Life is a school of probability... Walter Bagehot (English Economist) I don't believe in providence...

Life is a school of probability...Life is a school of probability... Walter Bagehot (English Economist)Walter Bagehot (English Economist)

I don't believe in providence and fate, as a I don't believe in providence and fate, as a technologist I am used to reckoning with the technologist I am used to reckoning with the formulae of probability…formulae of probability…

Max Frisch (German Architect and Novelist)Max Frisch (German Architect and Novelist)

AP StatisticsAP Statistics Sampling Distributions? Sampling Distributions?

Braser–Braser Chapter 7.1Braser–Braser Chapter 7.1

§ Review of Statistical TermsReview of Statistical Terms

– PopulationPopulation, from a statistical point of view, is , from a statistical point of view, is considered as a set of measurements or considered as a set of measurements or counts, existing or conceptualcounts, existing or conceptual

– SampleSample is a subset of measurements from is a subset of measurements from the population. Random Samples are the population. Random Samples are considered for this sectionconsidered for this section



§ Review of Statistical TermsReview of Statistical Terms

– ParameterParameter is a numerical descriptive measure is a numerical descriptive measure of a population. In statistical practice the value of a population. In statistical practice the value of a parameter is not know, it is not possible to of a parameter is not know, it is not possible to examine the entire populationexamine the entire population

– StatisticStatistic is a numerical descriptive measure is a numerical descriptive measure of a sample, not depending of any unknown of a sample, not depending of any unknown parameter. An statistic is used to estimate an parameter. An statistic is used to estimate an unknown parameterunknown parameter

§ Common Statistics and ParametersCommon Statistics and Parameters

MeasureMeasure StatisticStatistic



ParameterParameter

MeanMean

VarianceVariance

Standard DeviationStandard Deviation

x 2s 2s

ProportionProportion p̂ p

§ Why Sample?Why Sample?

At times, we’d like to know something about the At times, we’d like to know something about the population, but because our time, resources, population, but because our time, resources, and efforts are limited, we can take just a and efforts are limited, we can take just a sample to learn about the populationsample to learn about the population

Ex: Take a sample of voters to learn about Ex: Take a sample of voters to learn about probable election results (before the final probable election results (before the final count).count).



§ Why Sample?Why Sample?



InferenceInference is to draw conclusions for a entire is to draw conclusions for a entire population from the information of a samplepopulation from the information of a sample

So we must use measurements from a sample So we must use measurements from a sample instead. In such cases, we will use a statistic ( instead. In such cases, we will use a statistic ( , , s, or s, or ) to make ) to make inferencesinferences about corresponding about corresponding population parameters (population parameters (, , , or , or pp))

xp̂

§ Types of InferenceTypes of Inference

Estimation.Estimation. In this case, we In this case, we estimateestimate or or approximateapproximate the value of a population parameter the value of a population parameter

TestingTesting: In this case, we : In this case, we formulateformulate a decision a decision about a population parameterabout a population parameter



RegressionRegression: In this case, we make: In this case, we make predictions predictions or or forecastsforecasts about the value of a statistical about the value of a statistical variablevariable

§ Are Inferences Reliable?Are Inferences Reliable?

To evaluate the reliability of our inference, we To evaluate the reliability of our inference, we need to know about the probability distribution need to know about the probability distribution of the statistic we are usingof the statistic we are using



Typically, we are interested in the sampling Typically, we are interested in the sampling distributions for sample means and sample distributions for sample means and sample proportionsproportions

§ Sampling DistributionsSampling Distributions

A A Sampling DistributionSampling Distribution is a probability is a probability distribution of a sample statistic based on all distribution of a sample statistic based on all possible simple random samples of the possible simple random samples of the same same sizesize from the same population from the same population



§ Sampling Distributions ExampleSampling Distributions Example

In a rural community with a children’s fishing pond, In a rural community with a children’s fishing pond, are posted rules stating that all fish under 6 inches are posted rules stating that all fish under 6 inches must be returned to the pond, and the limit of five fish must be returned to the pond, and the limit of five fish per day may be kept. 100 random samples of five per day may be kept. 100 random samples of five trout are taken and recorded the lengths of the five trout are taken and recorded the lengths of the five trout. What is the average (mean) length of a trout trout. What is the average (mean) length of a trout taken from the pond (taken from the pond (textbook pp.362 table 7-1))



