Overview
This lecture covers applications of the normal distribution, including standardization, inverse calculations, binomial approximation, and hypothesis testing for the mean.
Calculating Probabilities & Inverse Normal
- To find probabilities for Y ~ N(μ, σ²), use your calculator's normal cumulative distribution function (NCD).
- For P(Y < a), input mean, standard deviation, upper value as a, lower value as a large negative number.
- For P(Y = a), probability is zero as the normal distribution is continuous.
- Use the inverse normal function to find values given a probability (area to the left).
Standardizing & Solving for μ or σ
- If μ or σ is unknown, standardize: Z = (X - μ)/σ, where Z ~ N(0, 1).
- Use inverse normal to find Z for a given probability, then solve for the unknown parameter.
- If both μ and σ are missing, set up simultaneous equations from two different probabilities.
Binomial Approximation Using Normal Distribution
- Normal approximation is suitable when n is large and p is close to 0.5.
- For Binomial(n, p): Mean μ = np, Variance σ² = np(1-p), Standard deviation σ = sqrt(np(1-p)).
- Apply a continuity correction by adjusting boundaries by 0.5 when switching from discrete to continuous.
- Rewrite probability ranges with <, >, ≤, ≥ and expand bounds before using the normal.
Hypothesis Testing for the Mean
- Null hypothesis (H₀): mean equals stated value; alternative (H₁): mean differs (one- or two-tailed).
- For sampling: sample mean distribution has mean μ and variance σ²/n.
- Calculate probability of observing the sample mean or more extreme under H₀.
- If probability < significance level, reject H₀ in context; otherwise, there is insufficient evidence.
Key Terms & Definitions
- Normal Distribution — A continuous probability distribution, symmetric around the mean, defined by μ (mean) and σ² (variance).
- Standardization — Converting X ~ N(μ,σ²) to Z ~ N(0,1) using Z = (X-μ)/σ.
- Inverse Normal — Calculator function to find x given cumulative probability.
- Continuity Correction — Adjusting discrete binomial boundaries by ±0.5 when approximating with normal.
- Hypothesis Testing — Statistical method to test claims about population mean using sample data.
- Significance Level (α) — Probability threshold (e.g., 0.05) for rejecting H₀.
Action Items / Next Steps
- Practice using normal and inverse normal functions on your calculator.
- Work through sample questions involving standardization and simultaneous equations.
- Review binomial to normal approximation problems and apply continuity corrections.
- Prepare for hypothesis testing by stating hypotheses clearly and interpreting results in context.