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Inference about a Population
Proportion
BPS chapter 19
© 2010 W.H. Freeman and Company
Sample proportion
Which of the following symbols represents the sample proportion?
a)
b)
c)
d)
P-value
Sample proportion (answer)
Which of the following symbols represents the sample proportion?
a)
b)
c)
d)
P-value
Sampling distribution
The theoretical sampling distribution of
a)
b)
c)
d)
p̂
Gives the values of p̂ from all possible samples of size n from the
same population.
Provides information about the shape, center, and spread of the
values in a single sample.
Can only be constructed from the results of a single random sample
of size n.
Is another name for the histogram of the values in a random sample
of size n.
Sampling distribution (answer)
The theoretical sampling distribution of
a)
b)
c)
d)
p̂
Gives the values of p̂ from all possible samples of size n from
the same population.
Provides information about the shape, center, and spread of the
values in a single sample.
Can only be constructed from the results of a single random sample
of size n.
Is another name for the histogram of the values in a random sample
of size n.
Sampling distribution
Which of the following is true?
a)
b)
c)
The shape of the sampling distribution of p̂ is always bell-shaped.
The shape of the sampling distribution of p̂ gets closer to the
shape of the population distribution as n gets large.
The shape of the sampling distribution of p̂ becomes
approximately normal as n gets large.
Sampling distribution (answer)
Which of the following is true?
a)
b)
c)
The shape of the sampling distribution of p̂ is always bell-shaped.
The shape of the sampling distribution of p̂ gets closer to the
shape of the population distribution as n gets large.
The shape of the sampling distribution of p̂ becomes
approximately normal as n gets large.
Sampling distribution
Which of the following best describes the mean of the sampling
distribution of p̂ ?
a)
b)
c)
As n increases, the mean of the sampling distribution of p̂ gets
closer to p.
As n increases, the mean of the sampling distribution of p̂ gets
closer to x .
Regardless of the value of n, the mean of the sampling distribution
of p̂ is equal to p.
Sampling distribution (answer)
Which of the following best describes the mean of the sampling
distribution of p̂ ?
a)
b)
c)
As n increases, the mean of the sampling distribution of p̂ gets
closer to p.
As n increases, the mean of the sampling distribution of p̂ gets
closer to x .
Regardless of the value of n, the mean of the sampling
distribution of p̂ is equal to p.
Sampling distribution
True or false: As n increases, the standard deviation of the sampling
distribution of p̂ gets smaller.
a)
b)
True
False
Sampling distribution (answer)
True or false: As n increases, the standard deviation of the sampling
distribution of p̂ gets smaller.
a)
b)
True
False
Sampling distribution
Ten percent of all customers of Cheap Foods regularly purchase GoodEnuf Brand Chicken Fingers. We plan to ask a random sample of
45 Cheap Foods customers if they regularly purchase Good-Enuf
Chicken Fingers. We will then calculate p̂ from the responses. Is
the shape of the sampling distribution of p̂ close enough to normal
to use the normal distribution to compute probabilities on p̂ ?
a)
b)
c)
d)
Yes, because n > 30.
No, because np = (45) (0.10) = 4.5 which is < 10.
No, because we only have data from one sample.
We cannot know the shape without knowing how many of the 45
customers purchase Good-Enuf Chicken Fingers.
Sampling distribution (answer)
Ten percent of all customers of Cheap Foods regularly purchase GoodEnuf Brand Chicken Fingers. We plan to ask a random sample of
45 Cheap Foods customers if they regularly purchase Good-Enuf
Chicken Fingers. We will then calculate p̂ from the responses. Is
the shape of the sampling distribution of p̂ close enough to normal
to use the normal distribution to compute probabilities on p̂ ?
a)
b)
c)
d)
Yes, because n > 30.
No, because np = (45) (0.10) = 4.5 which is < 10.
No, because we only have data from one sample.
We cannot know the shape without knowing how many of the 45
customers purchase Good-Enuf Chicken Fingers.
