Chapter 8

Sampling Variability and Sampling Distributions

Suppose we are interested in finding the true mean () fat content of quarter-pound hamburgers marketed by a national fast food chain. To learn something about , we could obtain a sample of n = 50 hamburgers and determine the fat content of each one.

Would the sample mean be a good estimate of ?

How close is the

sample mean to ?

Will other samples of n = 50 have the same sample mean?

To answer these questions, we will examine the sampling distribution,

which describes the long-run behavior of sample statistic.

Recall that the sample mean is a

statistic

Statistic

• A number that that can be computed from sample data

• Some statistics we will use includex – sample means – standard deviationp – sample proportion

• The observed value of the statistic depends on the particular sample selected from the population and it will vary from sample to sample.

This variability is called sampling

variability

The campus of Wolf City College has a fish pond. Suppose there are 20 fish in the pond. The lengths of the fish (in inches) are given below:

4.5 5.4 10.3

7.9 8.5 6.6 11.7

8.9 2.2 9.8

6.3 4.3 9.6 8.7 13.3

4.6 10.7

13.4

7.7 5.6We caught fish with lengths 6.3 inches, 2.2 inches, and 13.3 inches.x = 7.27 inches

Let’s catch two more

samples and look at the

sample means.

2nd sample - 8.5, 4.6, and 5.6 inches.x = 6.23 inches3rd sample – 10.3, 8.9, and 13.4 inches.x = 10.87 inches

Suppose we randomly catch a sample of 3 fish from this pond and measure their length. What would the mean length of the sample be?

This is an example

of sampling variability

This is a statistic!The true mean =

8.Notice that some

sample means are closer and some

farther away; some above and some below the mean.

Fish Pond Continued . . .

4.5 5.4 10.3

7.9 8.5 6.6 11.7

8.9 2.2 9.8

6.3 4.3 9.6 8.7 13.3

4.6 10.7

13.4

7.7 5.6

There are 1140 (20C3) different possible samples of size 3 from this population. If we were to catch all those different samples and calculate the mean length of each sample, we would have a distribution of all possible x.

This would be the sampling distribution of x.

Sampling Distributions of x

• The distribution that would be formed by considering the value of a sample statistic for every possible different sample of a given size from a population.

In this case, the sample statistic is the sample

mean x.

Fish Pond Revisited . . .

Suppose there are only 5 fish in the pond. The lengths of the fish (in inches) are given below:

x = 7.84

x = 3.262

What is the mean and standard

deviation of this population?

6.6 11.7

8.9 2.2 9.8 We will keep the population size small so that we can find ALL the

possible samples.

Fish Pond Revisited . . .

What is the mean and standard

deviation of these sample

means?

6.6 11.7

8.9 2.2 9.8

Pairs 6.6 & 11.7

6.6 & 8.9

6.6 & 2.2

6.6 & 9.8

11.7 & 8.9

11.7 & 2.2

11.7 & 9.8

8.9 & 2.2

8.9 & 9.8

2.2 &

9.8

x 9.15 7.75 4.4 8.2 10.3 6.95 10.75

5.55 9.35 6How many samples of size 2 are possible?

x = 7.84

x = 1.998

How do these values compare to the population

mean and standard

deviation?

x = 7.84 and x = 3.262

Let’s find all the samples of size

2.

These values determine the sampling distribution of x for samples of size 2.

Fish Pond Revisited . . .

6.6 11.7

8.9 2.2 9.8

Triples 6.6, 11.7, 8.9

6.6, 11.7, 2.2

6.6, 11.7, 9.8

6.6, 8.9, 2.2

6.6, 8.9, 9.8

6.6, 2.2, 9.8

11.7, 8.9, 2.2

11.7, 8.9, 9.8

11.7, 2.2, 9.8

8.9, 2.2, 9.8

x 9.067 6.833 9.367 5.9 8.433 6.2 7.6 10.133

7.9 6.967

How many samples of size 3 are possible?

x = 7.84

x = 1.332

How do these values compare to

the population mean and standard

deviation?

x = 7.84 and x = 3.262

Now let’s find all the samples of size 3.

What is the mean and standard

deviation of these sample

means?

These values determine the sampling distribution of x for samples of size 3.

What do you notice?

• The mean of the sampling distribution EQUALS the mean of the population.

