MDM4U – Grade 12 Data Management – Continuous Probability Distributions

  • Continuous Probability Distributions
    • a random variable that can assume all possible random values (ie city temperature)
    • Probability Density Function: a function that describes how likely this random variable will occur at a given point.
    • Height formula:  height = 1/(b-a) where b is the top range, and a is the bottom range given.

 

  • The Normal Distribution
    • used to solve continuous probabilities
    • symmetry about the mean
    • total area under the curve is 1
    • standard deviation is the distance from the mean to the point of inflection
    • Any normal distribution can be described as by the mean and the variance: so we often write N(mean, variance) to describe a distribution
    • The distribution chart shows area under the graph from the X value to the left end
    • Z-Scores can be calculated using Normal distributions
      • Z = x – mean / standard deviation
      • Sometimes, you will have to subtract the mean to equalize. This makes it so the mean is on the center.

 

  • Normal Approximation
    • Step 1: Check if a normal approximation is appropriate. Test if np > 5 and nq > 5.
    • Step 2: Estimate the mean and standard deviation (mean = nq, SD = √(npq) )
    • Step 3: Estimate the probability using z-score method from above.

 

  • Confidence Interval
    • x- z * (σ/√n) < μ < x + z * (σ/√n)
      • where x is the mean of the sample
      • z is the z score of acceptable error
      • μ is the mean of population
      • n is the size of sample
      • σ is the standard deviation
    • Confidence levels and z- scores are retrieved from a given chart below: