Chapter 3
What Do Samples Tell Us?
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Thought Question 1
During a medical exam, the doctor
measures your cholesterol two times.
Do you think both measurements would
be exactly the same? Why or why not?
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Thought Question 2
To estimate the percentage of all adults who
have an Internet connection in their homes,
a properly chosen sample of 1100 adults
across the U.S. was sampled, and 60% said
“yes”.
How close do you think that is to the
percentage of the entire country who have
an Internet connection? Within 30%? 10%?
5%? 1%? Exactly the same?
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Sampling Terminology
• Parameter
– fixed, unknown number that describes the population
– A parameter is a fixed number, but in practice we don’t
know its value
• Statistic
– known value calculated from a sample
– It can change from sample to sample
– a statistic is used to estimate a parameter
• Bias
– in repeated samples, the sample statistic consistently
misses the population parameter in the same direction
• Variability
– different samples from the same population may yield
different values of the Chapter
sample
statistic
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Figure 3.3 Bias and variability in shooting arrows at a target. Bias means the archer
systematically misses in the same direction. Variability means that the arrows are
scattered.
Sampling Strategy
• A good sampling method has both small
bias and small variability.
• To reduce bias, use random sampling.
• To reduce variability, use larger samples
– estimates from random samples will be closer to the
true values in the population if the samples are larger
– how close will they be?
• margin of error
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Proportions
• The proportion of a population that has
some outcome (“success”) is p.
• The proportion of successes in a sample is
measured by the sample proportion:
number of successes in the sample
p̂ = total number of observations in the sample
“p-hat”
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Example: Same-sex marriages
• In 2004 President Bush made the declaration: “Today I call
upon the Congress to promptly pass, and send to the states
for ratification, an amendment to our Constitution defining
and protecting marriage as a union of a man and woman as
husband and wife.”
• A Gallup Poll conducted from July 2003 to February 2004
asked the following question:
•
“Would you favor or oppose a constitutional amendment that
would define marriage as being between a man and a woman,
thus barring marriages between gay and lesbian couples?”
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Example: Same-sex marriages
• Gallup Poll found out that such an amendment is supported
by a very slim majority of Americans, 51%, with 45% opposed.
• Gallup talked with 2527 randomly selected adults to reach
these conclusions.
• How can 2527 people tell us about the opinions of the 307
million people?
• A sample won’t tell us the truth about the population, so
Gallup gives a margin of error. Here is Gallup’s claim: “For
results based on a sample of this size, one can say with 95%
confidence that the error attributable to sampling and other
random effects could be plus or minus 2 percentage points.”
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Margin of Error
• The amount by which the proportion obtained
from the sample ( p̂ ) will differ from the true
population proportion (p) rarely exceeds the
margin of error.
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What Margin of Error means
“Margin of Error plus or minus two percentage
points is shorthand for the statement:
If we took many samples using the same method we
used to get this one sample, 95% of the samples would
give a result within plus or minus 2 percentage points of
the truth about the population.
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Margin of Error
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Margin of Error and Sample Size
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Figure 3.1 The results of many SRSs have a regular pattern. Here, we draw 1000 SRSs
of size 100 from the same population. The population proportion is p = 0.5. The
sample proportions vary from sample to sample, but their values center at the truth
about the population.
Figure 3.2 Draw 1000 SRSs of size 2527 from the same population as in Figure 3.1.
The 1000 values of the sample proportion are much less spread out than was the case
for smaller samples.
Confidence statements
A confidence statement has two parts:
a margin of error and a level of confidence.
The margin of error says how close the sample statistic
lies to the population parameter. The level of confidence
says what percentage of all possible samples satisfy the
margin of error.
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Population size doesn’t matter
The variability of a statistic from a random sample
does not depend on the size of the population as
long as the population is at least 100 times larger
than the sample.
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Case Study
Guaranteed Health Insurance in the U.S.?
New York Times/CBS News Poll, January 2006
62% say it should be guaranteed
by the government.
same as in 2000, up 6 points from 1996
31% say it is not the responsibility
of the government.
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Case Study
How the Poll was Conducted
This New York Times/CBS News poll was based
on telephone interviews conducted January 20
through January 25, 2006 with 1,229 adults
throughout the United States.
The survey has a random sampling error of
approximately ±3 percent.
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Case Study
Conclusion (Confidence statement)
For the proportion of the population who favor
guaranteed health insurance, the sample
proportion was = .62 (62%) and the margin of
p̂
error was ±.03 (3%). We can then say that:
“we are 95% confident that the proportion of the
population who favor guaranteed health insurance
was between .59 and .65 (59% and 65%).”
