Chapter 6(4)

Chapter Overview

• MTH213: Business Statistics - Focus on Chapter 6: The Normal Distribution.

Outline

1. Continuous Probability Distribution

   • Definition of continuous variables.

   • Characteristics & functions of continuous probability distributions.

2. 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:

  1. 68.26% lie within 1 standard deviation of the mean (µ ± σ).

  2. 95.44% lie within 2 standard deviations (µ ± 2σ).

  3. 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

1. Recognize the normal distribution's symmetry.

2. Total area under the curve equals 1.

3. 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

1. 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:

  1. Sketch the normal curve.

  2. Identify mean and x values.

  3. Find cumulative area less than x.

  4. 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.