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BUSN 352: Statistics ReviewPowerPoint Presentation

BUSN 352: Statistics Review

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Professor Joseph Szmerekovsky. BUSN 352: Statistics Review. Probability Distributions. Probability Concepts. Let A be an event. Pr( A ) is then the probability that A will occur… If A never occurs, Pr( A ) = 0 If A is sure to occur, Pr( A ) = 1. Example: Find the probability.

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Probability Concepts

- Let A be an event. Pr(A) is then the probability that A will occur…
- If A never occurs, Pr(A) = 0
- If A is sure to occur, Pr(A) = 1

Example: Find the probability

- Selecting a black card from the standard deck of 52 cards
- Selecting a King
- Selecting a red King
- General Pattern: Pr(A)=

1/2

4/52 =1/13

2/52 =1/26

Example: Find probability

Throwing two fair dice find the probability that

- sum of the faces is equal to 2
- sum of the faces is equal to 4
- sum of the faces is equal to 7

Random Variables

- A random variable is a rule that assigns a numerical value to each possible outcome of an experiment.
- Discrete random variable -- countable number of values.
- Continuous random variable -- assumes values in intervals on the real line.

The Basics of Random Variables

- The probability distribution of a discrete random variable X gives the probability of each possible value of X.
- The sum of the probabilities must be 1.

Example: Probability distributions

- Fair coin toss

- Two fair coins

- Find Pr(X1)
- Find Pr(X=1.5)
- Find Pr(X1.5)

Expected Value

- The expected value (mean) of the random variable is the sum of the products of x and the corresponding probabilities

Example : Insurance Policy

Alice sells Ben a $10,000 insurance policy at an annual premium of $460.

If Pr(Ben dies next year) = .002, what is the expected profit of the policy?

E(X) = 460(.998) + (-9540)(.002) = $440

Example: Debbon Air “Seat Release”

- Debbon Air needs to make a decision about Flight 206 to Myrtle Beach.
- 3 seats reserved for last-minute customers (who pay $475 per seat), but the airline does not know if anyone will buy the seats.
- If they release them now, they know they will be able to sell them all for $250 each.

Debbon Air “Seat Release”

- The decision must be made now, and any number of the three seats may be released.
- Debbon Air counts a $150 loss of goodwill for every last-minute customer turned away.
- Probability distribution for X = # of last-minute customers requesting seats:

“Debbon Air” Seat Release

- What is Debbon Air’s expected net revenue (revenue minus loss of goodwill) if all three seats are released now?

- X = 0: Net Revenue = 3($250) = $750

- X = 1: Net Rev = 3($250) - $150 = $600

E (Net Revenue) =

750(.45) + 600(.30) + 450(.15) + 300(.10)

= $615.

“Debbon Air” Seat Release

- How many seats should be released to maximize expected net revenue?

Two seats should be released.

Variance and Standard Deviation of Random Variables

- The variance of a discrete R.V. X is

- The standard deviation is the square root of the variance.

f(x)

Area under the graph = Pr(a<X<b)

a

x

b

Continuous random variables- Continuous random variable -- assumes values in intervals on the real line.

Total area = 1

f(x)

1

x

0.2

0.5

0.6

1

Example: Uniform distribution- Is this a valid probability density function?

Yes

- Find Pr(0.2 < X < 0.5)

0.3·1 = 0.3

- Find Pr(X > 0.6)

0.4·1 = 0.4

The Normal Probability Model

- Importance of the Normal model
- Numerous phenomena seem to follow it, or can be approximated by it.
- It provides the basis for classical statistical inference through the Central Limit Theorem.
- It motivates the Empirical Rule.

The Normal Probability Model

- Crucial Properties
- Bell-shaped, symmetric
- Measures of central tendency (mean, median) are the same.
- Parameters are mean and standard deviation .

The Standard Normal Distribution

Normal with

Mean

SD

Standard Normal

with Mean 0

and SD 1

-2

-1

0

+1

+2

Table A.1: Standard Normal Distribution

- Standard Normal random variable Z
- E(Z) = 0 and SD(Z) = 1

- Table A.1 gives Standard Normal probabilities to four decimal places.

.4332

fZ(z)

z

0

1.50

Z Scores: Standardizing Normal Distributions

- Suppose X is
- Transformation Formula:

- For a given x, the Z score is the number of SD’s that x lies away from the mean.

Example: Tele-Evangelist Donations

- Money collected daily by a tele-evangelist, Y,is Normal with mean $2000, and SD $500.

- What is the chance that tomorrow’s donations will be less than $1500?

Convert to Z scores

Tele-Evangelist Donations

- Money collected is Normal with mean $2000 and SD $500.
- What is the probability that tomorrow’s donations are between $2000 and $3000?

- Let Y = $ collected tomorrow
- Y is Normal with mean 2000 and SD 500
- Need :

- Convert to Z scores:

= .4772

Tele-Evangelist Donations

- What is the chance that tomorrow’s donations will exceed $3000?

- Y is still Normal with mean 2000 and SD 500...

Convert to Z scores

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