PracticePractice

• Textbook Section 8.1 Problems: pp. 365 Textbook Section 8.1 Problems: pp. 365

Checking for UnderstandingChecking for Understanding

HM STAT Space (7.1)HM STAT Space (7.1)



Information: the negative reciprocal value Information: the negative reciprocal value of probability…of probability… Claude Shannon (American Mathematical Engineer)Claude Shannon (American Mathematical Engineer)

From principles is derived probability, but From principles is derived probability, but truth or certainty is obtained only from facts. truth or certainty is obtained only from facts.

Tom Stoppard (English Playwriths)Tom Stoppard (English Playwriths)

There is an old saying: All roads lead to Rome. In Statistics we can recast this saying: All probability distributions average out to the Normal distribution, (as the sample size increases)

§ The Central Limit Theorem (Normal) The Central Limit Theorem (Normal)

AP StatisticsAP Statistics The Central Limit Theorem The Central Limit Theorem


For a Normal Probability Distribution, let x be a random variable with a normal distribution whose mean is , and whose standard deviation is . Let be the sample mean corresponding to random samples of size n taken form the x distribution. Then the following is true:

x

– The distribution is a normal distributionThe distribution is a normal distributionx– The mean of the distribution is The mean of the distribution is x– The standard deviation of the distribution is:The standard deviation of the distribution is:x

n

§ Central Limit Theorem. Converting to Central Limit Theorem. Converting to zz

nn is the sample size is the sample size is the mean of the is the mean of the xx distribution distribution is the standard deviation of the is the standard deviation of the xx distribution distribution

x



x

We can convert the distribution to the standard normal z distribution using the following formulas

x

nx

n

xxz

x

x



§ Sample Size ConsiderationsSample Size Considerations

For the Central Limit Theorem (CLT) to be applicable:For the Central Limit Theorem (CLT) to be applicable:

– If the If the xx distribution is symmetric or reasonably distribution is symmetric or reasonably symmetric, symmetric, nn ≥ 30 ≥ 30 should suffice should suffice

– If the If the xx distribution is highly skewed or unusual, distribution is highly skewed or unusual, even larger sample sizes will be requiredeven larger sample sizes will be required

– If possible, make a graph to visualize how the If possible, make a graph to visualize how the sampling distribution is behavingsampling distribution is behaving



§ Central Limit Theorem. (Any Distribution)Central Limit Theorem. (Any Distribution)

If If xx posses any distribution with mean posses any distribution with mean and and standard deviation standard deviation , then the sample mean , then the sample mean based on a random sample of size based on a random sample of size nn will have a will have a normal distribution that approaches the normal distribution that approaches the distribution of a normal random variable with distribution of a normal random variable with mean mean and standard deviation , as and standard deviation , as nn increases without limitincreases without limit

x

n



§ Finding Probabilities Using Central Limit TheoremFinding Probabilities Using Central Limit Theorem

Given a probability distribution of Given a probability distribution of xx values with values with sample size sample size n,n, mean mean , and standard deviation , and standard deviation : :

– If the If the xx distribution is normal, then thedistribution is normal, then the

– Even if the Even if the x x distribution is not normal, if the distribution is not normal, if the sample of the size is sample of the size is nn 3030, then by , then by CLTCLT, , thethe

x

x

distribution is normaldistribution is normal

distribution is approximately normaldistribution is approximately normal




Given a probability distribution of Given a probability distribution of xx values with values with mean mean , standard deviation , standard deviation , and sample of size , and sample of size nn

– ConvertConvert

x

xxz

n

x

x to to zz using the formula: using the formula:




Given a probability distribution of Given a probability distribution of xx values with values with mean mean , standard deviation , standard deviation , and sample of size , and sample of size nn

– Use the standard normal distribution to find the Use the standard normal distribution to find the corresponding probability for the events corresponding probability for the events regardingregarding x



§ Central Limit Theorem. ExampleCentral Limit Theorem. Example

The heights of 18-year old men are approximately The heights of 18-year old men are approximately normally distributed, with a mean normally distributed, with a mean = 68 inches = 68 inches and a standard deviation and a standard deviation = 3 inches = 3 inches

a. What is the probability that a randomly a. What is the probability that a randomly selected man is taller than 72 inches?selected man is taller than 72 inches?

