BPS - 5th Ed. Chapter 11 1 Sampling Distributions

BPS - 5th Ed.BPS - 5th Ed. Chapter 11Chapter 11 11

Chapter 11Chapter 11

Sampling Distributions Sampling Distributions

Chapter 11Chapter 11 22BPS - 5th Ed.BPS - 5th Ed.

Sampling TerminologySampling Terminology ParameterParameter

fixed, unknown number that describes the fixed, unknown number that describes the populationpopulation

StatisticStatistic known value calculated from a sampleknown value calculated from a sample a statistic is often used to estimate a parametera statistic is often used to estimate a parameter

VariabilityVariability different samples from the same population may different samples from the same population may

yield different values of the sample statisticyield different values of the sample statistic Sampling DistributionSampling Distribution

tells what values a statistic takes and how often it tells what values a statistic takes and how often it takes those values in repeated samplingtakes those values in repeated sampling


Parameter vs. StatisticParameter vs. Statistic

A properly chosen sample of 1600 people across the United States was asked if they regularly watch a certain television program (Real Housewives of New Jersey), and 24% said yes. The parameter of interest here is the true proportion of all people in the U.S. who watch the program, while the statistic is the value 24% obtained from the sample of 1600 people.


Parameter vs. StatisticParameter vs. Statistic

The mean of a population is denoted by µ – this is a parameter.

The mean of a sample is denoted by – this is a statistic. is used to estimate µ.

xx

The true proportion of a population with a certain trait is denoted by p – this is a parameter.

The proportion of a sample with a certain trait is denoted by (“p-hat”) – this is a statistic. is used to estimate p.

p̂ p̂


The Law of Large NumbersThe Law of Large Numbers

Consider sampling at random from a Consider sampling at random from a population with true mean population with true mean µµ. As the . As the number of (independent) number of (independent) observations sampled increases, the observations sampled increases, the mean of the sample gets closer and mean of the sample gets closer and closer to the true mean of the closer to the true mean of the population.population.( gets closer to µ ) x


The Law of Large NumbersThe Law of Large NumbersGamblingGambling

The “house” in a gambling operation is not The “house” in a gambling operation is not gambling at allgambling at all the games are defined so that the gambler has a the games are defined so that the gambler has a

negative expected gain per play (the true mean negative expected gain per play (the true mean gain after all possible plays is negative)gain after all possible plays is negative)

each play is independent of previous plays, so each play is independent of previous plays, so the the law of large numberslaw of large numbers guarantees that the guarantees that the average winnings of a large number of average winnings of a large number of customers will be close the (negative) true customers will be close the (negative) true averageaverage


Sampling DistributionSampling DistributionThe The sampling distributionsampling distribution of a of a

statistic is the distribution of values statistic is the distribution of values taken by the statistic in all possible taken by the statistic in all possible samples of the same size (samples of the same size (nn) from ) from the same populationthe same populationto describe a distribution we need to to describe a distribution we need to

specify the shape, center, and spreadspecify the shape, center, and spreadwe will discuss the distribution of the we will discuss the distribution of the

sample mean (x-bar) in this chaptersample mean (x-bar) in this chapter


Case StudyCase Study

Does This Cheese Smell Bad?

As cheese ages bacteria grows causing “off-odors”. Cheese makers want to know the odor threshold – the lowest concentration of a smell that the human nose can detect. Different people have different thresholds, and of interest is the mean threshold in the population of all adults.



Suppose the mean threshold of all adults is =25 micrograms of bacteria per ounce of cheese, with a standard deviation of =7 micrograms per ounce and the threshold values follow a bell-shaped (normal) curve.



Where should 95% of all Where should 95% of all individualindividual threshold values fall?threshold values fall?

mean plus or minus two standard mean plus or minus two standard deviationsdeviations

25 25 2(7) = 11 2(7) = 11

25 + 2(7) = 3925 + 2(7) = 39

95% should fall between 11 & 3995% should fall between 11 & 39

What about the What about the meanmean (average) of a sample (average) of a sample of of n n adults? What values would be expected?adults? What values would be expected?