• As the sample size increases, the standard deviation of the sampling distribution decreases.

x =

as n x

General Properties of Sampling Distributions of x

Rule 1:

Rule 2:

This rule is exact if the population is infinite, and is approximately correct if the population is finite and no more than 10% of the population is included in the sample

x

nx

Note that in the previous fish pond

examples this standard deviation

formula was not correct because the sample sizes were more than 10% of

the population.

The paper “Mean Platelet Volume in Patients with Metabolic Syndrome and Its Relationship with Coronary Artery Disease” (Thrombosis Research, 2007) includes data that suggests that the distribution of platelet volume of patients who do not have metabolic syndrome is approximately normal with mean = 8.25 and standard deviation = 0.75.

We can use Minitab to generate random samples from this population. We will

generate 500 random samples of n = 5 and compute the sample mean for each.

Platelets Continued . . .

Similarly, we will generate 500 random samples of n = 10, n = 20, and n = 30. The density histograms below display the resulting 500 x for each of the given sample sizes.

What do you notice about the means of these

histograms?

What do you notice about the

standard deviation of these

histograms?

What do you notice about the shape of these

histograms?

General Properties Continued . . .

Rule 3: When the population distribution is normal, the sampling distribution of x is also normal for any sample size n.

The paper “Is the Overtime Period in an NHL Game Long Enough?” (American Statistician, 2008) gave data on the time (in minutes) from the start of the game to the first goal scored for the 281 regular season games from the 2005-2006 season that went into overtime. The density histogram for the data is shown below.

Let’s consider these 281 values as a population. The distribution is strongly positively skewed with mean = 13 minutes and with a median of 10 minutes.

Using Minitab, we will generate 500 samples of

the following sample sizes from this distribution:

n = 5, n = 10, n = 20, n = 30.

These are the density histograms for the 500 samples

Are these histograms centered at approximately

= 13?

What do you notice about the standard deviations of these

histograms?

What do you notice about the shape of these histograms?

General Properties Continued . . .

Rule 4: Central Limit TheoremWhen n is sufficiently large, the sampling distribution of x is well approximated by a normal curve, even when the population distribution is not itself normal.

CLT can safely be applied if n exceeds 30.

How large is “sufficiently large” anyway?

A soft-drink bottler claims that, on average, cans contain 12 oz of soda. Let x denote the actual volume of soda in a randomly selected can. Suppose that x is normally distributed with = .16 oz. Sixteen cans are randomly selected, and the soda volume is determined for each one. Let x = the resulting sample mean soda.

If the bottler’s claim is correct, then the sampling distribution of x is normally distributed with:

04.16

16.

12

nx

x

.9772 - .1587 = .8185

Soda Problem Continued . . .

04.16

16.

12

nx

x

What is the probability that the sample mean soda volume is between 11.96 ounces and 12.08 ounces?

P(11.96 < x < 12.08) =

To standardized these endpoints, use

n

xxz

x

x

104.

1296.11*

a 2

04.1208.12

*

b

Look these up in the table and subtract the

probabilities.

A hot dog manufacturer asserts that one of its brands of hot dogs has a average fat content of 18 grams per hot dog with standard deviation of 1 gram. Consumers of this brand would probably not be disturbed if the mean was less than 18 grams, but would be unhappy if it exceeded 18 grams. An independent testing organization is asked to analyze a random sample of 36 hot dogs. Suppose the resulting sample mean is 18.4 grams. Does this result suggest that the manufacturer’s claim is incorrect?

Since the sample size is greater than 30, the Central Limit Theorem applies.So the distribution of x is

approximately normal with 1667.

36

1and18 xx

Hot Dogs Continued . . .

Suppose the resulting sample mean is 18.4 grams. Does this result suggest that the manufacturer’s claim is incorrect?

P(x > 18.4) =

1667.36

1and18 xx

40.21667.

184.18

z

1 - .9918 = .0082

Values of x at least as large as 18.4 would be observed only about .82% of

the time.The sample mean of 18.4 is large enough

to cause us to doubt that the manufacturer’s claim is correct.

Let’s explore what happens with in distributions of sample proportions (p). Have students perform the following experiment.

•Toss a penny 20 times and record the number of heads. •Calculate the proportion of heads and mark it on the dot plot on the board.

What shape do you think the dot plot will have?

The dotplot is a partial graph of the sampling distribution of all

sample proportions of sample size 20.

This is a statistic!

What would happen to the dotplot if we flipped the penny 50 times and recorded the proportion of

heads?

Sampling Distribution of p

The distribution that would be formed by considering the value of a sample statistic for every possible different sample of a given size from a population.

