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REVIEW Central Limit Theorem and The t Distribution

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# REVIEW Central Limit Theorem and The t Distribution - PowerPoint PPT Presentation

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

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

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

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

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?
• _ 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

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
• 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
• 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)
• _

Case 3: P(X > 21000)

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

_

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

• 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 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
• 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

_

• 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