Sampling distribution
A potential candidate for president has stated that she will run for office
if at least 30% of Americans voice support for her candidacy. To
make her decision, she draws a random sample of 500 Americans.
Suppose that in fact 35% of all Americans support her candidacy.
The mean for the sampling distribution of p̂ is
a) 0.30
b) 0.35
c) Cannot be determined without more information.
Sampling distribution (answer)
A potential candidate for president has stated that she will run for office
if at least 30% of Americans voice support for her candidacy. To
make her decision, she draws a random sample of 500 Americans.
Suppose that in fact 35% of all Americans support her candidacy.
The mean for the sampling distribution of p̂ is
a) 0.30
b) 0.35
c) Cannot be determined without more information.
Confidence interval
The purpose of a confidence interval for p is
a)
b)
c)
d)
To give a range of reasonable values for the level of confidence.
To give a range of reasonable values for the sample proportion.
To give a range of reasonable values for the population proportion.
To give a range of reasonable values for the difference between the
sample proportion and the population proportion.
Confidence interval (answer)
The purpose of a confidence interval for p is
a)
b)
c)
d)
To give a range of reasonable values for the level of confidence.
To give a range of reasonable values for the sample proportion.
To give a range of reasonable values for the population
proportion.
To give a range of reasonable values for the difference between the
sample proportion and the population proportion.
Confidence interval
The confidence interval formula for p does NOT include
a)
b)
c)
d)
e)
The sample proportion.
The z* value for specified level of confidence.
The margin of error.
The sample size.
The population size.
Confidence interval (answer)
The confidence interval formula for p does NOT include
a)
b)
c)
d)
e)
The sample proportion.
The z* value for specified level of confidence.
The margin of error.
The sample size.
The population size.
Standard error
The standard error
a)
b)
c)
Is the true standard deviation of the sampling distribution of p̂ .
Is an estimate, using sample data, of the standard deviation of the
sampling distribution of p̂ .
Measures the maximum difference expected between p and p̂ at a
specified level of confidence.
Standard error (answer)
The standard error
a)
b)
c)
Is the true standard deviation of the sampling distribution of p̂ .
Is an estimate, using sample data, of the standard deviation of
the sampling distribution of p̂ .
Measures the maximum difference expected between p and p̂ at a
specified level of confidence.
Confidence interval
In 1993, presidential candidate Ross Perot appeared on television to
voice his opinions on government reform. To gauge public opinion,
Perot urged viewers to fill out the survey appearing in that week’s
issue of TV Guide. Of the approximately 1.4 million respondents,
98% agreed with Perot’s platform on health care reform. Are the
assumptions met for computing a confidence interval for the
proportion of U.S. adults that agree with Ross Perot’s health care
reform platform?
a)
b)
c)
d)
Yes, because the sample size is large.
Yes, because the population is at least 10 times larger than the
sample.
Both of the above answers are correct.
No, because the data are from a voluntary response sample.
Confidence interval (answer)
In 1993, presidential candidate Ross Perot appeared on television to
voice his opinions on government reform. To gauge public opinion,
Perot urged viewers to fill out the survey appearing in that week’s
issue of TV Guide. Of the approximately 1.4 million respondents,
98% agreed with Perot’s platform on health care reform. Are the
assumptions met for computing a confidence interval for the
proportion of U.S. adults that agree with Ross Perot’s health care
reform platform?
a)
b)
c)
d)
Yes, because the sample size is large.
Yes, because the population is at least 10 times larger than the
sample.
Both of the above answers are correct.
No, because the data are from a voluntary response sample.
Margin of error
The margin of error covers
a)
b)
c)
d)
Random sampling error.
Undercoverage error.
Non-response error.
All of the above.
Margin of error (answer)
The margin of error covers
a)
b)
c)
d)
Random sampling error.
Undercoverage error.
Non-response error.
All of the above.
Margin of error
When an opinion poll states with 95% confidence the margin of error for
the sample percentage is plus or minus 3 percentage points, this
means that
a)
b)
c)
d)
Between 92% and 98% of the people chosen for the sample were
contacted.