Sampling DistributionSampling Distribution What about the What about the meanmean (average) of a sample of (average) of a sample of

n n adults? What values would be expected?adults? What values would be expected?

Answer this by thinking: Answer this by thinking: “What would “What would happen if we took many samples of happen if we took many samples of nn subjects from this population?”subjects from this population?” (let’s say (let’s say that n=10 subjects make up a sample)that n=10 subjects make up a sample) take a large number of samples of take a large number of samples of nn=10 =10

subjects from the populationsubjects from the population calculate the sample mean (x-bar) for each calculate the sample mean (x-bar) for each

samplesample make a histogram (or stemplot) of the values make a histogram (or stemplot) of the values

of x-barof x-bar examine the graphical display for shape, examine the graphical display for shape,

center, spreadcenter, spread



Mean threshold of all adults is =25 micrograms per ounce, with a standard deviation of =7 micrograms per ounce and the threshold values follow a bell-shaped (normal) curve.

Many (1000) repetitions of sampling n=10 adults from the population were simulated and the resulting histogram of the 1000x-bar values is on the next slide.



Case StudyCase StudyDoes This Cheese Smell Bad?


Mean and Standard Deviation of Mean and Standard Deviation of Sample MeansSample Means

OR OR “Means of the Means”“Means of the Means”

If numerous samples of size n are taken from

a population with mean and standard

deviation , then the mean of the sampling

distribution of is (the population mean)

and the standard deviation is:

( is the population s.d.) n

X


Mean and Standard Deviation of Mean and Standard Deviation of Sample MeansSample Means

Since the mean of is , we say that is

an unbiased estimator of X X

Individual observations have standard deviation , but sample means from samples of size n have standard deviation

. Averages are less variable than individual observations.

X

n


Sampling Distribution ofSampling Distribution ofSample MeansSample Means

If individual observations have the N(µ, )

distribution, then the sample mean of n

independent observations has the N(µ, / )

distribution.

X

n

“If measurements in the population follow a Normal distribution, then so does the sample mean.”

When do I use the adjusted standard When do I use the adjusted standard deviation? deviation?

When evaluating Samples: When evaluating Samples: When evaluating an When evaluating an individual:individual:

n

We can compare an individual to the entire population. For example, what is the probability that one student from MHS will get a 2000 on the SAT?

We can compare a sample to the performance of past samples of the population. For example: what is the probability that a sample of 20 students from MHS will get an average of 1800 on the SAT?



Mean threshold of all adults is =25 with a standard deviation of =7, and the threshold values follow a bell-shaped (normal) curve.


(Population distribution)


Central Limit TheoremCentral Limit Theorem

“No matter what distribution the population values follow, the sample mean will follow a Normal distribution if the sample size is large.”

If a random sample of size n is selected from ANY population with mean and standard

deviation , then when n is large the sampling distribution of the sample mean is approximately Normal:

is approximately N(µ, / )

X

nX


Central Limit Theorem:Central Limit Theorem:Sample SizeSample Size

How large must How large must nn be for the CLT to be for the CLT to hold?hold?depends on how far the population depends on how far the population

distribution is from Normaldistribution is from Normalthe further from Normal, the larger the the further from Normal, the larger the

sample size neededsample size neededa sample size of a sample size of 25 or 3025 or 30 is typically large is typically large

enough for any population distribution enough for any population distribution encountered in practice encountered in practice

recall: if the population is Normal, any recall: if the population is Normal, any sample size will work (sample size will work (nn≥1)≥1)


Central Limit Theorem:Central Limit Theorem:Sample Size and Distribution of Sample Size and Distribution of x-x-

barbar

n=1

n=25n=10

n=2


Statistical Process ControlStatistical Process Control

Goal is to make a process stable over Goal is to make a process stable over time and keep it stable unless there time and keep it stable unless there are planned changesare planned changes

