9'3: Central Limit Theorem

1
9.3 Central Limit Theorem

• Many populations do indeed have an approximately
  normal distribution. But, what about
  distributions that are skewed or not normal???
  If the sampling distribution of x-bar is not
  approximately normal, we have no way of using
  x-bar to compare with the population parameter.

2

• Look at your 25 pennies. If we were to plot the
  age of your 25 pennies, what kind of distribution
  do you think it would resemble? Symmetric or
  skewed?
• Fill out the table to find the distribution of
  your coins. Remember we are using age as the
  x-variable.

3

• Put your 25 pennies in a cup. Randomly select 5
  pennies. Find the average age of your sample of
  five. Record this as x-bar(5). Repeat this to
  get another random sample of size 5.
• Now, randomly select 10 pennies. Find the
  average age and record this as x-bar(10). Repeat
  this process to get another random sample of size
  10.

4

• Find the average age of all 25 of your coins.
  Record this as x-bar(25).
• Now, we will look at a histogram of all the
  distributions. You are going to come up and put
  a dot to represent each pennys age in your
  population.

5

• Population Distribution

6

• Sampling Distribution for n 5

7

• Sampling Distribution n 10

8

• Sampling Distribution n 25

9

• Central Limit Theorem Simulation

10

• Central Limit Theorem Draw an SRS of size n
  from any population with mean µ and finite std.
  dev. s. When n is large enough the sampling
  distribution of the sample mean, x-bar, is close
  to the normal distribution.
• How large a sample is needed for x-bar to be
  close to normal depends on the population
  distribution. More observations are required for
  distributions farther from normal.

11

• Ex. The time a technician requires to perform
  preventative maintenance on an A/C unit is
  described by the exponential distribution with µ
  1 and s 1. You have 70 units in your office
  building, what is the probability that their
  average maintenance time is more than 50 minutes?