Central limit theorem

Central limit theorem

Lessons

The distribution of sampling means is normally distributed
\cdot μx=μ\mu_{\overline{x}}=\mu
\cdot σx=σn\sigma_{\overline{x}}=\frac{\sigma}{\sqrt{n}}

Central Limit Theorem:
Z=xμxσx=xμσnZ=\frac{\overline{x}-\mu_{\overline{x}}}{\sigma_{\overline{x}}}=\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}
Typically n30n \geq 30
  • Introduction
    a)
    The distribution of sampling means is normally distributed

    b)
    Formula for the Central Limit Theorem


  • 1.
    Comparing the Individual Z-Score to the Central Limit Theorem
    A population of cars has an average weight of 1350kg with a standard deviation of 200 kg. Assume that these weights are normally distributed.
    a)
    Find the probability that a randomly selected car will weigh more than 1400kg.

    b)
    What is the probability that a group of 30 cars will have an average weight of more than 1400kg?

    c)
    Compare the two answers found in the previous parts of this question.


  • 2.
    Applying the Central Limit Theorem
    Skis have an average weight of 11 lbs, with a standard deviation of 4 lbs. If a sample of 75 skis is tested, what is the probability that their average weight will be less than 10 lbs?

  • 3.
    Increasing Sample Size
    At the University of British Columbia the average grade for the course "Mathematical Proofs" is 68%. This grade has a standard deviation of 15%.
    a)
    If 20 students are randomly sampled what is the probability that the average of their mark is above 72%?

    b)
    If 50 students are randomly sampled what is the probability that the average of their mark is above 72%?

    c)
    If 100 students are randomly sampled what is the probability that the average of their mark is above 72%?