Statistics: Population Parameters, Sampling Distributions, and Hypothesis Testing

What does a population parameter describe?

The entire population (e.g., μ, σ, p). Parameters describe true population values, not samples.

What does a statistic describe?

A sample. It estimates population parameters like μ and σ.

What does the Central Limit Theorem (CLT) state?

As sample size n increases, the sampling distribution of the mean approaches normal, regardless of population shape.

What is the mean of the sampling distribution of the sample mean?

Equal to the population mean (μ).

What is the standard error (SE) of the mean?

SE = σ / √n. It measures how much sample means vary from the population mean.

When can we use a normal approximation for proportions?

When np ≥ 10 and n(1 - p) ≥ 10.

How does increasing sample size affect the standard error?

SE decreases — larger samples give more precise estimates.

When does the finite population correction (FPC) apply?

When sampling without replacement and sample > 5% of population.

What is a sampling distribution?

The distribution of a statistic (like x̄) from all possible samples of the same size.

What is the Law of Large Numbers?

As n increases, the sample mean (x̄) approaches the true population mean (μ).

What is a point estimate?

A single number used to estimate a population parameter (like x̄ or p̂).

What is a confidence interval (CI)?

A range of values that likely contains the true population parameter.

How is a confidence interval interpreted?

95% confidence means that if we repeated sampling many times, 95% of intervals would contain μ.

How does higher confidence affect the interval?

Higher confidence → wider interval (more certainty, less precision).

How does larger sample size affect the interval?

Larger n → smaller margin of error → narrower CI.

When do you use a z-distribution for a confidence interval?

When σ is known or n is large (≥ 30).

When do you use a t-distribution for a confidence interval?

When σ is unknown and sample size is small (< 30).

What are the degrees of freedom for a one-sample t-interval?

df = n - 1.

What is the standard error for a proportion?

SE = √[p̂(1 - p̂)/n].

How does sample size affect margin of error (MOE)?

As n increases, MOE decreases.

What are the critical z-values for 90%, 95%, and 99% confidence?

1.645, 1.96, and 2.576.

If a 95% CI for μ is (10, 14), what is the point estimate?

12 — the midpoint of the interval.

If a 95% CI for μ excludes the null value, what's the test result?

Reject H₀ at α = 0.05 — the difference is significant.

What does a wider t-distribution mean compared to z?

More uncertainty — t has heavier tails when n is small.

What does a null hypothesis (H₀) usually state?

No change, no difference, or equality (status quo).

What is the alternative hypothesis (H₁)?

The researcher's claim (change, difference, or inequality).

What is a Type I error?

Rejecting a true H₀ (false positive). Probability = α.

What is a Type II error?

Failing to reject a false H₀ (false negative). Probability = β.

What is the significance level (α)?

The probability of making a Type I error (rejecting a true H₀).

What is the power of a test?

1 - β, the probability of correctly rejecting a false H₀.

What is the decision rule for p-values?

If p ≤ α → reject H₀. If p > α → fail to reject H₀.

What does "fail to reject H₀" mean?

There's not enough evidence against H₀ — not proof that H₀ is true.

What are the three types of alternative hypotheses?

Left-tailed (μ < μ₀), right-tailed (μ > μ₀), and two-tailed (μ ≠ μ₀).

What are common critical z-values for α = 0.05?

One-tailed = 1.645, two-tailed = ±1.96.

What happens to power when sample size increases?

Power increases — easier to detect a true effect.

What happens to power when α increases?

Power increases, but so does risk of Type I error.

What does a small p-value mean?

Strong evidence against H₀ — reject H₀.

What does a large p-value mean?

Weak evidence against H₀ — fail to reject H₀.

What does the p-value represent?

The probability of getting a result as extreme as the sample's, assuming H₀ is true.

Why do we never "accept" H₀?

Because failing to reject doesn't prove it's true — it just means not enough evidence.

When is a result "statistically significant"?

When p < α.

What test statistic is used when σ is known?

z = (x̄ - μ₀) / (σ / √n).

What test statistic is used when σ is unknown?

t = (x̄ - μ₀) / (s / √n).

What test statistic is used for proportions?

z = (p̂ - p₀) / √[p₀(1 - p₀)/n].

What is the formula for the margin of error?

MOE = critical value × SE.

What happens if α is reduced from 0.05 to 0.01?

Less chance of Type I error, but more chance of Type II error (less power).

What does "two-tailed test" mean?

Testing for any difference (μ ≠ μ₀), not just increase or decrease.

What does "one-tailed test" mean?

Testing in one direction only (greater or less).

What does the finite population correction (FPC) formula do?

Adjusts SE when sampling > 5% of a finite population.

What does "sampling without replacement" mean?

Once selected, an element can't be chosen again — requires FPC if pop small.

What Excel function finds a right-tail p-value from z?

=1-NORM.S.DIST(z,TRUE)

What Excel function gives the critical z-value for a given probability?

=NORM.S.INV(probability)

What Excel function gives a critical t-value for a given α?

=T.INV(1-α/2,df)

What Excel function simulates random sampling?

=RAND()