We will use:p for the population proportion

and p for the sample proportion

In this case, we will use

np

sample the in successes of numberˆ

Suppose we have a population of six students: Alice, Ben, Charles, Denise, Edward, & Frank

We are interested in the proportion of females. This is called

What is the proportion of females?

Let’s select samples of two from this population.

How many different samples are possible?

the parameter of interest

6C2 =15

1/3

We will keep the population small so that we can find ALL

the possible samples of a given size.

Find the 15 different samples that are possible and find the sample proportion of the number of females in each sample.

Alice & Ben .5Alice & Charles .5Alice & Denise 1Alice & Edward .5Alice & Frank .5Ben & Charles 0Ben & Denise .5Ben & Edward 0

Ben & Frank 0Charles & Denise .5Charles & Edward 0Charles & Frank 0Denise & Edward .5Denise & Frank.5Edward & Frank 0

Find the mean and standard deviation of these sample proportions.

29814.0and31

ˆˆ pp

How does the mean of the sampling

distribution compare to the population parameter (p)?

General Properties for Sampling Distributions of p

Rule 1:

Rule 2:

This rule is exact if the population is infinite, and is approximately correct if the population is finite and no more than 10% of the population is included in the sample

pp

ˆ

npp

p

)1(ˆ

Note that in the previous student

example this standard deviation

formula was not correct because the

sample size was more than 10% of

the population.

In the fall of 2008, there were 18,516 students enrolled at California Polytechnic State University, San Luis Obispo. Of these students, 8091 (43.7%) were female. We will use a statistical software package to simulate sampling from this Cal Poly population.

We will generate 500 samples of each of the following sample sizes: n = 10, n = 25, n = 50, n = 100 and compute the proportion of females for each sample.

The following histograms display the distributions of the sample proportions for the 500 samples of each sample size.

Are these histograms centered around the

true proportion p = .437?

What do you notice about the standard deviation of these

distributions?

What do you notice about the shape of these distributions?

The development of viral hepatitis after a blood transfusion can cause serious complications for a patient. The article “Lack of Awareness Results in Poor Autologous Blood Transfusions” (Health Care Management, May 15, 2003) reported that hepatitis occurs in 7% of patients who receive blood transfusions during heart surgery. We will simulate sampling from a population of blood recipients. We will generate 500 samples of each of the following sample sizes: n = 10, n = 25, n = 50, n = 100 and compute the proportion of people who contract hepatitis for each sample.

The following histograms display the distributions of the sample proportions for the 500 samples of each sample size.

Are these histogram

s centered around the true proportion p = .07?

What happens to the shape of these

histograms as the sample

size increases?

General Properties Continued . . .

Rule 3: When n is large and p is not too near 0 or 1, the sampling distribution of p is approximately normal.

The farther the value of p is from 0.5, the larger n must be for the sampling distribution of p to be approximately normal.A conservative rule of thumb:

If np > 10 and n (1 – p) > 10, then a normal distribution provides a reasonable approximation to the sampling distribution of p.

Blood Transfusions Revisited . . .Let p = proportion of patients who contract

hepatitis after a blood transfusionp = .07

Suppose a new blood screening procedure is believed to reduce the incident rate of hepatitis. Blood screened using this procedure is given to n = 200 blood recipients. Only 6 of the 200 patients contract hepatitis. Does this result indicate that the true proportion of patients who contract hepatitis when the new screening is used is less than 7%?

To answer this question, we must consider the

sampling distribution of p.

Blood Transfusions Revisited . . .Let p = .07 p = 6/200 = .03

Is the sampling distribution approximately normal?np = 200(.07) = 14 > 10n(1-p) = 200(.93) = 186 > 10

What is the mean and standard deviation of the sampling distribution?

Yes, we can use a normal approximation.

018.200

)93(.07.

07.

ˆ

ˆ

p

p

Blood Transfusions Revisited . . .Let p = .07 p = 6/200 = .03

Does this result indicate that the true proportion of patients who contract hepatitis when the new screening is used is less than 7%?

018.200

)93(.07.

07.

ˆ

ˆ

p

p

P(p < .03) =

Assume the screening procedure is not

effective and p = .07.

22.2

200)93(.07.

07.03.

z

.0132

This small probability tells us that it is unlikely that a

sample proportion of .03 or smaller would be observed if the

screening procedure was ineffective.

This new screening procedure

appears to yield a smaller incidence rate for hepatitis.