Hypothesis Testing and Decision Rules

P-value and Significance Level (α)

In hypothesis testing, the P-value is compared to the significance level (α) to determine whether to reject the null hypothesis (H₀). This rule is fundamental for statistical inference.

• P-value ≤ α: Reject H₀. There is sufficient evidence for the alternative hypothesis (H₁).

• P-value > α: Do NOT reject H₀. There is not sufficient evidence for H₁.

• Example: If P = 0.527 and α = 0.05, then 0.527 > 0.05 → do not reject H₀.

Identifying Statistical Procedures n,

Keywords and Their Statistical Implications

Recognizing keywords in a problem helps determine the correct statistical test or interval.

• Mean / average: Use t procedures (t-test, t-interval).

• Proportion / percentage / percent: Use z procedures (z-test, z-interval).

• Before and after, same people, matched pairs: Dependent samples (paired t-test).

• Two different groups: Independent samples (independent-samples t-test).

Alternative Hypothesis (H₁) and Test Tails

The wording of the claim determines the form of the alternative hypothesis and the direction of the test.

• "Different", "changed", "not equal": (two-tailed)

• "Greater", "higher", "more": (right-tailed)

• "Less", "lower", "decreased": (left-tailed)

• Example: "X more expensive than Y" with means .

Confidence Intervals

Structure and Interpretation

A confidence interval estimates a population parameter with a specified level of confidence.

• General form:

• Lower limit:

• Upper limit:

• Sample mean: Use t-value

• Sample proportion: Use z-value

Confidence level affects interval width:

• 90% CI: narrower, more precise

• 95% CI: moderate width

• 99% CI: wider, less precise

• Reason: Higher confidence requires a wider interval to ensure the true parameter is captured.

Dependent vs. Independent Samples

Definitions and Examples

Understanding whether samples are dependent or independent is crucial for selecting the correct test.

• Dependent samples: Observations are paired or matched (e.g., same subjects measured twice).

• Independent samples: Observations are from separate, unrelated groups.

Example (Dependent): 29 Chicago cameras measured on Wednesday and Saturday. Each camera has paired observations.

• Calculate differences:

• One-sample t-test on differences:

• Null hypothesis:

Example (Independent): Compare average exam scores of students from two different colleges.

Exam Problem Recognition Guide

Stepwise Approach

• Step 1: Identify variable type (mean or proportion).

• Step 2: Determine number of populations (one or two).

• Step 3: For two means, check if samples are paired (dependent) or separate (independent).

• Step 4: Identify the claim's direction (different, greater, less).

• Step 5: Compare P-value to α for decision.

Types of Errors and Power

Type I and Type II Errors

Errors in hypothesis testing arise from incorrect decisions about H₀.

• Type I Error: Reject H₀ when H₀ is true (false alarm). Probability = α.

• Type II Error: Do not reject H₀ when H₀ is false (missed detection). Probability = β.

• Power: ; measures the test's ability to detect a real effect.

Example: If β = 0.20, then Power = 0.80 (80%).

Chapter Distinctions and Formula Selection

Quick Reference Table

Use the following table to select the correct procedure based on the problem's context:

Additional info: Table entries inferred from context and standard statistics curriculum.

Recognizing Symbols and Formula Selection

• \hat p: Proportion; use formulas containing \hat p.

• \bar{x}, s, n: Mean; one group → Chapter 10, two groups → Chapter 11.

• \bar{x}_1, \bar{x}_2, s_1, s_2, n_1, n_2: Two independent means.

• d, \bar d, s_d: Matched/dependent pairs.

Confidence Interval and Test Statistic Formulas

One Proportion Confidence Interval

• Lower:

• Upper:

• Common z* values: 90% = 1.645, 95% = 1.96, 99% = 2.575

One Mean Confidence Interval

• Lower:

• Upper:

Matched-Pairs (Dependent Samples) Test Statistic

• Null hypothesis:

Independent Samples Test Statistic

• For two means: Use independent-samples t-test (Welch's t-test).

• Calculate sample means and standard deviations for each group.

Critical Values and Tails

• Left-tailed: Use α; answer negative.

• Right-tailed: Use α; answer positive.

• Two-tailed: Use α/2; answers ±.

• H₁ determines the tail: < = left, > = right, ≠ = two-tailed.

Observational Studies vs. Experiments

Definitions and Recognition

• Observational study: Researcher observes/measures without assigning treatments. Can show association, not causation.

• Experiment: Researcher assigns treatments (randomly or otherwise). Can establish cause-and-effect if properly designed.

• Keywords: "observed", "surveyed", "measured", "records reviewed" → observational; "assigned", "randomly assigned", "treatment", "placebo" → experiment.

Boxplots and Data Visualization

Interpreting Boxplots

• Boxplot outlier: a dot/point beyond the whisker.

• No dots beyond whiskers: no outliers.

• Comparing groups: Shifted box indicates higher/lower central tendency.

Summary Exam Checklist

• Mean or proportion? Mean → t, Proportion → z

• One or two populations?

• If two means: paired or separate?

• Claim direction: different → ≠, greater → >, less → <

• Compare P to α: P ≤ α → REJECT, P > α → DON'T REJECT

• Type I = false alarm (α), Type II = missed it (β), Power = 1 − β

• More confidence = wider interval

Additional Academic Context

• Retrospective study: Looks back at existing/past data.

• Prospective study: Follows subjects forward and collects future results.

• Randomized experiment: Researcher randomly assigns treatments.

• Response variable: The outcome being measured.

• Independent-sample conditions: Random, independent, each sample ≤5% of its population; n ≥ 30 counts as a large sample.

• Order matters: If the problem defines μ₁ − μ₂, keep that order when subtracting sample means.