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Key Concepts
•
•
•
•
Parameter versus Statistic
Bias and Variability
Margin of Error
Confidence Statements
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Exercise 3.13
Which data have high/low bias, high/low variability?
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Exercises 3.3-3.6
• Determine whether the highlighted number is a parameter or a statistic.
• The Bureau of Labor Statistics announces that last month it interviewed all
members of the labor force in a sample of 55,000 households; 4.7% of the
people were unemployed.
• A carload lot of ball bearings has an average diameter of 2.503
centimeters (cm). This is within the specifications for acceptance of the lot
by the purchaser. The inspector happens to inspect 100 bearings from the
lot with an average diameter of 2.515 cm. This is outside the specified
limits, so the lot is mistakenly rejected.
• Voter registration records show that 15.5% of all voters in Philadelphia are
registered as Republicans. However, a radio talk show host in Philadelphia
found that of 20 local residents who called the show recently, 60% were
registered Republicans.
• A national polling organization uses a random digit dialing device to dial
residential phone numbers in the U.S. Of the first 100 numbers dialed, 43
are unlisted numbers. This is not surprising, because 40% of all residential
phones in the U.S. are unlisted.
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Exercise 3.9
• Canada’s national health care. The Ministry of Health in the Canadian
province of Ontario wants to know whether the national health care
system is achieving its goals in the province. Much information about
health care comes from patient records, but that resource doesn’t allow
us to compare people who use health services with those who don’t. So
the ministry of Health conducted the Ontario Health Survey, which
interviewed a random sample of 61,239 people who live in the province of
Ontario.
a) What is the population for this sample survey? What is the sample?
b) The survey found that 76% of males and 86% of females in the sample had
visited a general practitioner at least once in the past year. Do you think
these estimates are close to the truth about the entire population? Why?
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Exercise 3.11
Sampling variability. In thinking about Gallup’s sample of size 2527,
we asked “Could it happen that one random sample finds that 51% of
adults favor an amendment and a second random sample finds that
only 37% favor one? ”.
Look at the following figure which shows the results of 1000 samples
of this size when the population truth is p=0.5 or 50%. Would you be
surprised if a sample from this population gave 51%? Would you be
surprised if a sample gave 37%?
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Exercise 3.14
Is a larger sample size always better? In February 2004, USA Today
conducted an online poll. Visitors to their website were asked the
following question: “Should the U.S. pass a constitutional
amendment banning gay marriage?” Visitors could vote by clicking a
button. The results as of 3:30 PM on February 25 were that
68.61% voted “No” and 31.39% voted “Yes”
A total of 63,046 votes had been recorded. Using our quick method,
we find that the margin of error for 95% confidence for a sample of
this size is roughly equal to four-tenths of one percentage points.
Is it correct to say that, based on this USA Today online poll, we are
95% confident that 68.61 plus or minus 0.4% of American adults are
opposed to having the U.S. pass a constitutional amendment
banning gay marriage?
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Exercise 3.22
The death penalty. The Gallup Poll asked a sample of 1010 adults,
“Are you in favor of the death penalty for a person convicted of
murder?” The proportion who said in favor was 69%.
a) How many of the 1010 people interviewed said they were in favor
of the death penalty for person convicted of murder?
b) Gallup says that the margin of error for this poll is plus or minus 3
percentage points. Explain to someone who knows no statistics what
“margin of error plus or minus 3 percentage points” means.
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Exercise 3.24, and 3.28
Find the margin of error. Exercise 3.22 concerns a Gallup
Poll sample of 1010 people.
a)Use the quick method to estimate the margin of error
for statements about the population of all adults.
b)Is your result close to the 3% margin of error
announced by Harris?
c)Suppose you want a margin of error half as large as
the one you found in (a). How many people must you
interview?
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Exercise 3.29
Satisfying the Congress. Exercise 3.12 describes a sample survey of
1004 adults, with margin of error plus or minus 4 percentage points
for 95% confidence.
a) A member of the Congress thinks that 95% confidence is not
enough. He wants to be 99% confident. How would the margin of
error for 99% confidence based on the sample compare with the
margin of error for 95% confidence.
b)Another member of the Congress is satisfied with 95% confidence,
but she wants a smaller margin of error than plus or minus 4
percentage points. How can we get a smaller margin of error, still
with 95% confidence?
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