Objectives (BPS chapter 11) Sampling distributions Parameter versus statistic The law of large...

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Objectives (BPS chapter 11) Sampling distributions Parameter versus statistic The law of large numbers What is a sampling distribution? The sampling distribution of The central limit theorem Statistical process control x

Transcript of Objectives (BPS chapter 11) Sampling distributions Parameter versus statistic The law of large...

Page 1: Objectives (BPS chapter 11) Sampling distributions  Parameter versus statistic  The law of large numbers  What is a sampling distribution?  The sampling.

Objectives (BPS chapter 11)

Sampling distributions

Parameter versus statistic

The law of large numbers

What is a sampling distribution?

The sampling distribution of

The central limit theorem

Statistical process control

x

Page 2: Objectives (BPS chapter 11) Sampling distributions  Parameter versus statistic  The law of large numbers  What is a sampling distribution?  The sampling.

Reminder: Parameter versus statistic Sample: the part of the

population we actually examine

and for which we do have data.

A statistic is a number

describing a characteristic of a

sample. We often use a statistic

to estimate an unknown

population parameter.

Population: the entire group of

individuals in which we are

interested but can’t usually

assess directly.

A parameter is a number

describing a characteristic of

the population. Parameters

are usually unknown.

Population

Sample

Page 3: Objectives (BPS chapter 11) Sampling distributions  Parameter versus statistic  The law of large numbers  What is a sampling distribution?  The sampling.

The law of large numbersLaw of large numbers: As the number of randomly-drawn observations (n) in a sample increases,

the mean of the sample ( ) gets closer and closer to the population mean (quantitative variable).

the sample proportion ( ) gets closer and closer to the population proportion p (categorical variable).

x p̂

Page 4: Objectives (BPS chapter 11) Sampling distributions  Parameter versus statistic  The law of large numbers  What is a sampling distribution?  The sampling.

What is a sampling distribution?The sampling distribution of a statistic is the distribution of all

possible values taken by the statistic when all possible samples of a

fixed size n are taken from the population. It is a theoretical idea—we

do not actually build it.

The sampling distribution of a statistic is the probability distribution of

that statistic.

Note: When sampling randomly from a given population,

the law of large numbers describes what happens when the sample size n

is gradually increased.

The sampling distribution describes what happens when we take all

possible random samples of a fixed size n.

Page 5: Objectives (BPS chapter 11) Sampling distributions  Parameter versus statistic  The law of large numbers  What is a sampling distribution?  The sampling.

Sampling distribution of (the sample

mean)

We take many random samples of a given size n from a population

with mean and standard deviation

Some sample means will be above the population mean and some

will be below, making up the sampling distribution.

Sampling distribution of “x bar”

Histogram of some sample

averages

x

Page 6: Objectives (BPS chapter 11) Sampling distributions  Parameter versus statistic  The law of large numbers  What is a sampling distribution?  The sampling.

Sampling distribution of

√n

For any population with mean and standard deviation :

The mean, or center of the sampling distribution of , is equal to the

population mean .

The standard deviation of the sampling distribution is /√n, where n

is the sample size.

x

x

Page 7: Objectives (BPS chapter 11) Sampling distributions  Parameter versus statistic  The law of large numbers  What is a sampling distribution?  The sampling.

Mean of a sampling distribution of :

There is no tendency for a sample mean to fall systematically above or

below even if the distribution of the raw data is skewed. Thus, the mean of

the sampling distribution of is an unbiased estimate of the population

mean —it will be “correct on average” in many samples.

Standard deviation of a sampling distribution of :

The standard deviation of the sampling distribution measures how much the

sample statistic varies from sample to sample. It is smaller than the

standard deviation of the population by a factor of √n. Averages are less

variable than individual observations.

x

x

x

x

Page 8: Objectives (BPS chapter 11) Sampling distributions  Parameter versus statistic  The law of large numbers  What is a sampling distribution?  The sampling.

For normally distributed populationsWhen a variable in a population is normally distributed, then the sampling distribution of for all possible samples of size n is also normally distributed.

If the population is N(), then the sample means distribution is N(/√n).

Population

Sample means

x

Page 9: Objectives (BPS chapter 11) Sampling distributions  Parameter versus statistic  The law of large numbers  What is a sampling distribution?  The sampling.

IQ scores: population vs. sample

In a large population of adults, the mean IQ is 112 with standard deviation 20.

Suppose 200 adults are randomly selected for a market research campaign.

The distribution of the sample mean IQ is 

A) exactly normal, mean 112, standard deviation 20. 

B) approximately normal, mean 112, standard deviation 20. 

C) approximately normal, mean 112 , standard deviation 1.414. 

D) approximately normal, mean 112, standard deviation 0.1.

C) approximately normal, mean 112, standard deviation 1.414. 

Population distribution: N ( = 112; = 20)

Sampling distribution for n = 200 is N ( = 112; /√n = 1.414)

Page 10: Objectives (BPS chapter 11) Sampling distributions  Parameter versus statistic  The law of large numbers  What is a sampling distribution?  The sampling.

Application

Hypokalemia is diagnosed when blood potassium levels are low, below

3.5mEq/dl. Let’s assume that we know a patient whose measured potassium

levels vary daily according to a normal distribution N( = 3.8, = 0.2).