33.13

6872

zGet the z score:

Find Probability: P(z>72) = 1 – P(z<72) = .0918




The heights of 18-year old men are approximately The heights of 18-year old men are approximately normally distributed, with a mean normally distributed, with a mean = 68 in and a = 68 in and a standard deviation standard deviation = 3 in = 3 in

b. What’s the b. What’s the probabilityprobability that the average height of that the average height of 2 randomly selected men is greater than 72 in?2 randomly selected men is greater than 72 in?

Using CLT with n = 2:

68 xx 3

2.1212

xn

72 681.89

2.121z

( 72) ( 1.89) 0.0294P x P z




The heights of 18-year old men are approximately The heights of 18-year old men are approximately normally distributed, with a mean normally distributed, with a mean = 68 in and a = 68 in and a standard deviation standard deviation = 3 in = 3 in

c. What is the probability that the average height of c. What is the probability that the average height of 16 randomly selected men is greater than 72 in?16 randomly selected men is greater than 72 in?

68 xx 75.016

3

nx

33.575.0

6872

z 000.0)33.5()72( zPxP

Using CLT with n = 16:

PracticePractice

• Textbook Section 7.2 Problems: pp. 373 – 379 Textbook Section 7.2 Problems: pp. 373 – 379





The scientific imagination always restrains The scientific imagination always restrains itself within the limits of probability...itself within the limits of probability... Thomas Huxley (English Biologist)Thomas Huxley (English Biologist)

A property in the 100-year floodplain has a 96 A property in the 100-year floodplain has a 96 percent chance of being flooded in the next percent chance of being flooded in the next hundred years without global warming. The hundred years without global warming. The fact that several years go by without a flood fact that several years go by without a flood does not change that probability...does not change that probability...

Earl Blumenauer (Oregon Representative)Earl Blumenauer (Oregon Representative)

Many issues in life come down to success or failure. In most cases, we will not be successful all the time, so proportions of successes are very important. What is the probability sampling distributions for proportions?…

The annual crime rate in the Capital Hill of Denver is 111 victims per 1000 residents. (111 out of 1000 residents have been victim of a least one crime). These crimes range from minor crimes (stolen hubcaps or purse snatching) to major crimes (murder). The Arms is an apartment building on this neighborhood that has 50 year-round residents. Consider each of the n = 50 residents as a binomial trial. The random variable r, (1 = r = 50), represents the number of victims of a least one crime next year.

(a) What is the population probability p that a resident a resident in the Capital Hill neighborhood will be / will not be a victim of a crime?

(b) What is the probability that between 10% and 20% of the Arms residents will be victims of a crime next year?

Hint: Use the binomial distribution. Use the normal approach to the binomial. Compare answers

§ Sampling Distribution for the ProportionSampling Distribution for the Proportion

Given:Given: nn = number of binomial trials (constant) = number of binomial trials (constant)

If If npnp > > 55 and and nqnq > > 55, then the random variable, then the random variable

AP StatisticsAP Statistics Sampling Distributions for Proportions Sampling Distributions for Proportions


nrp ˆ

rr = number of successes = number of successes

pp = probability of success on each trial = probability of success on each trial

qq = = 1 1 – p– p = probability of failure on each trial = probability of failure on each trial

can be approximated by a normal random variable can be approximated by a normal random variable xx with with

pp ˆn

pqp ˆ

nrp ˆ

§ Continuity CorrectionsContinuity Corrections

Since is discrete, but Since is discrete, but xx is continuous, we have to is continuous, we have to make a continuity correction; for a small make a continuity correction; for a small nn, the , the correction is meaningfulcorrection is meaningful



