Post on 26-Dec-2015
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Section VII, Measure -
Probability
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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
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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
50.2
1
11
1
tailsand heads #
heads #)(
# #
#
#
#)(
of
ofheadP
failuresofsuccessesof
successesof
iespossibilitof
successesofsuccessP
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
)!(!
!
!
!
rrn
nC
rn
nP
rn
rn
Number of Ways
r
n
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%
-1s-2s-3s +1s +2s +3s0
s± ≈68%1
s± 99.73%3
s± ≈95%2
z value = distance from the mean measured in standard deviations
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.
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.
Remember when we talked about 3? The 3 is the z value.
s
xxz
xz
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.
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