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REVIEW Central Limit Theorem and The t Distribution. 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. _ The Random Variables X and X.

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review central limit theorem and the t distribution
REVIEW

Central Limit Theorem

and

The t Distribution

random variables
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
the random variables x and x
_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 = σ

_ X = a random variable designating the average outcome of n measurements of the event

_ _‗Mean of X = µ; Standard deviation of X = σ/√n

THIS IS ALWAYS TRUE AS LONG AS σ IS KNOWN!

example
Example

Attendance at a basketball game averages 20000 with a standard deviation of 4000.

X = Attendance at a game

µ = 20000; σ= 4000

_ X = Average attendance at n games

_ Mean of X = 20000

‗ ‗ Standard deviation of X = 4000/√n

the central limit theorem
The Central Limit Theorem

1

2

Central

Limit

Theorem

  • Assume that the standard deviation of the random variable X, σ, 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.

what is a large enough sample size
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.
examples
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.
answers assuming attendance has a normal distriubtion
Answers Assuming Attendance Has a Normal Distriubtion
  • If X, attendance, has a normal distribution since σ is known to = 4000, THEN
  • _ Average attendance, X, is normal with: ‗ Standard deviation of X =

__

4000/√16 = 1000 in case 2

and __

4000 /√64 = 500 in case 3.

calculations
Calculations
  • Case 1: P(X > 21000)

Here, z = (21000-20000)/4000 = .25

So P(X > 21000) = 1 - .5987 = .4013

  • _

Case 2: P(X > 21000)

Here, z = (21000-20000)/1000 = 1.00

_

So P(X > 21000) = 1 - .8413 = .1587

calculations continued
Calculations (Continued)
  • _

Case 3: P(X > 21000)

Here, z = (21000-20000)/500 = 2.00

_

So P(X > 21000) = 1 - .9772 = .0228

answers assuming attendance does not have a normal distriubtion
Answers Assuming Attendance Does Not Have a Normal Distriubtion
  • Case 1 : Since X is not normal we cannot evaluate P(X > 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 > 21000)

Here, z = (21000-20000)/500 = 2.00

_

So P(X > 21000) = 1 - .9772 = .0228

what happens when is unknown t distribution
What Happens When σ 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/√n
  • 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 σ is unknown.
when to use z and when to use t
When to use z and When to use t
  • USEz
  • Large n or sampling from a normal distribution
  • σ is known
  • USEt
  • Large n or sampling from a normal distribution
  • σ is unknown
  • z and t distributions are used in hypothesis testing and confidence intervals

_ These are determined by the distribution of X.

review
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 σ is known