Chapter 6(4)
Chapter Overview
MTH213: Business Statistics - Focus on Chapter 6: The Normal Distribution.
Outline
Continuous Probability Distribution
Definition of continuous variables.
Characteristics & functions of continuous probability distributions.
Normal Distribution
Key properties of normal distribution.
Computing normal probabilities.
Learning Objectives
Compute probabilities from the normal distribution.
Apply the normal distribution to solve business problems.
Use normal probability plots to assess data normality.
Continuous Probability Distribution
Definition:
A continuous variable can assume any value in a continuum (uncountable values).
Examples: time to complete a task, starting salaries, stock prices, ages.
Probability Distribution Function
Defined by a function f(x) with the following properties:
f(x) ≥ 0 for all x.
Total area under f(x) equals 1.
Probability Density Function (pdf)
Assigns probabilities to intervals of values.
Distribution Shapes
Normal Distribution:
Symmetrical, bell-shaped curve with values clustering around the mean.
Uniform Distribution:
Symmetrical, rectangular shape.
Skewed Distribution:
Either left or right skewed.
Probability Properties
For a continuous random variable:
The area under f(x) from x = a to b represents the probability that x falls within that range.
P(X = c) = 0.
P(a ≤ X ≤ b) = P(a < X < b) = P(X < b) − P(X < a).
Properties of Normal Distribution
Significance:
Many variables (e.g., heights, weights) can be modeled as normally distributed.
Cornerstone of statistical inference.
Definition:
A random variable X is normally distributed with mean µ and standard deviation σ, denoted N(µ, σ²).
Probability density function:
f(x) = (1 / (√(2πσ))) * e^(-(x−µ)² / (2σ²)) for -∞ < x < ∞.
Characteristics of Normal Distribution
Bell-shaped curve concentrating near mean µ.
Symmetrical distribution (Mean = Median).
Variance σ² measures variability.
Total area under the curve is 1.
Standard Normal Distribution:
If µ = 0 and σ = 1, follows N(0, 1).
Empirical Rule: The 68.26-95.44-99.74 Rule
Properties of normally distributed variables:
68.26% lie within 1 standard deviation of the mean (µ ± σ).
95.44% lie within 2 standard deviations (µ ± 2σ).
99.74% lie within 3 standard deviations (µ ± 3σ).
Example with IQ: Mean = 100, SD = 16.
68% between 84 and 116.
95% between 68 and 132.
99.74% between 52 and 148.
Standard Normal Distribution Transformation
Transform any normal variable X into Z:
Z = (X − µ) / σ.
Computing Normal Probabilities
Steps to Find Normal Probabilities
Recognize the normal distribution's symmetry.
Total area under the curve equals 1.
Standard normal distribution transformation:
Finding P(a < X < b): Translate x-values to z-values and use the standard normal table.
Calculator Suggestion
Online Normal Distribution Calculator for probability calculations.
Examples of Normal Probability Problems
Hybrid Car Example:
Mean distance: 65 miles, SD: 4 miles.
Calculate:
P(X < 60) = 0.1056.
P(X > 75) = 0.0062.
P(55 < X < 70) = 0.8881.
Exercises
Determine various probabilities related to work commute times and customer browsing durations.
Utilize calculations based on mean and standard deviation to answer business-related probability questions.
Backward Normal Calculations
To find the observed value x for a given percentile:
Sketch the normal curve.
Identify mean and x values.
Find cumulative area less than x.
Use normal table for necessary z-value and unstandardize: X = µ + Zσ.
Additional Exercises
Logistics and planning scenarios derived from normal distributions across various contexts such as utility bills and temperature ranges.
Additional Resources
Excel Functions:
Use NORM.DIST and NORM.INV for calculations.
Online calculators for further utility and ease of computations.