Chapter 10 Sampling and Sampling Distributions

• 10.1 Random sampling

• 10.4 Stratified sampling

• 10.6 Sampling distribution

• 10.7 The standard error of the mean

• 10.8 The central limit theorem

10.1 Random sampling

• Example: At a parts depot the inventory shows 1000 parts in stock. What percentage of those are actually in stock?– Let p = true %– To estimate p, take a sample of n parts to check

out in the supply room.– Then – How do we choose the parts to check?

ˆ sample proportionp

Random sample

• A random sample is an insurance policy to protect against bias.

• A simple random sample gives each of the

possible sample choices the same chance of being selected.

N

n

Random sampling

In other words, for a finite population of N sample points:

A sample of size n is random if any n sample points have a probability

to be selected.

n

N1

To generalize

For an infinite population, or a finite population but sampling with replacement.

A value is observed according to a probability distribution.

A random sample of size n consists of observed values that are independent and have the same distributions.

10.2 &10.3 Skip

10.4 Stratified sample

• A simple random sample is not always the best option.

• Consider the following example. We want to estimate the average number of vireos per area.

Conifer Desiduous

Then the population mean and sample mean are

total red-eyed vireos

total area

total observed red-eyed vireos

total observed areax

Question: if we use simple random sampling, is always a good estimator of ?

x

• Answer: NO.

• Reasoning: A simple random sample of n=10 locations might all end up Conifer. We would be better off putting n1 samples in conifer and n2 samples in Desiduous.

Stratified sampling

• A stratified random sample breaks the population into strata and samples randomly within each stratum.

• In our previous example:– Stratum 1 = Deciduous– Stratum 2 = Conifer

Notations

N1=number of units in stratum 1

N2=number of units in stratum 2

N = N1+N2 = number of units in entire population

n1=number of units sampled from stratum 1

n2=number of units sampled from stratum 2

n = n1+n2 = number of sampled units

Optimal allocation• How do we decide n1 and n2 (when n is fixed)?

N1=# of possible sampling locations in conifer

N2=# in desiduous

σ1=standard deviation of bird counts in conifer

σ2=standard deviation of bird counts in desiduous

Optimal allocation: (Problem 10.29)1 1

11 1 2 2

2 22

1 1 2 2

nNn

N N

nNn

N N

Sample more in strata, with--more units (area)--higher variability

Proportional allocation

• Proportional allocation doesn’t consider σ and makes the sample number from each stratum proportional to the size of the stratum.

1 21 2

1 2 1 2

N N

n n n nN N N N

Example of proportional allocation

If

N1=100 N2=300 n=40

Then1

1 2

1

2

100 1

400 4

140 10

43

40 304

N

N N

n

n

A comparison between simple random sampling and stratified sampling

Example: Population: Weights of rocks

4, 6, 10, 12. So N=4 and =8.

Sample n=2.

Possible results for simple random samplesSample Probability sample mean

Probability

4 6 1/6 5 1/6

4 10 1/6 7 1/6

4 12 1/6 8

6 10 1/6 8

6 12 1/6 9 1/6

10 12 1/6 11 1/6

2/6

Example Continued

• Possible results for a stratified sample. Choose 1 from the smaller rocks and choose 1 from the larger rocks.

Sample Probability sample mean Probability

4 10 1/4 7 1/4

4 12 1/4 8

6 10 1/4 8

6 12 1/4 9 1/4

1/2

0

0

1/5

1/3

2/5

5 6 7 8 9 10 11

Series2

0

1/5

2/5

3/5

7 8 9

Series2

SRS Stratified

The stratified sample is more likely to be close to the true population value, here.

A comparison between simple random sampling and stratified sampling

10.6 Sampling Distributions10.6 Sampling Distributions

A statistic (e.g. sample mean) from a random sample or randomized experiment is a random variable and its probability distribution is a sampling distribution.

The population distribution of a variable is the distribution of its values for all members of the population. The population distribution is also the probability distribution of the variable when choosing one subject at random from the population.

Results of many random samples of size n=100, from a population where it is known that 60% of the people hate to shop for clothes, population proportion (parameter) p = .6) and sample proportion (statistic).

