistic (such as X̄ or P̂) over all possible samples of size n.

If population is Normal, what is X̄’s distribution?

X̄ ~ N(μ, σ/√n) for any n.

If population not Normal?

By CLT, X̄ ≈ N(μ, σ/√n) when n ≥ 30.

Parameters of sampling distribution of X̄?

Mean = μ; SD = σ/√n (standard error).

Conditions for CLT to apply to X̄?

iid observations, finite σ, n large (≥ 30).

Define sample proportion P̂.

P̂ = X/n where X ~ Binomial(n, p).

Distribution of P̂ (large n)?

P̂ ≈ N(p, p(1−p)/n).

Check CLT for P̂?

np ≥ 10 and n(1−p) ≥ 10.

Standard error of P̂?

√[p(1−p)/n] (or use P̂ if p unknown).

What does CLT guarantee?

Sampling means and proportions ≈ Normal as n grows.

As n ↑, what happens to variability?

SE ↓ → more precise estimates.

Purpose of a confidence interval (CI)?

Estimate a population parameter with a range of plausible values.

General form of CI?

Point Estimate ± Margin of Error (E).

Margin of Error depends on?

Confidence level (z/t), sample size (n), and σ (or s).

Case 1 Formula?

X̄ ± zₐ/₂ (σ/√n)

When is case 1 valid?

Population not Normal but n large (CLT applies).

Case 2 Formula?

X̄ ± tₐ/₂, (n−1) (s/√n)

When is Case 2 valid?

Population ≈ Normal or n large.

T vs. Z shape difference?

t is bell-shaped with heavier tails; as df ↑, t → z.

Confidence ↑ → width?

wider. 

n ↑ → width?

narrower.

95% CI [64.7, 71.3] means what?

The procedure yields intervals capturing μ 95% of the time; we’re 95% confident this one does.

What are the five steps in hypothesis testing?

1. State H₀ and Hₐ.

2. Select significance level α

3. Compute test statistic.

4. Find p-value or critical region.

5. Decide & interpret in context

Define Type I, Type II errors, and Power.

• Type I (α): Reject H₀ when true.

• Type II (β): Fail to reject H₀ when false.

• Power = 1 − β = P(reject H₀ | Hₐ true)

How does sample size affect power?

Larger n → smaller SE → higher power to detect true differences

How does σ (variability) affect power?

Larger σ spreads the sampling distribution → lower power to detect the same effect

How are two-sided Z-tests and CIs related?

For a two-sided Z-test at α, H₀ is rejected iff μ₀ is outside the (1 − α) CI

What are the two-sample design types?

• Independent samples: different subjects per group.

• Paired (dependent): same or matched subjects measured twice

What is the goal of a two-sample test?

Compare population means μ₁ and μ₂; test H₀ : μ₁ = μ₂ ⇔ μ₁ − μ₂ = 0

What are the assumptions for an independent-sample t-test?

1. Independence within and between groups.

2. Approx Normality of each population.

3. Equal variance for pooled test.

When to use Wilcoxon Rank-Sum test?

For two independent groups when n small and Normality fails; tests median difference, not mean