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Section VII, Measure -

Probability

2222

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?

Central Limit Theorem

x

x

VII-5

3333

Central Limit Theorem

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

Central Limit Theorem Explanation

4444

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

Probability

50.21

111

tailsand heads #heads #)(

# # #

# #)(

ofofheadP

failuresofsuccessesofsuccessesof

iespossibilitofsuccessesofsuccessP

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

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

VII-7

5555

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

Example

VII-7

6666

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?

Compound Example

VII-8

7777

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

Non-Mutually Exclusive Example

VII-11

8888

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

Dependent Example

VII-12

9999

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

)!(!!

!!

rrnnC

rnnP

rn

rn

Number of Ways

rn

VII-13

10101010

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

6! = 6x5x4x3x2x1=720

Note: 0!=1

Factorial (!)

VII-13

11111111

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?

Permutation Example

VII-13

12121212

How many 3 letter groupings can be found from the word C A T?

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

Combination Example

VII-13

13131313

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

Binomial Probability Distribution Example

VII-14

14141414

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

15151515

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 .

Poisson Probability Distribution Example

VII-17

16161616

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

Normal Distribution

Basics

x,s,

VII-18

17171717

Normal Distribution

50%50%

-1-2-3 +1 +2 +30

± ≈68%1

± 99.73%3± ≈95%2

z value = distance from the mean measured in standard deviations

LCL UCL

See XII-2

VII-18

18181818

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)

Normal Curve Theory

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

VII-18

19191919

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

z Value

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

sxxzxz

See VIII-22-24

VII-19

20202020

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”?

z Value Example

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

21212121

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?

z Table Exercise

VII-19