Lecture on Standard Normal Distribution

Overview

• Discussion on Standard Normal Distribution
• Review of important equations and formulas
• Worked examples to demonstrate application of formulas

Key Concepts

Normal Distribution

• Bell Curve shape
• Random Variable X
• Parameters:
  • Mean (μ)
  • Standard Deviation (σ)
• Notation:
  • Mean is centered on the bell curve
  • 1 standard deviation from mean: Z = 1
  • Z-scores: Positive when X > Mean, Negative when X < Mean
  • Z-Score Formula: Z = \frac{X - \text{mean}}{\text{standard deviation}}
  • X Formula: X = \text{mean} + Z \times \text{standard deviation}

Probability Density Function (PDF)

• Normal Distribution PDF: f(X) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2} \left(\frac{X - \text{mean}}{\sigma}\right)^2}
• e is approximately 2.718
• Generally not needed unless using calculus

Empirical Rule

• 68-95-99.7 Rule
  • 68% within 1 standard deviation
  • 95% within 2 standard deviations
  • 99.7% within 3 standard deviations
• Area under the Curve
  • Total = 1
  • Calculating specific regions:
    • Region within 1 SD: 34% each side
    • Region within 2 SDs: 13.5% each side
    • Region within 3 SDs: 2.35% each side
    • Beyond 3 SDs: 0.15% each side

Worked Examples

Example 1

1. Given Values:
   • Mean = 50, Standard Deviation = 10
2. Z-score Calculation:
   • Given Z = 1.4, calculate X
   • Use formula: X = 50 + 1.4 \times 10 = 64
3. X Calculation:
   • Given X = 30, calculate Z
   • Use formula: Z = \frac{30 - 50}{10} = -2
4. Positive vs. Negative Z-scores:
   • Positive Z: Above mean
   • Negative Z: Below mean

Example 2

• Statistics Class Scores:
  • Mean = 74, SD = 8, Students = 2000
  • Calculate using empirical rule:
    • Less than 58: 2.5%
    • Between 66 and 82: 68%
    • At most 90: 97.5%
    • At least 66: 84%
    • More than 98: 0.15% chance for 3 students

Advanced Calculations

Non-Whole Z-Scores

• Z-Table Usage: For Z-scores like 1.56
• Standard Z Table usage example:
  • Z = 1.56: Area = 0.94062
  • Z = -0.43: Area = 0.3336

Additional Examples

• IQ Scores:
  • Mean = 100, SD = 15
  • Less than 80: P = 9.18%
  • Greater than 136: P = 0.82%
  • Between 95 and 110: 37.787%
  • 90th Percentile: IQ = 119.2
  • Middle 30%: Between 94.2 and 105.8

Company Tire Example

• Manufacturing Defects:
  • Mean Defects = 10, SD = 3.13
  • Less than 8 Defects: P = 26.1%
  • More than 15 Defects: P = 5.48%
  • Between 7 and 14 Defects: P = 73.12%

Conclusion

• Emphasis on understanding the standard normal distribution and empirical rule
• Application of Z-table for non-standard Z-scores
• Practical examples to illustrate concepts

Notes

• Always check areas under the curve to ensure understanding of probabilities
• Practice with Z-tables and empirical rules to solidify understanding
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