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BA 275Quantitative Business Methods

Quiz #1

Experiencing Random Behavior Normal Probability Distribution Normal Probability Table

Agenda

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0 10 20 30 40 50 600

0.01

0.02

0.03

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Review Question: Warranty Level

Mean = 30,000 miles STD = 5,000 miles

Q1: If the level of warranty is set at 15,000 miles, about what % of tires will be returned under the warranty?Q2: If we can accept that up to 2.5% of tires can be returned under warranty, what should be the warranty level?

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0 10 20 30 40 50 600

0.01

0.02

0.03

0.04

The Empirical Rule is not Enough

Mean = 30,000 miles STD = 5,000 miles

Q1: If the level of warranty is set at 12,000 miles, about what % of tires will be returned under the warranty?Q2: If we can accept that up to 3% of tires can be returned under warranty, what should be the warranty level?

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The Normal Probability Distribution

A specific curve that is symmetric and bell-shaped with two parameters m and s2.

It has been used to describe variables that are too cumbersome to be consider as discrete (i.e., continuous variable). For example, Physical measurements of members of a biological

population (e.g., heights and weights), IQ and exam scores, amounts of rainfall, scientific measurements, etc.

It can be used to describe the outcome of a binomial experiment when the number of trials is large.

It is the foundation of classical statistics. Central Limit Theorem

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Standard Normal Probabilities (Table A)

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Standard Normal Probabilities (Table A)

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Example 1

m = 0s = 1a = 1.96

A

a

Prob = ???

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Example 2

m = 0s = 1a = ?????

C

a

Prob = 0.0793

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Example 3

m = 0s = 2a = 2.00b = ??????

D

a b

Prob = 0.1005

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Sampling Distribution (Section 4.4)

A sampling distribution describes the distribution of all possible values of a statistic over all possible random samples of a specific size that can be taken from a population.

000,000,000,169,3

25

45

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Central Limit Theorem (CLT)

The CLT applied to Means

If ),(~ 2smNX , then ),(~2

nNX

sm .

If ~X any distribution with a mean m, and variance s2,

then ),(~2

nNX

sm given that n is large.

CLT demo

Example 1: X ~ a normal distribution with the mean 16, and variance 25.

Example 2: X ~ a distribution with the mean 8.08, and variance 38.6884.

With a sample of size n = 25, can we predict the value of the sample mean?

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0 10 20 30 40 50 600

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0.02

0.03

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Answer: Review Question: Warranty Level

Mean = 30,000 miles STD = 5,000 miles

Q1: If the level of warranty is set at 15,000 miles, about what % of tires will be returned under the warranty? => 0.15%Q2: If we can accept that up to 2.5% of tires can be returned under warranty, what should be the warranty level? => 20,000 miles

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0 10 20 30 40 50 600

0.01

0.02

0.03

0.04

Answer: The Empirical Rule is not Enough

Mean = 30,000 miles STD = 5,000 miles

Q1: If the level of warranty is set at 12,000 miles, about what % of tires will be returned under the warranty? => almost 0.0000Q2: If we can accept that up to 3% of tires can be returned under warranty, what should be the warranty level? => 20,600 miles

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Answer: Example 1

m = 0s = 1a = 1.96

A

a

Prob = ???

Prob = 0.025

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Answer: Example 2

m = 0s = 1a = ?????

C

a

Prob = 0.0793

a = -1.41

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Answer: Example 3

m = 0s = 2a = 2.00b = ??????

D

a b

Prob = 0.1005

b = 3.14