Review-Ch 5 Statistics

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49 Terms

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Probabilities

the long run frequency of an event

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Probability Rules

  1. ANY Probability is 0ā‰¤p(x)ā‰¤1

  2. SUM of ALL probabilities is p(sum)=1

  • complement event, eventā€™=1-p(event)

    p(xc)=1-p(x)

  • the ā€˜ and power of c means the notation for the even not occurring

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Law of Large numbers

  • in the long run, a cumulative relative frequency is closer and closer to the true probability of an event

  • random sampling, the larger the sample, the closer the proportion of success will be to the proportion of the population.

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Simulation Process

  1. Assumptions

    ______ selected independently of EACH other

    ____%

  2. Assign digits & Model

  • Number 01-99, ignore 00 and excess

  • Numbers 00-99

  • single run

  • WHICH/HOW summary statistics will be recorded

  1. Stimulate multiple/many repetitions

  • Record results in a frequency distribution

  1. conclusion

  • Make sure it is in context

  • estimated probability as a decimal

*Try to write EVERY possible outcome

*flip sign and +- SAME constant from meanā†’ SAME probability

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Multiplication Principle

  • The TOTAL possible outcome na wants x nb way

  • Sampling with/without replacement

Math

Independent: P(A) P(B)

Dependent: P(A) * P(B | A)

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Conditional Probability

  • May NEED to restrict in denominator values because of the ā€œgiven thatā€ ā†’ shrinks the population of interest

Math

P(__| __): given that

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Mutually Exclusive

  • Events that cannot occur simultaneously

  • doesnā€™t imply independence

  • can ADD Probabilities

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Independant events

  • Results donā€™t affect EACH other

  • can multiply probabilities

  • can be assumed for large populations

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Dependent events

  • EXPLICITLY write down EACH value/trial because results affect each other

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Formula for mutually exclusive or not

Not mutually exclusive

Mutually exclusive

p(AUB)=p(A)+p(B)-p(Aāˆ©B)

p(AUB)=p(A)+p(B)

the mutually exclusive is because p(Aāˆ©B)=0 because events canā€™t happen at the SAME time

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p(Aāˆ©B)

Probability of A and B

Independent:p(Aāˆ©B)=p(A)*p(B)

Dependent: (Aāˆ©B)=p(A)*p(A|B)

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p(A U B)

Probability of A or B

Independent: p(A)+p(B)-p(Aāˆ©B)

Dependent: NONE

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p(A|B)

Probability of A given that B

Independent: p(A)

Dependent: p(Aāˆ©B)/p(B)

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Probability distribution

  • the number of practical outcomes of a CHANCE process

  • follows probability rules

  • set number of trials

  • x variable =number of success

  • Scenarios

    • if not ALL values are EXPLICITLY shown for probability ā†’ calculate what you can to solve

    • only several values, not allā†’ consider minimum, mean/median, maximum

<ul><li><p>the number of <strong>practical outcomes</strong> of a CHANCE process </p></li><li><p>follows <strong>probability</strong> rules </p></li><li><p><strong>set number</strong> of trials </p></li><li><p>x variable =<strong>number</strong> of success </p></li><li><p>Scenarios </p><ul><li><p>if <span style="color: red">not</span> ALL values are EXPLICITLY shown for <strong>probability</strong> ā†’ calculate what you <mark data-color="green">can</mark> to solve </p></li><li><p>only several values, <span style="color: red">not</span> allā†’ consider minimum, mean/median, maximum </p></li></ul></li></ul>
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Expected value

  • the net value on average

Math

ux=E(x)=np

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ux

  • the mean of x variable

Math

ux=Ī£xi *p(xi)

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Ļƒ x

  • the stndard deviation of x variable

Ļƒ x= āˆš (Ī£xi - ux )Ā²*p(xi)

sq root all

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( Ļƒ x)Ā²

  • variance

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finiding z score through norrmal distribution

  1. invnorm

  • given the % of area

  • using (o,1) as (u,Ļƒ ) for the default

  1. Z formula

  • z=(x-u)/Ļƒ

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C (Linear transformation )

C=shift in +-

  • only affects the mean

Math

uc+dx=|d|ux+c

Ļƒ c+dx=|d|Ļƒx

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D(Linear transformation )

  • A transformation of *

  • affects BOTH the mean and the standard deviation

Math

uc+dx=|d|ux+c

Ļƒ c+dx=|d|Ļƒx

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u mean (Combining sets)

  • u can also mean E(x), the expected vlalue

Math

ux+y=ux+uy

ux-y=ux-uy

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Ļƒ standard deviation (Combining sets)

  • for the formulas, need to be āˆš and have Ā² inside because standard deviations are ALWAYS positive

Math

Ļƒx+y=āˆš(ĻƒĀ²x+ĻƒĀ²y)

Ļƒx-y=āˆš(ĻƒĀ²x-ĻƒĀ²y)

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ĻƒĀ² varience (Combining sets)

  • Usually, a NEW variable is created which represents a total

  • remove the āˆš because variance is standard deviation Ā²

Math

ĻƒĀ²x+y=(ĻƒĀ²x+ĻƒĀ²y)

ĻƒĀ²x-y=(ĻƒĀ²x-ĻƒĀ²y)

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Linear Combination

  • when constants are multiplied in

  • for mean, just multiply in

  • for sd and variance, need to square values

Math

u= aux+bux

Ļƒ=āˆšaĀ²ĻƒxĀ²+ bĀ²ĻƒyĀ²

ĻƒĀ²=aĀ²ĻƒxĀ²+ bĀ²ĻƒyĀ²

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Binomial Distribution (Definition)