Most samples of size n = 100 gave estimates close to .6, but some were far off. On average, they centered on .6, they are variable, but unbiased.

Sampling VariabilitySampling VariabilityApproximating the Sampling DistributionApproximating the Sampling Distribution

p̂

Results of many random samples of size n=2500, from the same population, with population proportion (parameter) p = .6) and sample proportion (statistic).

Larger samples are more precise (have less variability) AND are unbiased.

Sampling Variability (cont.)Sampling Variability (cont.)Approximating the Sampling DistributionApproximating the Sampling Distribution

p̂

Sample size

What advantage is there of taking a larger sample? Larger n?

xTaking a larger sample decreases the potential deviation of away from .

xx

Let be the standard deviation of the sampling distribution of , then the larger the sample size is, the smaller is.x

Unbiased estimators

• P.77 “Estimators having the desirable property that their values will on average equal the quantity they are supposed to estimate are said to be unbiased.”

• is an unbiased estimator of .

• Another example of unbiased estimator is s2 for 2.

• Choosing non-random samples can introduce bias.

x then If x

So, as n increases, decreases. If the sample size is multiplied by 4, the “standard deviation” (standard error of the mean) will be divided in half for infinite populations.

If x is the mean of a sample of size n from a population having mean and standard deviation . The mean and standard deviation of x are:

(standard error of the mean)

for finite populations

10.7 The Standard Error of the Mean10.7 The Standard Error of the Mean

xn

x

1or

N

nN

nx

x

Standard DeviationStandard Deviationvs.vs.

Standard Error of the MeanStandard Error of the Mean

• The standard error of the mean is smaller than the standard deviation by a factor of the square root of the sample size.

• The standard deviation describes the variability of individuals.

• The standard error of the mean describes the accuracy of means of a given size or the potential error in sample mean as a guess at .

Examples

• Population size N=4, sample size n=2,

025.20

131

110 112

150 152

What is the standard error of the mean?

56.1114

24

2

025.20

x

Examples

• Infinite Population

5

62

P(3)=P(5)=

P(7)=P(9)=1/4

What is the standard error of the mean?

58.12

5x

Example• Governor’s Poll

p̂ s1' offraction observed x1, 1, 1 0, 1, 00, …

Estimate guess

What is the standard error of the mean?

n

pp

pp

pppp

pppp

PP

ps

p

)1(

xbinomial a of variance toRelated )1(

]1)[1(

)1()1()0(

)1()1()0()0(

1 offraction

ˆx

2

22

222

If a population has the N(,) distribution, then the sample mean x of n independent observations has the distribution.

For ANY population with mean and standard deviation the sample mean of a random sample of size n is approximately

when n is LARGE.

10.8 Central Limit Theorem10.8 Central Limit Theorem

( , )Nn

( , )Nn

x

Central Limit Theorem

If the population is normal or sample size is large, sample mean follows a normal distribution

and

follows a standard normal distribution.

x( , )N

n

n

xxz

x

x

• The closer x’s distribution is to a normal distribution, the smaller n can be and have the sample mean nearly normal.

• If x is normal, then the sample mean is normal for any n, even n=1.

n=1 n=2

n=10 n=25

Central Limit Theorem At WorkCentral Limit Theorem At Work

• Usually n=30 is big enough so that the sample mean is approximately normal unless the distribution of x is very asymmetrical.

• If x is not normal, there are often better procedures than using a normal approximation, but we won’t study those options.

Example• X=ball bearing diameter• X is normal distributed with =1.0cm and =0.02cm• =mean diameter of n=25• Find out what is the probability that will be off by

less than 0.01 from the true population mean.

%76.98)5.25.2(

)004.0

0.101.1

004.0

0.199.0()01.199.0(

004.025

02.0

0.1

zP

zPxP

nx

x

xx

Exercise

The mean of a random sample of size n=100 is going to used to estimate the mean daily milk production of a very large herd of dairy cows. Given that the standard deviation of the population to be sampled is s=3.6 quarts, what can we assert about he probabilities that the error of this estimate will be more then 0.72 quart?