The percentage of people who said “Yes” to the question was
between 92% and 98%.
95% of the time, the sample percentage differs from the true
population value by exactly 3 percentage points.
95% of all samples chosen using the same method will give a
sample percent within 3 percentage points of the true population
value.
Margin of error (answer)
When an opinion poll states with 95% confidence the margin of error for
the sample percentage is plus or minus 3 percentage points, this
means that
a)
b)
c)
d)
Between 92% and 98% of the people chosen for the sample were
contacted.
The percentage of people who said “Yes” to the question was
between 92% and 98%.
95% of the time, the sample percentage differs from the true
population value by exactly 3 percentage points.
95% of all samples chosen using the same method will give a
sample percent within 3 percentage points of the true
population value.
Hypotheses
Suppose we are interested in testing
H 0 : p  p0
H a : p  p0
What is
a)
b)
c)
p0?
The sample proportion.
The population proportion.
The hypothesized value of the population proportion.
Hypotheses (answer)
Suppose we are interested in testing
H 0 : p  p0
H a : p  p0
What is
a)
b)
c)
p0?
The sample proportion.
The population proportion.
The hypothesized value of the population proportion.
P-value
For a test of hypotheses for p, the P-value is
a)
b)
c)
The probability that p equals p0.
The probability of getting a p̂ equal to our observed value of p̂ or
more extreme computed assuming the null hypothesis is true.
The probability of getting a p̂ equal to our observed value of p̂ or
more extreme computed assuming the alternative hypothesis is
true.
P-value (answer)
For a test of hypotheses for p, the P-value is
a)
b)
c)
The probability that p equals p0.
The probability of getting a p̂ equal to our observed value of p̂
or more extreme computed assuming the null hypothesis is
true.
The probability of getting a p̂ equal to our observed value of p̂ or
more extreme computed assuming the alternative hypothesis is
true.
Margin of error
If you increase the acceptable margin of error from 0.2 to 0.3, the
required sample size will
a)
b)
c)
d)
Decrease.
Remain the same.
Increase.
Either increase or decrease because the sample sizes vary
according to chance.
Margin of error (answer)
If you increase the acceptable margin of error from 0.2 to 0.3, the
required sample size will
a)
b)
c)
d)
Decrease.
Remain the same.
Increase.
Either increase or decrease because the sample sizes vary
according to chance.
Margin of error
Suppose you want to estimate the proportion of adults in Vermont
(population 0.6 million) that approve of the new health care bill. You
also want to estimate the proportion of adults in New York
(population 19 million) that approve the new health care bill. To
achieve the same margin of error in a 95% confidence interval for p
in Vermont and New York, the required sample size of the Vermont
sample will be
a)
b)
c)
Smaller than the New York sample.
Larger than the New York sample.
Approximately the same as the New York sample.
Margin of error (answer)
Suppose you want to estimate the proportion of adults in Vermont
(population 0.6 million) that approve of the new health care bill. You
also want to estimate the proportion of adults in New York
(population 19 million) that approve the new health care bill. To
achieve the same margin of error in a 95% confidence interval for p
in Vermont and New York, the required sample size of the Vermont
sample will be
a)
b)
c)
Smaller than the New York sample.
Larger than the New York sample.
Approximately the same as the New York sample.
Non-response
You sent out the number of surveys required by the sample size
formula for an acceptable margin of error and your non-response
rate is 50%. What affect will this have on your results?
a)
b)
c)
d)
Decrease the margin of error, but not affect the accuracy of the
results.
Increase the margin of error, but not affect the accuracy of the
results.
Bias the results if the non-response is due to some characteristic of
the population.
Bias the results and decrease the margin of error.
Non-response (answer)
You sent out the number of surveys required by the sample size
formula for an acceptable margin of error and your non-response
rate is 50%. What affect will this have on your results?
a)
b)
c)
d)
Decrease the margin of error, but not affect the accuracy of the
results.
Increase the margin of error, but not affect the accuracy of the
results.
Bias the results if the non-response is due to some
characteristic of the population.
Bias the results and decrease the margin of error.