BINS

B: Binary(2 outcomes ONLY)

I: Independent(outcomes donā€™t affect impact EACH other)

N: Number of trials is FIXED

S: Success probability (is constant)

  • as n, the number of trials, increasesā†’ shape of distribution becomes MORE normal

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Binomial Distribution (steps)

STEPS

  1. Name the distribution(ie. binomialā€¦)

  2. Set Parameters(ie. n=, p=)

  3. Boundaries(eg. at leastā€¦)

  4. Direction(inequalities signs)

  5. Correct probability

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Binomial Distribution (formulaā€™s)

Math

ux=np

Ļƒxāˆš=np(1-p)

ĻƒxĀ²=np(1-p)

  • at p=0.5ā†’ variance increases as n increases

  • fixed n value, V is the maximum when p=0.5

<p><u>Math</u></p><p>u<sub>x</sub>=np</p><p>Ļƒ<sub>x</sub><mark data-color="yellow">āˆš</mark>=np(1-p)</p><p>Ļƒ<sub>x</sub>Ā²=np(1-p)</p><ul><li><p>at p=0.5ā†’ variance <span style="color: green">increases</span> as n <span style="color: green">increases</span></p></li><li><p>fixed n value, V is the <mark data-color="green">maximum</mark> when p=0.5</p></li></ul>
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Geometric Distribution (Definition)

BINS

B: Binary(2 outcomes ONLY)

I: Independent(outcomes donā€™t affect impact EACH other)

N: Number of trials is not FIXED

S: Success probability (is constant)

  • random variable, x, COUNTS trial NO. until the first success

    *most likely value is 1

  • expected trials till nth success = np

  • the probability of EACH success value decreases by a factor of q= 1-p

  • Shape: ALWAYS unimodal and skewed right

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Geometric Distribution (steps)

STEPS

  1. Name the distribution(ie. geometricā€¦)

  2. Set Parameters(ie. p=)

  3. The trial on which the first success occurs(x=)

  4. Correct probability

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Geometric Distribution (formulaā€™s)

Math

  • q=fail, p=success

q=1-p

ux= 1/p

Ļƒx= (āˆš 1-p)/p

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Probability that the 1st success occurs on the xth outcome

p(x=_)=geompdf(p= ,x=)

  • p=probability of a single trial, = first success occurrence

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more than x tries till first success(calculator)

p(x>_)=1- geomcdf(p= ,x=)

  • the 1- and the x=number means everything ā€œABOVEā€

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cumulative probability of x or BEFORE xth trail

p(x=_)=1geomcdf(p= ,x=)

  • p=probability of a single trial, = first success occurrence

  • n is not givenā†’leave empty

  • ā€œFirst success occurrence ON or BEFORE xth trailā€

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Probability of exactly x success

binompdf(n,p,x)

  • n=total no. of trial

  • p=probability of success of a single trial

  • x=number of wanted successā€™s

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Cumulative probability of f or FEWER success in n ways

binomcdf(n,p,x)

  • n=total no. of trial

  • p=probability of success of a single trial

  • x=c or fewer success

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At leastā€¦ binom

p=(xā‰„number)

1-binomcdf(n=,p=,x=number-1)

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fewer thanā€¦

(x<number)

binomcdf(n=,p=,x=number-1)

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<p>interpret the formula for binomial </p>

interpret the formula for binomial

(n on top of p) = n!/(k!q!)

  • n=total trial no.

  • k=wanted

  • p=probability

  • q=p-1

<p>(n on top of   p) = n!/(k!q!) </p><ul><li><p>n=total trial no.</p></li><li><p>k=wanted</p></li><li><p>p=probability </p></li><li><p>q=p-1</p></li></ul>
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Sample space

  • set of ALL possible outcomes

  • ie {H, T}

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Event

  • Outcome OR set of outcomes of a random thing

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Probability model

  • Assign probabilities to EACH event inn sample space

  • example

    • p(tails)=0.5

    • p(heads)=0.5

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Random

  • A regular pattern can be inferred if long repetitious trials are run, but for individual outcomesā†’this is uncertain

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Tree diagram

  • used for multi step problems

  • Label the start and then < grow on

<ul><li><p>used for multi step problems </p></li><li><p>Label the <strong>start</strong> and then &lt; grow on </p></li></ul>
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Venn diagrams

  • intersections overlapā†’ donā€™t ā€œrepeatā€ them, so may need to use subtraction

  • outside circle: ā€œnone of aboveā€

  • rectangle box=sample space

<ul><li><p>intersections overlapā†’ <span style="color: red">donā€™t</span> ā€œrepeatā€ them, so may need to use <span style="color: red">subtraction </span></p></li><li><p>outside circle: ā€œnone of aboveā€</p></li><li><p>rectangle box=<strong>sample space </strong></p></li></ul>
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U

add, union

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āˆ©

  • or

  • 1 event or BOTH occurring

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Relative frequency

  • proportion of times the event happened

number of times event happened/total number of trials

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Cumulative probability distribution

  • A function, table, or graph that links outcomes with the probability of less than or equal to that outcome occurring.

<ul><li><p>A function, table, or graph that links <strong>outcomes</strong> with the probability of <strong>less than or equal</strong> to that outcome occurring.</p></li></ul>