Normal Distribution & Central Limit Theorem

Understanding the Normal Distribution

• Normal distributions are bell-shaped and symmetric, representing data clustered around a mean.
• The mean (μ) indicates the center of the distribution.
• The standard deviation (σ) controls the spread and width.
• Notation: X ~ N(μ, σ), e.g., Verbal SAT ~ N(580, 70).
• Empirical Rule: 68% of data within 1σ, 95% within 2σ, 99.7% within 3σ of the mean.

Density Curve Concepts

• A density curve is a smooth version of a histogram.
• The total area under the curve equals 1.
• The proportion in an interval is the area over that interval.

Standard Normal Distribution (Z ~ N(0, 1))

• Standardization converts any normal variable X ~ N(μ, σ) to standard normal Z ~ N(0,1) using the formula: Z=(X−μ)/σ
• Reverse Standardization: X=μ+Z⋅σ (Finding a value from Z-score).

The Central Limit Theorem (CLT)

• For large random samples, sample statistics (proportions, means) follow a normal distribution, even if the population isn't normal.
• Allows using normal distribution to approximate bootstrap and randomization distributions.
• Enables inference using z-scores without simulation.

Computing a P-Value Using the Normal Distribution

• Calculate the standardized test statistic (z-score): Z=(statistic−null value)/SE
• P-value Interpretation: |Z| > 2 indicates an extreme result (p-value < 0.05).

Computing a Confidence Interval (CI) Using the Normal Distribution

• Formula: CI=statistic±z∗⋅SE
• z* is the critical value from the standard normal distribution based on the confidence level.
• Common z* Values: 80% (1.282), 90% (1.645), 95% (1.960), 98% (2.326), 99% (2.575).

Summary of Key Terms

• μ (mu): Population mean.
• σ (sigma): Population standard deviation.
• SE: Standard Error.
• Z-score: Number of standard errors a value is from the mean.
• P-value: Probability of observing a result as extreme as, or more extreme than, what was observed under the null.
• CI (Confidence Interval): Range of values likely to contain the true population parameter.

When to Use the Normal Distribution (CLT for Proportions)

• Appropriateness Conditions: both np≥10 and n(1−p)≥10 must be true.

One Proportion vs. Two Proportions

• One Proportion: A single group answering Yes/No; inference focuses on one population proportion p.
• Two Proportions: Comparison between two categorical groups; inference focuses on the difference p1−p2.
• Two separate variables: a response and an explanatory variable.

Key Formulas from the Central Limit Theorem

• Confidence Interval: statistic±z∗×SE
• Standardized Test Statistic: Z=(statistic−null value)/SE

Standard Error (SE)

• One Proportion (p known): SE=√(p(1−p)/n)
• One Proportion (p unknown, for CI): SE=√(p^(1−p^)/n)
• Two Proportions (CI): SE=√(p^1(1−p^1)/n1+p^2(1−p^2)/n2)
• Two Proportions (Hypothesis Test): Pooled p^=(x1+x2)/(n1+n2); SE=√(p^(1−p^)(1/n1+1/n2))

Confidence Intervals

• CI for One Proportion: p^±z∗⋅SE
• CI for Two Proportions: (p^1−p^2)±z∗⋅SE
• If the CI for p1−p2 includes 0, there’s no evidence of a difference between groups.

Hypothesis Tests

• One Proportion: H₀: p=p0; Test Statistic: Z=(p^−p0)/SE, where SE=√(p0(1−p0)/n)
• Two Proportions: H₀: p1=p2; use pooled proportion p^pooled=(x1+x2)/(n1+n2); Z=(p^1−p^2)/SE
• If p-value < α, reject H₀; if p-value > α, fail to reject H₀.

When to Use the t-Distribution

• Use the t-distribution when the population standard deviation (σ) is unknown.
• Conditions: If n ≥ 30, t-distribution is safe regardless of shape; if n < 30, population data must be approximately normal.
• Use t for means, z for proportions.

Central Limit Theorem (CLT) for a Mean

• With large enough n or normally distributed data, the sampling distribution of xˉ is approximately normal.
• Standard Error (SE): SE=s/√n (where s = sample standard deviation); as n increases, SE decreases.

Confidence Intervals for Means

• One Sample Mean: xˉ±t∗⋅SE
• xˉ±t∗⋅: t-multiplier based on confidence level and degrees of freedom (df = n - 1).
• Two Sample Means: (xˉ1−xˉ2)±t∗⋅SE
• Standard Error: SE=√(s12/n1+s22/n2)
• Degrees of Freedom: Conservative approach: use min(n1,n2)−1
• CI that includes 0 → no significant difference; CI that excludes 0 → evidence of a difference.

Hypothesis Testing for Means

• One Mean: H₀: μ=μ0; Test statistic: t=(xˉ−μ0)/SE, where SE=s/√n
• Two Means: H₀: μ1=μ2; Test statistic: t=(xˉ1−xˉ2)/SE
• Same SE and df as confidence interval

Visualizing the t-Distribution

• Looks like z-distribution but with fatter tails.
• As df ↑, t becomes more like standard normal (z).
• n ≥ 30 → t ≈ z

Recognizing Paired Data vs. Independent Samples

• Paired Data: Two measurements per case; analyze the difference per pair.
• Independent Samples: Two separate groups of individuals.
• Rule of thumb: If it's "two measurements per unit", it’s paired. If it’s “two different units,” it’s independent.

Why Use Paired Data?

• Advantage: Reduced variability; leads to smaller standard error, more statistical power

Analyzing Paired Data

• Steps: Calculate differences, treat differences as a single quantitative variable, and apply inference for one mean.

Inference with Paired Data (Means)

• Use t-distribution with df=nd−1
• Assumptions: nd≥30 OR differences are approximately normal

Formulas (Paired Data)

• CI for paired mean: dˉ±t∗⋅SE
• SE: SE=sd/√nd
• Test statistic (t): t=(dˉ−μ0)/SE
• df: nd−1

Synthesis: Choosing the Right Inference

• Single Proportion: Z distribution, np≥10, n(1−p)≥10
• Two Proportions: Z distribution, all counts ≥ 10
• One Mean: t-distribution, df = n − 1, n ≥ 30 or normal data
• Two Independent Means: t-distribution, df = min(n₁−1, n₂−1), n₁ ≥ 30 or normal, same for n₂
• Paired Means: t-distribution, df = nd − 1, nd ≥ 30 or differences ~ normal

Hypothesis Testing & Confidence Intervals: Full Process

• State Hypotheses: H₀: No difference, Hₐ: Difference exists