Statistical Inference

Grade 12 · Math · Study Guide

A rigorous overview of hypothesis testing, confidence intervals, p-values, and the logic of drawing conclusions from sample data.

Statistical inference uses sample data to make conclusions about a population. Two core tools are confidence intervals and hypothesis tests.

A confidence interval gives a range of plausible values for a population parameter. For a population mean with known standard deviation: CI = x̄ ± z*(σ/√n), where z* is the critical value from the standard normal distribution (e.g., 1.96 for 95% confidence). Interpreting correctly: a 95% CI means that 95% of all such intervals constructed from repeated samples would contain the true parameter — NOT that there is 95% probability the parameter is in this particular interval.

Hypothesis testing begins with a null hypothesis H₀ (typically equality) and an alternative hypothesis H₁. We compute a test statistic and a p-value, which is the probability of observing results as extreme as ours if H₀ were true. If p < α (significance level, typically 0.05), we reject H₀. A Type I error is rejecting a true H₀ (false positive, probability = α). A Type II error is failing to reject a false H₀ (false negative, probability = β). Power = 1 - β is the probability of correctly rejecting a false H₀.

The Central Limit Theorem guarantees that for large samples (n ≥ 30), the sampling distribution of x̄ is approximately normal, enabling inference even when the population is not normal.

Key terms

Null Hypothesis (H₀)
The default claim being tested, typically stating no effect or no difference; rejected only when evidence strongly suggests otherwise
P-value
The probability of obtaining a test statistic at least as extreme as observed, assuming H₀ is true; smaller p-values provide more evidence against H₀
Confidence Interval
A range of values computed from sample data that, with a specified confidence level, contains the true population parameter
Type I Error
Rejecting a true null hypothesis; the false positive rate; its probability equals the significance level α
Central Limit Theorem
For large n, the sampling distribution of x̄ is approximately N(μ, σ²/n) regardless of the population's shape
Statistical Power
The probability that a test correctly rejects a false null hypothesis; Power = 1 - β

Flashcards

Hypothesis Test Steps
1) State H₀ and H₁. 2) Choose α. 3) Compute test statistic. 4) Find p-value. 5) Compare to α. 6) State conclusion in context.
Z-Test Statistic for Mean
z = (x̄ - μ₀) / (σ/√n), used when σ is known and/or n ≥ 30
T-Test Statistic
t = (x̄ - μ₀) / (s/√n) with n-1 degrees of freedom; used when σ is unknown
95% Confidence Interval for μ
x̄ ± 1.96 · (σ/√n); uses z* = 1.96 for large samples
Margin of Error
z* · (σ/√n) or t* · (s/√n); half the width of a confidence interval
Type I vs Type II Error
Type I: reject true H₀ (probability = α). Type II: fail to reject false H₀ (probability = β)
Central Limit Theorem
x̄ ~ N(μ, σ²/n) for large n; allows normal-based inference for any population shape
One-Tailed vs Two-Tailed Test
One-tailed tests directional alternatives (H₁: μ > μ₀); two-tailed tests non-directional (H₁: μ ≠ μ₀); two-tailed splits α between both tails

Questions

  1. A study finds a p-value of 0.03 with a significance level of α = 0.05. What is the correct conclusion?

    1. Fail to reject H₀
    2. Reject H₀; there is statistically significant evidence against it
    3. Accept H₀
    4. The study is inconclusive
    Show answer

    Answer: Reject H₀; there is statistically significant evidence against it

    Since p = 0.03 < α = 0.05, we reject H₀. There is statistically significant evidence against the null hypothesis.

  2. A 95% confidence interval means there is a 95% probability that the true population mean lies within that specific interval.

    Show answer

    Answer: False

    The parameter is fixed (not random). The correct interpretation: if we repeated the procedure many times, 95% of the resulting intervals would capture the true parameter. Any specific interval either does or does not contain it.

  3. Which error occurs when you reject a null hypothesis that is actually true?

    1. Type II Error
    2. Type I Error
    3. Sampling Error
    4. Standard Error
    Show answer

    Answer: Type I Error

    Type I Error = false positive = rejecting a null hypothesis that is in fact true. Its probability is controlled by the significance level α.

  4. As sample size increases, the width of a confidence interval ___.

    Show answer

    Answer: decreases

    Margin of error = z*(σ/√n). As n increases, √n increases, so σ/√n decreases, making the interval narrower.

  5. A z-test yields z = 2.45 for a two-tailed test at α = 0.05 (critical value ±1.96). What is the conclusion?

    1. Fail to reject H₀
    2. Reject H₀
    3. Cannot determine without the p-value
    4. Accept H₁ with certainty
    Show answer

    Answer: Reject H₀

    Since |z| = 2.45 > 1.96 (critical value), z falls in the rejection region. We reject H₀ at α = 0.05.

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