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Section VII, Measure - Probability. Central Limit Theorem. States that any distribution of sample means from a large population approaches the normal distribution as n increases to infinity The mean of the population of means is always equal to the mean of the parent population.

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Central limit theorem
Central Limit Theorem

  • States that any distribution of sample means from a large population approaches the normal distribution as n increases to infinity

    • The mean of the population of means is always equal to the mean of the parent population.

    • The standard deviation of the population of means is always equal to the standard deviation of the parent population divided by the square root of the sample size (n).

  • If you chart the values, the values will have less variation than the individual measurements

  • This is true if the sample size is sufficiently large.

  • What does this mean?

VII-5


Central limit theorem1
Central Limit Theorem

Central Limit Theorem Explanation

For almost all populations, the sampling distribution of the mean can be closely approximated by a normal distribution, provided the sample is sufficiently large.

Collect many x children, (assumption is infinite number of samples), create histograms.

VII-6


Probability
Probability

  • A number expressing the likelihood that a specific event will occur, expressed as the ratio of the number of actual occurrences to the number of possible occurrences

Example: If a fair coin is tossed, what is the probability of a head occurring?

P(n) = probability of n occurrencesp= proportion success (what you are looking for)q= proportion failures (what you are not looking for)

VII-7


Example
Example

  • You are rolling a fair six-sided die. What are the odds that you will roll a 3?

VII-7


Compound example
Compound Example

  • You are rolling a fair six-sided die. What are the odds that you will roll a 2 or a 4? A 2 and then a 4?

VII-8


Non mutually exclusive example
Non-Mutually Exclusive Example

  • You have a standard deck of cards. What are the odds that you draw a 3 or club?

VII-11


Dependent example
Dependent Example

  • You have a standard deck of cards. What are the odds that you will draw four aces without replacement?

VII-12


Number of ways
Number of Ways

  • Number of ways is a listing of possible successes

  • Permutation: PN,n, P(n,r), nPr, the number of arrangements when order is a concern – think ‘word’

  • Combination: CN,n, , nCr, the number of arrangements when order is not a concern

VII-13


Factorial
Factorial (!)

  • The product of a number and all counting numbers descending from it to 1

    6! = 6x5x4x3x2x1=720

    Note: 0!=1

VII-13


Permutation example
Permutation Example

  • How many 3 letter arrangements can be found from the word C A T? How about 2 letter arrangements?

  • Three lottery numbers are drawn from a total of 50. How many arrangements can be expected?

VII-13


Combination example
Combination Example

  • How many 3 letter groupings can be found from the word CAT?

  • Three lottery numbers are drawn from a total of 50. How many combinations can be expected?

VII-13


Binomial probability distribution example
Binomial Probability Distribution Example

  • A single six-sided die is tossed five times. Find the probability of rolling a four, three times.

VII-14


Poisson probability distribution
Poisson Probability Distribution

  • Refers to the probability distribution for defect count

  • Each unit of measure can have 0, 1, or multiple errors, defects, or some other type of measured occurrence.

  • Consider the following scenarios:

    • The number of speeding tickets issued in a certain county per week.

    • The number of calls arriving at an emergency dispatch station per hour.

    • The number of typos per page in a technical book.

  • Calculated by:

    • x = number of occurrences per unit interval (time or space)

    • λ = average number of occurrences per unit interval

VII-16


Poisson probability distribution example
Poisson Probability Distribution Example

  • The average number of homes sold by the Acme Realty company is two homes per day. What is the probability that exactly three homes will be sold tomorrow?

  • μ = 2; since 2 homes are sold per day, on average.

  • x = 3; since we want to find the likelihood that 3 homes will be sold tomorrow.

  • e = 2.71828; since e is a constant equal to approximately 2.71828.

  • We plug these values into the Poisson formula as follows:

  • Solution:

  • P(x; μ) = (e^-μ) (μ^x) / x!

  • P(3; 2) = (2.71828^-2) (2^3) / 3!

  • P(3; 2) = (0.13534) (8) / 6

  • P(3; 2) = 0.180

  • Thus, the probability of selling 3 homes tomorrow is 0.180 .

VII-17


Normal distribution basics
Normal Distribution Basics

  • Symmetrical, Bell-Shaped

  • Extends from Minus Infinity to Plus Infinity

  • Two Parameters

    • Mean or Average ( )

    • Standard Deviation ( )

  • Space under the entire curve is 100% of the data

  • Mean, median and mode are the same

VII-18


Normal distribution

±

±

±

s

s

s

2

1

3

≈95%

99.73%

≈68%

Normal Distribution

See XII-2

50%

50%

LCL

UCL

-3s

-2s

-1s

0

+1s

+2s

+3s

z value = distance from the mean measured in standard deviations

VII-18


Normal curve theory
Normal Curve Theory

  • Normal Curve theory tells us that the probability of a defect is smallest if you

    • stabilize the process (control)

    • make sigma as small as possible (reduce variation)

    • get Xbar as close to target as possible (center)

So… we first want to stabilize the process, second we will reduce variation and last thing is to center the process.

VII-18


Z value
z Value

See VIII-22-24

  • Specifies the areas under the normal curve

  • Represents the distance from the center measured in standard deviations

  • Values found on the normal table

    Population Sample

Remember when we talked about 3? The 3 is the z value.

VII-19


Z value example
z Value Example

  • The known average human height is 5’8” tall with a standard deviation of 5 inches. What are the z values for 6’2” and 4’8”?

A positive value indicates a z value to the right of the mean and a negative indicates a z value to the left of the mean.

VII-19


Z table exercise
z Table Exercise

  • From our answers from the last exercise, what is the values for:

    • P(Area > 6’2”)?

    • P(Area < 4’8”)?

    • P(4’8”< Area < 6’2”)?

    • Prove area under the normal curve at 1s, 2s, 3s?

VII-19


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