REVIEW

1

• REVIEW
• Central Limit Theorem
• and
• The t Distribution

2
Random Variables

• A random variable is a quantitative experiment
  whose outcome is not known in advance.
• All random variables have three things
• A distribution
• A mean
• A standard deviation

3
_The Random Variables X and X

• X a random variable designating the outcome
  of a single event
• Mean of X µ Standard deviation of X s
• _
  X a random variable
  designating the average outcome of n
  measurements of the event
• _
  _ Mean of X µ Standard
  deviation of X s/vn
• THIS IS ALWAYS TRUE AS LONG AS s IS KNOWN!

4
Example

• Attendance at a basketball game averages 20000
  with a standard deviation of 4000.
• X Attendance at a game
• µ 20000 s 4000
• _
  X Average attendance
  at n games
• _
  Mean of
  X 20000
• 
  
  Standard deviation of X 4000/vn

5
The Central Limit Theorem

• Assume that the standard deviation of the random
  variable X, s, is known.
• Two cases for the distribution of X
• X (Attendance at a game) is normal. THEN
• _
  
  X (Average attendance at n games)
  has a normal
  distribution for any sample size, n
• X (Attendance at a game) is not normal. THEN
• _
  
  X (Average attendance at n games) has an
  unknown distribution.
• But the larger the value of n, the closer it is
  approximated by a normal distribution.

6
What is a Large Enough Sample Size?

• _ To
  determine whether or not X can be approximated by
  a normal distribution, typically n 30 is used
  as a breakpoint.
• In most cases, smaller values of n will provide
  satisfactory results, particularly if the random
  variable X (attendance at a game) has a
  distribution that is somewhat close to a normal
  distribution.

7
Examples

• Attendance at a basketball game averages 20000
  with a standard deviation of 4000.
• Assuming that attendance at a game follows a
  normal distribution, what is the probability
  that
• Attendance at a game exceeds 21000?
• Average attendance at 16 games exceeds 21000?
• Average attendance at 64 games exceeds 21000?
• Repeat the above when you cannot assume
  attendance follows a normal distribution.

8
Answers Assuming Attendance Has a Normal
Distriubtion

• If X, attendance, has a normal distribution since
  s is known to 4000, THEN
• _
  
  Average attendance, X, is normal with
  
  Standard
  deviation of X
• __
  
  
  
• 4000/v16 1000 in case 2
• and __
• 4000 /v64 500 in case 3.

9
Calculations

• Case 1 P(X gt 21000)
• Here, z (21000-20000)/4000 .25
• So P(X gt 21000) 1 - .5987 .4013
• _
• Case 2 P(X gt 21000)
• Here, z (21000-20000)/1000 1.00
• _
  
  
• So P(X gt 21000) 1 - .8413 .1587

10
Calculations (Continued)

• _
• Case 3 P(X gt 21000)
• Here, z (21000-20000)/500 2.00
• _
  
  
• So P(X gt 21000) 1 - .9772 .0228

11
Answers Assuming Attendance Does Not Have a
Normal Distriubtion

• Case 1 Since X is not normal we cannot
  evaluate P(X gt 21000)
• 
  _
  
  Case 2 Since X is not normal, n is small, X has
  an unknown distribution. Thus we cannot evaluate
  this probability either.
• Since n is large, case 3 can be evaluated in the
  same manner as when X was assumed to be normal
  _
• Case 3 P(X gt 21000)
• Here, z (21000-20000)/500 2.00
• _
  
  
• So P(X gt 21000) 1 - .9772 .0228

12
What Happens When s Is Unknown? -- t Distribution

• This is the usual case.
• 
  _
• If X has a normal distribution, X will have a t
  distribution with
• n-1 degrees of freedom _
• Standard deviation s/vn
• But the t distribution is robust meaning we can
  use it even when X is only roughly normal a
  common assumption.
• From the central limit theorem, it can also be
  used with large sample sizes when s is unknown.

13
When to use z and When to use t

• z and t distributions are used in hypothesis
  testing and confidence intervals
• 
  _
  
  These are determined by the
  distribution of X.

14
REVIEW

• 
  _
• The random variables X and X
• Mean and standard deviation
• Central Limit Theorem
• _
• Probabilities For the Random Variable X
• t Distribution
• When to use z and When to Use t
• Depends only on whether or not s is known