Sampling DistributionsDay 1 and Day 2

9.1

Many investigations and research projects try to draw

conclusions about how the values of some variable x

are distributed in a population. Often, attention is

focused on a single characteristic of that distribution.

Examples include:

1. x = fat content (in grams) of a quarter-pound

hamburger, with interest centered on the mean fat

content μ of all such hamburgers

2. x = fuel efficiency (in miles per gallon) for a 2003

Honda Accord, with interest focused on the variability

in fuel efficiency as described by σ, the standard

deviation for the fuel efficiency population

distribution

3. x = time to first recurrence of skin cancer for a patient

treated using a particular therapy, with attention

focused on p, the proportion of such individuals whose

first recurrence is within 5 years of the treatment.

Parameter:

Statistic:

A number that describes the

____________. This number is typically

unknown.

A number that describes the ________.

We use this number to ___________ the

______________.

Population Sample

Mean

Standard

Deviation

Proportion

Standard

deviation of the

proportion

Parameter Statistic

Is the boldfaced number a parameter of a statistic. Use the proper notation to describe the number.

•The Bureau of Labor Statistics announces that last month it interviewed all members of the labor force in a sample of 50,000 households; 4.5% of the people interviewed are unemployed.

•The ball bearing in a large container have mean diameter 1.35 centimeters. This is within the specification for acceptance of the container by the purchaser. By chance, an inspector chooses 100 bearings from the container that have mean diameter 1.37 cm.

Sampling Distribution:

The distribution of all values taken by the statistic

in _____ ___________ ______of the ________

________from the _____________ _____________

Sampling Variability:

The ______________ between each ____________ of

samples of the ___________size.

If I compare many different samples and the statistic

is very __________ in each one, then the ___________

_____________ is _______. If I compare many

different samples and the statistic is very ___________

in each one, then the ______________ ____________is

___________.

Unbiased:

When the statistic is __________to the _________ value

of the parameter

Unbiased Estimator:

The unbiased _____________

Look at the following four histograms. (Use the bull’s-eye example on for reference.)

In these histograms, what represents variability?

In these histograms, what represents bias?

________ bias, ________ variability

________ bias, ________ variability

________ bias, ________ variability

________ bias, ________ variability

How sampling works:

1. Take a _______ number of samples from the

________ population.

2. Calculate the sample _________ or sample

_______________ for each sample

3. Make a _______________ of the values of the

statistics

4. Examine the ________________

Facts about Samples:

If the population mean ( ) and the population standard

deviation ( ) are unknown, we can use x to estimate

and use to estimate . These estimates may or may

not be reliable.

x

•

• If I chose a different sample, it would still represent

the same population. A different sample ________

_________produces different _____________.

• A statistic can be _____________ and still have high

______________. To avoid this, ___________ the size of

the sample. ___________ samples give smaller spread.

Example #1: Classify each underlined number as a

parameter or statistic. Give the appropriate notation

for each.

a. Forty-two percent of today’s 15-year-old girls will

get pregnant in their teens.

b. The National Center for Health Statistics reports

that the mean systolic blood pressure for males 35 to

44 years of age is 128 and the standard deviation is

15. The medical director of a large company looks at

the medical records of 72 executives in this age group

and finds that the mean systolic blood pressure for

these executives is 126.07.

Example #1: Classify each underlined number as a parameter or statistic.

Give the appropriate notation for each.

Example #2: Suppose you have a population in which

60% of the people approve of gambling.

You want to take many samples of size 10 from this population to

observe how the sample proportion who approve of gambling

vary in repeated samples.

b. Describe the design of a simulation using the partial random

digits table below to estimate the sample proportion who approve

of gambling. Label how you will conduct the simulation. Then

carry out five trials of your simulation. What is the average of

the samples? How close is it to the 60%?

c. The sampling distribution of is the distribution of

from all possible SRSs of size 10 from this

population. What would be the mean of this

distribution if this process was repeated 100 times?

p̂

p̂

d. If you used samples of size 20 instead of size 10,

which sampling distribution would give you a better

estimate of the true proportion of people who

approve of gambling? Explain your answer.

e. Make a histogram of the sample distribution.

Describe the graph.

Sampling Proportions

Sampling Distribution of

• If our sample is an SRS of size n, then the following statements describe the sampling model for :

(2) The standard deviation is

p̂

Sampling Distribution of a Sample Proportion:

(1) The mean is exactly _________.

(2) The standard deviation is

Rule of Thumb #1:

You can only use if the population is ________

the ___________ __________. A census should be

impractical!

p̂

N 10nwhen

Rule of Thumb #2:

Only use the _____________ approximation of the

sampling distribution of when: p̂

and

Conclusion:

If p is the population proportion then,

If is the sample proportion then, p̂

ONLY if and

So, to calculate a Z-score for this!

statistic parameterStandardized test statistic:

standard deviation of statistic

Or

Example #1

Suppose you are going to roll a fair six-sided die

60 times and record , the proportion of times that

a 1 or a 2 is showing.

a. Where should the distribution of the 60 -

values be centered?

b. What is the standard deviation of the

sampling distribution of , the proportion of

all rolls of the die that show a 1 or a 2 out of the

60 rolls?

p̂

c. Describe the shape of the sampling distribution of

Justify your answer.

p̂

Example #2

According to government data, 22% of American

children under the age of 6 live in households with

incomes less than the official poverty level. A study of

learning in early childhood chooses an SRS of 300

children. What is the probability that more than 20%

of the sample are from poverty households?

b. How large a sample would be needed to guarantee

that the standard deviation of is no more than 0.01?

Explain.