Class 11

Week 7: Session 11 Overview

Topics Covered

  • Standard Normal Distribution (6.1)

  • Real Applications (6.2)

  • Assessing Normality (6.5)

Probability Distributions

Continuous Random Variable

  • Defined by a probability density function (pdf)

    • Probability of observing any single value is 0

    • Interest is in intervals of values

    • P(x) ≥ 0 for all x

    • Total area under the curve (AUC) = 1

Discrete Random Variable

  • Defined by a probability mass function (pmf)

    • Probabilities are assigned for discrete values

    • 0 ≤ P(x) ≤ 1 for all x

    • Sum of probabilities equals 1

Characteristics of Normal Distribution

  • Symmetric about its mean (μ)

  • Highest point occurs at x = μ

  • Total area under the curve equals 1

  • Area under the curve to the right of μ = 0.5

  • As x increases or decreases indefinitely, the curve approaches 0

  • Empirical Rule: Approximately 68% of data falls within one standard deviation of the mean, 95% within two, and 99.7% within three.

Normal Probability Distribution Formula


  • Normal Distribution:[ f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x - \mu)^2}{2\sigma^2}} ]

Standard Normal Distribution

  • If X ~ N(μ, σ), then Standardization gives: [ Z = \frac{X - \mu}{\sigma} ]

  • Z ~ N(0,1) is the standard normal distribution.

Using Technology (JMP)

Data Distribution Calculators

  • Download the "Student Add-In" for JMP

  • Main functions include:

    • Distribution Calculator

    • Confidence Interval calculations

    • Hypothesis Tests

Distribution Characteristics

  • Normal Distribution Parameters:

    • Mean and standard deviation can be adjusted in calculations

    • Calculate probabilities and critical values based on inputs.

Applications of Normal Distribution

Example: Length of Human Pregnancies

  • Mean (μ) = 266 days

  • Standard deviation (σ) = 16 days


  • Probability of more than 280 days:[ P(X > 280) = 0.1908 ]

Identifying Extreme Preterm Pregnancy

  • Bottom 2% threshold for length of pregnancy:

    • Determine length of pregnancy using Z-scores for given probabilities.

Range Rule of Thumb for Significant Values

  • Calculate significant ranges:

    • Significantly low: μ - 2σ = 234

    • Significantly high: μ + 2σ = 298

  • Evaluating normal pregnancies at 280 days is not considered significantly high.

Assessing Normality

Methods

  1. Histogram: A bell-shaped histogram indicates normality; otherwise, it might not be normal.

  2. Outliers: Presence of multiple outliers suggests non-normal distribution.

  3. Normal Quantile Plot (Q-Q Plot): Points must align closely to a straight line for normality.

Conclusion on Normality Assessment

  • Non-normal distribution confirmed by:

    • Histogram is not bell-shaped.

    • Q-Q plot does not align closely to straight line.