If only one measurement is made, what's the probability that this patient will be

misdiagnosed hypokalemic?

2.0

8.35.3)(

x

z z = 1.5, P(z < 1.5) = 0.0668 ≈ 7%

If instead measurements are taken on four separate days, what is the probability of such a misdiagnosis?

42.0

8.35.3)(

n

xz

z = 3, P(z < 1.5) = 0.0013 ≈ 0.1%

Note:

Make sure to standardize (z) using the standard deviation for the sampling distribution.

Page 11: Objectives (BPS chapter 11) Sampling distributions  Parameter versus statistic  The law of large numbers  What is a sampling distribution?  The sampling.

Practical note

Large samples are not always attainable.

Sometimes the cost, difficulty, or preciousness of what is studied limits

drastically any possible sample size.

Blood samples/biopsies: no more than a handful of repetitions

acceptable. Often we even make do with just one.

Opinion polls have a limited sample size due to time and cost of

operation. During election times, though, sample sizes are increased

for better accuracy.

Not all variables are normally distributed. Income is typically strongly skewed for example.

Is still a good estimator of then?

x

Page 12: Objectives (BPS chapter 11) Sampling distributions  Parameter versus statistic  The law of large numbers  What is a sampling distribution?  The sampling.

The central limit theorem

Central Limit Theorem: When randomly sampling from any population

with mean and standard deviation , when n is large enough, the

sampling distribution of is approximately normal: N(/√n).

Population with strongly skewed

distribution

Sampling distribution of

for n = 2 observations

Sampling distribution of

for n = 10 observations

Sampling distribution of for n = 25 observations

x

x

x

x

Page 13: Objectives (BPS chapter 11) Sampling distributions  Parameter versus statistic  The law of large numbers  What is a sampling distribution?  The sampling.

Income distribution

Let’s consider the very large database of individual incomes from the Bureau of

Labor Statistics as our population. It is strongly right-skewed.

We take 1000 SRSs of 100 incomes, calculate the sample mean for

each, and make a histogram of these 1000 means.

We also take 1000 SRSs of 25 incomes, calculate the sample mean for

each, and make a histogram of these 1000 means.

Which histogram

corresponds to the

samples of size

100? 25?

Page 14: Objectives (BPS chapter 11) Sampling distributions  Parameter versus statistic  The law of large numbers  What is a sampling distribution?  The sampling.

In many cases, n = 25 isn’t a huge sample. Thus,

even for strange population distributions we can

assume a normal sampling distribution of the

mean, and work with it to solve problems.

How large a sample size?

It depends on the population distribution. More observations are

required if the population distribution is far from normal.

A sample size of 25 is generally enough to obtain a normal sampling

distribution from a strong skewness or even mild outliers.

A sample size of 40 will typically be good enough to overcome extreme

skewness and outliers.

Page 15: Objectives (BPS chapter 11) Sampling distributions  Parameter versus statistic  The law of large numbers  What is a sampling distribution?  The sampling.

Statistical process control

Industrial processes tend to have normally distributed variability, in part

as a consequence of the central limit theorem applying to the sum of

many small influential factors. Random samples taken over time can

thus be used to easily verify that a given process is not getting out of

“control.”

What is statistical control?

A variable that continues to be described by the same distribution when

observed over time is said to be in statistical control, or simply in

control.

Page 16: Objectives (BPS chapter 11) Sampling distributions  Parameter versus statistic  The law of large numbers  What is a sampling distribution?  The sampling.

Process-monitoring

What are the required conditions?

We measure a quantitative variable x that has a normal distribution.

The process has been operating in control for a long period, so that we

know the process mean µ and the process standard deviation σ that

describe the distribution of x as long as the process remains in control.

An control chart displays the average of samples of size n taken

at regular intervals from such a process. It is a way to monitor the

process and alert us when it has been disturbed so that it is now out of

control. This is a signal to find and correct the cause of the

disturbance.

x

Page 17: Objectives (BPS chapter 11) Sampling distributions  Parameter versus statistic  The law of large numbers  What is a sampling distribution?  The sampling.

control charts

For a process with known mean µ standard deviation σ, we calculate

the mean of samples of constant size n taken at regular intervals.

Plot (vertical axis)

against time (horizontal axis).

Draw a horizontal center

line at µ.

Draw two horizontal

control limits at µ ± 3σ/√n

(UCL and LCL).

x

x

x

Page 18: Objectives (BPS chapter 11) Sampling distributions  Parameter versus statistic  The law of large numbers  What is a sampling distribution?  The sampling.

A machine tool cuts circular pieces. A sample of four pieces is

taken hourly, giving these average measurements (in 0.0001

inches from the specified diameter).

Because measurements are made from the specified diameter,

we have a given target µ = 0 for the process mean. The process

standard deviation σ = 0.31. What is going on?

Sample 1 −0.142 0.093 0.174 0.085 −0.176 0.367 0.308 0.199 0.4810 0.2911 0.4812 0.5513 0.5014 0.3715 0.6916 0.4717 0.5618 0.7819 0.7520 0.4921 0.79

The process mean has drifted. Maybe the cutting blade is getting dull, or a

screw got a bit loose.

For the chart, the

center line is 0 and

the control limits are

±3σ/√4 = ± 0.465.

x x x

x

xx x

x