All processes have variationAll processes have variationStatistical description of stability over Statistical description of stability over

time: the time: the pattern of variationpattern of variation remains stable (does not say that remains stable (does not say that there is there is nono variation) variation)


Statistical Process ControlStatistical Process Control A variable described by the same A variable described by the same

distribution over time is said to be distribution over time is said to be in controlin control To see if a process has been disturbed and To see if a process has been disturbed and

to signal when the process is to signal when the process is out of controlout of control, , control chartscontrol charts are used to monitor the are used to monitor the processprocessdistinguish natural variation in the process from distinguish natural variation in the process from

additional variation that suggests a changeadditional variation that suggests a changemost common application: industrial processesmost common application: industrial processes


ChartsCharts There is a true mean There is a true mean that describes the that describes the

center or aim of the processcenter or aim of the process Monitor the process by plotting the meansMonitor the process by plotting the means

(x-bars) of small samples taken from the (x-bars) of small samples taken from the process at regular intervals over timeprocess at regular intervals over time

Process-monitoring conditionsProcess-monitoring conditions:: measure quantitative variable measure quantitative variable xx that is Normal that is Normal process has been operating in control for a long process has been operating in control for a long

periodperiod know process mean know process mean and standard deviation and standard deviation

that describe distribution of that describe distribution of xx when process is in when process is in controlcontrol

x


Control ChartsControl ChartsPlot the means (x-bars) of regular Plot the means (x-bars) of regular

samples of size n against timesamples of size n against timeDraw a horizontal Draw a horizontal center linecenter line at at

x

Draw horizontal Draw horizontal control limitscontrol limits at at ± 3 ± 3//n almost all (99.7%) of the values of x-bar should almost all (99.7%) of the values of x-bar should

be within the mean plus or minus 3 standard be within the mean plus or minus 3 standard deviationsdeviations

Any x-bar that does not fall between the Any x-bar that does not fall between the control limits is evidence that the control limits is evidence that the process is out of controlprocess is out of control



Need to control the tension in millivolts Need to control the tension in millivolts (mV) on the mesh of fine wires behind the (mV) on the mesh of fine wires behind the surface of the screen. surface of the screen.

Proper tension is Proper tension is 275275 mV ( mV (target mean target mean ))When in control, the standard deviation of When in control, the standard deviation of

the tension readings is the tension readings is =43=43 mV mV

Making Computer Monitors



Proper tension is Proper tension is 275275 mV ( mV (target mean target mean ). When in control, ). When in control, the standard deviation of the tension readings is the standard deviation of the tension readings is =43=43 mV. mV.


Take samples of Take samples of nn=4 screens and =4 screens and calculate the means of these samplescalculate the means of these samples

the control limits of the x-bar control chart the control limits of the x-bar control chart would bewould be

339.5 and 210.5

64.527532753μ4

43n

σ




(data)



Making Computer Monitors( chart)x

(in control)


Case StudyCase StudyMaking Computer Monitors

(examples of out of control processes)


Natural TolerancesNatural Tolerances

For x-bar charts, the For x-bar charts, the control limitscontrol limits for for the the meanmean of the process are of the process are ± 3 ± 3//n almost all (99.7%) of the values of almost all (99.7%) of the values of x-barx-bar should should

be within the mean plus or minus 3 standard be within the mean plus or minus 3 standard deviationsdeviations

When monitoring a process, the When monitoring a process, the natural natural tolerancestolerances for for individualindividual products are products are ± 3± 3 almost all (99.7%) of the almost all (99.7%) of the individual individual

measurementsmeasurements should be within the mean plus should be within the mean plus or minus 3 standard deviationsor minus 3 standard deviations

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BPS - 5th Ed. Chapter 11 1 Sampling Distributions