How to make corrections toHow to make corrections to

1.1. If If rr//nn is the right end point of a is the right end point of a

p̂ intervalsintervals

addadd 0.5/ 0.5/nn to get the corresponding right end to get the corresponding right end point of the point of the xx interval interval

interval, weinterval, wep̂

§ Continuity CorrectionsContinuity Corrections

Since is discrete, but Since is discrete, but xx is continuous, we have to is continuous, we have to make a continuity correction; for a small make a continuity correction; for a small nn, the , the correction is meaningfulcorrection is meaningful



How to make corrections toHow to make corrections to

2.2. If If rr//nn is the left end point of a is the left end point of a

p̂ intervalsintervals

subtractsubtract 0.5/ 0.5/nn to get the corresponding left end to get the corresponding left end point of the point of the xx interval interval

interval, weinterval, wep̂

§ Proportion Sampling Distribution. ExampleProportion Sampling Distribution. Example

Suppose the annual crime rate in Denver is Suppose the annual crime rate in Denver is pp = 0.111 = 0.111 If 50 people live in an apartment complex, what is the If 50 people live in an apartment complex, what is the probability that between 10% and 20% of the residents probability that between 10% and 20% of the residents will be victims of crimes next year?will be victims of crimes next year?



p̂

Checking conditions (Checking conditions (npnp >5, >5, nqnq > 5 > 5):):

npnp = (50)(.111) = 5.55 = (50)(.111) = 5.55 nq nq = (50)(.889) = 44.45 = (50)(.889) = 44.45

can be approximated with a normal distribution

n = n = 5050, p = , p = 0.1110.111, q = , q = 11 – p = – p = 1 1 –– 0.111 0.111 = = 0.8990.899





n = n = 5050, p = , p = 0.1110.111, q = , q = 0.8990.899

111.0ˆ pp

044.050

)889.0)(111.0(ˆ

n

pqp





n = n = 5050, p = , p = 0.1110.111, q = , q = 0.8990.899

Continuity Correction (Continuity Correction (0.50.5/n/n):): 0.5/50 = 0.010.5/50 = 0.01

)20.0ˆ10.0( pP )21.009.0( xP





n = n = 5050, p = , p = 0.1110.111, q = , q = 0.8990.899

Using z-scores:Using z-scores:

)21.009.0( xP 6722.0)25.248.0( zP

= 0.111= 0.111 , , = 0.044= 0.044

25.2044.0

111.021.02

z48.0

044.0

111.009.01

z





n = n = 5050, p = , p = 0.1110.111, q = , q = 0.8990.899

6722.0)20.0ˆ10.0( pP

Thus, there is about a 67% chance that between 10% and 20% of the residents will be victims of a crime next year.

§ Control Charts for ProportionsControl Charts for Proportions

Used to examine an attribute or quality of an Used to examine an attribute or quality of an observation (rather than a measurement).observation (rather than a measurement).



– Then use the normal approximation of the Then use the normal approximation of the sample proportion to determine the control limitssample proportion to determine the control limits

– Select a fixed sample size, n, at fixed time Select a fixed sample size, n, at fixed time intervals, and determine the sample proportions intervals, and determine the sample proportions at each intervalat each interval

How to use it:How to use it:

§ How to Make a P-ChartHow to Make a P-Chart

1. Estimate1. Estimate



3. Control limits are located at:3. Control limits are located at:

2. Take the center line of control chart as:2. Take the center line of control chart as:

p , the overall proportion of successes, the overall proportion of successes

samplesallintrialsofnumberTotal

samplesallinsuccessesobservedofnumberTotalp

pp ˆ

n

qpp 2

n

qpp 3andand

§ P-Chart. Out of Control SignalsP-Chart. Out of Control Signals

Signal 1.Signal 1.



At least two out of three consecutive At least two out of three consecutive points are beyond the control limitspoints are beyond the control limits

Run of nine consecutive points on one Run of nine consecutive points on one side of the center lineside of the center line

Any point beyondAny point beyond

pp ˆ

n

qpp 2

n

qpp 3 control limitcontrol limit

Signal 2.Signal 2.

Signal 3.Signal 3.

