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3.4.3 Examples of Normal Distribution

3.4.3 Examples of Normal Distribution

Key Concepts

  • Normal Distribution: Characterized by its bell-shaped curve, the normal distribution is defined by two parameters—mean (μ) and standard deviation (σ).
  • Applications: Used extensively in fields ranging from finance to science for modeling and predicting behaviors.

Example 1: Investment Returns

Scenario: Munaf Khan, a stock broker, analyzes returns on large-cap stock funds, which are normally distributed with a mean return of 12% and a standard deviation of 3%.

Categories of Investment:

  • Excellent: Returns > 10%
  • Moderate: Returns between 5% and 10%
  • Unsatisfactory: Returns < 5%

Probability Calculations:

  • Probability of unsatisfactory investment: Using the cumulative distribution function.
  • Probability of an excellent investment: Calculating the tail probability for returns over 10%.

Example 2: Screen Time Study

Scenario: Dr. Bhavani Kapadia studies screen time among school children, which is normally distributed with a mean of 8.4 hours/day and a standard deviation of 2.5 hours/day.

Questions:

  • Probability of screen time between 6 to 12 hours.
  • Top and bottom 20% of screen time durations: Using inverse probability calculations to determine cutoff points.

Example 3: Exam Duration

Scenario: Professor Hirav Joshi from Maharaja Sayajirao University observes the duration students take to complete his exams, following a normal distribution with a mean of 150 minutes and a standard deviation of 20 minutes.

Calculations:

  • Students completing within 2 hours.
  • Duration between 2 to 3 hours.
  • More than 3 hours duration.

Using Excel for Normal Distribution:

  • Implement NORM.DIST for direct probability calculations.
  • Use NORM.INV for inverse calculations to find critical points.
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