Lecture Notes: Using Z-Scores and Probability in Normal Distribution

Introduction

• Objective: To learn how to convert a value to a Z-score and find the probability of that value.
• Scenario: Analyze total cholesterol levels among men.
• Statistics:
  • Mean cholesterol level: 209 mg/dL
  • Standard deviation: 35 mg/dL
• Assumption: Cholesterol levels follow a normal distribution.

Key Concepts

Normal Distribution

• Defined by mean and standard deviation.
• Mean (midpoint) is 209 mg/dL.
• Standard deviation unit changes the value by 35 mg/dL.

Z-Score Calculation

• Formula: [ Z = \frac{X - \text{Mean}}{\text{Standard Deviation}} ]
• Example: Cholesterol level of 225 mg/dL
  • Calculate Z-score.
  • Verify sign (positive if above mean, negative if below mean).
  • Estimate number of standard deviations away.

Scenarios

Probability of Cholesterol Less Than 225 mg/dL

• Calculate Z-score for 225 mg/dL: 0.46
• Estimate Probability:
  • More than 0.5 (mean is 0.5)
  • Between 0.5 and 0.84 using empirical rule
• Z-Table Lookup: Probability is 0.6772
• Percentage Conversion: Approximately 68% of men have levels less than 225 mg/dL.

Proportion of Cholesterol More Than 225 mg/dL

• Same Z-score: 0.46
• Area to the Right:
  • Total area under curve is 1
  • Subtract area to the left (0.6772) from 1
• Proportion Result: 0.3228
• Interpretation: About 32% of men have levels greater than or equal to 225 mg/dL.

Probability Between Two Values: 174 mg/dL and 244 mg/dL

• Z-Scores: 174 mg/dL = -1; 244 mg/dL = +1
• Empirical Rule: 68% of data falls within one standard deviation.
• Probability Calculation:
  • From Z < -1 to Z > +1 = 0.6826
  • Approximation: 68%

Real-Life Example

• Tallest and Shortest Men:
  • Sultan (Tallest): 251 cm, Z-score of 9.5
  • Probability is extremely low.
  • He (Shortest): 74 cm, Z-score of -12.6
• Application: Used in growth percentiles for children.

Additional Notes

• Empirical Rule:
  • 95% within 2 standard deviations more precisely 1.96 standard deviations.
  • When calculating probabilities to four decimal places, 1.96 is used for accuracy.

Conclusion

• Understanding Z-scores and normal distribution are crucial for statistical analysis in real-life scenarios, such as health statistics and developmental progress measurements.