§ P-Chart. Out of Control SignalsP-Chart. Out of Control Signals



If no out-of-control signals occur, we say that the If no out-of-control signals occur, we say that the process is in control, while keeping a watchful process is in control, while keeping a watchful eye on what occurs nexteye on what occurs next

In some P-Charts the value ofIn some P-Charts the value ofIn this case, the control limits may drop below 0 or In this case, the control limits may drop below 0 or rise above 1. If this happens, follow the convention rise above 1. If this happens, follow the convention of rounding negative control limits to 0 and control of rounding negative control limits to 0 and control limits above 1 to 1limits above 1 to 1

may be near 0 or 1 may be near 0 or 1 p

§ Control Charts for Proportions. ExampleControl Charts for Proportions. Example ((pp. 384)pp. 384)

(a) Estimate the overall proportion of successes(a) Estimate the overall proportion of successes



samplesallintrialsofnumberTotal

samplesallinsuccessesobservedofnumberTotalp

175.0840

147

)60(14

10...8129

p


(b) Calculate (b) Calculate



p̂p̂and and

175.0ˆ ppp

049.60

)825.0)(175.0(ˆ

n

qp

n

pqp


(c) Estimate(c) Estimate



andandpn qn

5.10)175.0(60 pn 5.49)825.0(60 qn

Both are greater than 5, this means the normal Both are greater than 5, this means the normal distribution should be reasonable gooddistribution should be reasonable good


(d) Estimate the control limits of the P-Chart(d) Estimate the control limits of the P-Chart



)49.0(2175.02 n

qpp 273.0077.0 and


(d) Estimate the control limits of the P-Chart(d) Estimate the control limits of the P-Chart



)49.0(3175.03 n

qpp 322.0028.0 and




The Proportion of A’s given in class is in statistical The Proportion of A’s given in class is in statistical control, with exception of the one unusually good class control, with exception of the one unusually good class two semesters agotwo semesters ago

Out of control signalsOut of control signals

Signal 1. Semester 12 above 3s level (Very good class!)Signal 1. Semester 12 above 3s level (Very good class!)

Signal 2. Not presentSignal 2. Not present

Signal 3. Not presentSignal 3. Not present

PracticePractice

• Textbook Section 7.3 Problems: pp. 387 – 389 Textbook Section 7.3 Problems: pp. 387 – 389






Braser–Braser Chapter 7.3 pp 389Braser–Braser Chapter 7.3 pp 389

Custom Shows

§ Sampling Distributions ExampleSampling Distributions Example



AP StatisticsAP Statistics Normal Approximation to Binomial DistributionNormal Approximation to Binomial Distribution


§ Normal Approximation to Binomial ErrorNormal Approximation to Binomial Error

The error of the normal approximation to the binomial The error of the normal approximation to the binomial distribution decreases and becomes negligible as the distribution decreases and becomes negligible as the number of trials number of trials nn increases increases

However, if the number of trials is not big, the error in However, if the number of trials is not big, the error in this approximation can not be ignored…this approximation can not be ignored…



Binomial Probability

Normal Approach

?)105( rP n = 50

p = 0.111

q = 0.889

= 6.555

= 2.221

5 10

+ Continuity Correction



Binomial Probability

Normal Approach

0.1 0.2

+ Continuity Correction

?)20.0ˆ10.0( pP

044.0

111.0

889.ˆ

111.0ˆ

50

ˆ

ˆ

p

p

q

p

n

nrp /ˆ



§ Normal Approximation to BinomialNormal Approximation to BinomialContinuity Correction forContinuity Correction for

Step 1.Step 1. If If pp is a is a left-pointleft-point of an interval, of an interval, subtractsubtract 0.5/0.5/nn to obtain the corresponding random variable to obtain the corresponding random variable xx::

Step 2.Step 2. If If pp is a is a right-pointright-point of an interval, of an interval, addadd 0.5/0.5/nn to to obtain the corresponding random variable obtain the corresponding random variable xx::

np

n

rx

5.0ˆ

5.0

np

n

rx

5.0ˆ

5.0

nrp /ˆ

^

^

Life is a school of probability... Walter Bagehot (English Economist) I don't believe in providence...

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Transcript of Life is a school of probability... Walter Bagehot (English Economist) I don't believe in providence...