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Statistics 2 Unit 1: Review of Inference and Sampling Distributions

Every method in Statistics 2 rests on the same machinery: a statistic, its sampling distribution, a standard error, and the logic of intervals and tests. This unit sharpens that toolkit before you apply it to more complex models.

Covers: Sampling distributions and the CLT, Standard error, t confidence intervals for a mean, Hypothesis tests and p-values, Type I and II errors and power, Inference for a proportion, Checking conditions

Study guide

A statistic such as x̄ or p̂ varies from sample to sample, and its sampling distribution describes that variation. For a random sample of size n, x̄ has mean μ and standard deviation σ/√n, called the standard error when we estimate σ with s. The Central Limit Theorem adds that for large n, x̄ is approximately normal no matter how the population is shaped. Similarly, p̂ has mean p and standard deviation √(p(1 − p)/n) and is approximately normal when the expected counts of successes and failures are both at least 10. Because standard errors shrink like 1/√n, you must quadruple a sample to halve its uncertainty.

A confidence interval has the form estimate ± critical value × standard error. For a mean with unknown σ, use x̄ ± t*·s/√n with n − 1 degrees of freedom; the t distribution has heavier tails than the normal to account for estimating σ. The confidence level describes the method: if we repeated the sampling many times, about 95% of 95% intervals would capture μ. A single computed interval either contains μ or it does not, and the interval is about the parameter, not about individual observations. Higher confidence or more variability widens the interval; a larger sample narrows it.

A hypothesis test starts with a null hypothesis H₀ (no effect, no difference) and an alternative Hₐ. We compute a test statistic, usually (estimate − null value)/SE, and its p-value: the probability, if H₀ were true, of a result at least as extreme as the one observed. A small p-value means the data would be surprising under H₀, so we reject H₀ when p ≤ α. A large p-value is not proof that H₀ is true; it only says the data are consistent with it. A two-sided test at level α rejects exactly those null values that fall outside the (1 − α) confidence interval.

Two errors are possible. A Type I error rejects a true H₀ and happens with probability α. A Type II error fails to reject a false H₀ and happens with probability β; power = 1 − β is the chance of detecting a real effect. Power rises with larger samples, larger true effects, less variability and larger α. With a fixed sample size, lowering α raises β, so researchers plan n in advance to achieve, say, 80% or 90% power. Running many tests inflates the chance of at least one false positive, which motivates the multiple-comparison methods you will meet in ANOVA.

Every procedure carries conditions. Data should come from a random sample or randomized experiment, observations should be independent (sample under 10% of the population when sampling without replacement), and the sampling distribution must be close to normal. For means, that means a large sample or, for small n, a roughly symmetric population without outliers. For proportions, check np̂ ≥ 10 and n(1 − p̂) ≥ 10. Finally, keep statistical and practical significance separate: a huge sample can make a trivial effect significant, so always report the estimate and its interval, not just the p-value.

Key ideas

• A statistic's sampling distribution describes how it varies across repeated random samples.
• Standard errors shrink like 1/√n, so precision grows slowly with sample size.
• Confidence intervals are estimate ± critical value × SE, and the level describes the method's long-run success rate.
• A p-value is computed assuming H₀ is true; it is not the probability that H₀ is true.
• Type I error probability is α; power is the probability of rejecting H₀ when a specific alternative is true.
• A two-sided test at level α and a (1 − α) confidence interval always agree.
• Statistical significance does not imply practical importance.

Formulas and rules

Common mistakes

• Reading a p-value as the probability that H₀ is true.
• Using z* = 1.96 instead of t* when σ is unknown and n is small.
• Saying a 95% CI contains 95% of the data values.
• Forgetting to double a one-tail area for a two-sided p-value.
• Rounding a required sample size down instead of up.

Before the test, make sure you can

• Calculate the standard error of x̄ and of p̂.
• Construct and interpret a one-sample t confidence interval.
• Carry out a one-sample t test and a one-proportion z test, including the p-value.
• Explain what a p-value does and does not mean.
• Identify Type I and Type II errors in context and explain how α, n and effect size affect power.
• Use the link between two-sided tests and confidence intervals.
• Check the conditions for t and z procedures.
• Compute a required sample size for a desired margin of error.

Words to know

Sampling distribution

The distribution of a statistic across all possible random samples of the same size.

Standard error

The estimated standard deviation of a statistic, such as s/√n for x̄.

Central Limit Theorem

For large n, the sample mean is approximately normal regardless of the population's shape.

t distribution

A symmetric, heavy-tailed distribution used when σ is estimated by s; indexed by degrees of freedom.

Degrees of freedom

The number of independent pieces of information in an estimate; n − 1 for a one-sample t.

Confidence level

The long-run proportion of intervals from the method that capture the parameter.

Margin of error

The critical value times the standard error; half the width of a confidence interval.

Null hypothesis

The claim of no effect or no difference that a test tries to find evidence against.

p-value

The probability, assuming H₀ is true, of a statistic at least as extreme as the observed one.

Significance level α

The cutoff for rejecting H₀ and the probability of a Type I error.

Type I error

Rejecting a true null hypothesis (a false positive).

Type II error

Failing to reject a false null hypothesis (a missed effect).

Power

The probability of rejecting H₀ when a particular alternative is true; equals 1 − β.

Large-counts condition

For proportion inference, at least 10 expected successes and 10 expected failures.

10% condition

When sampling without replacement, keep n under 10% of the population so observations are nearly independent.

Practical significance

Whether an effect is large enough to matter in context, separate from its p-value.

Sample practice questions

30 more questions like these, from level 1 to 5, are in the practice set.