Sampling Distributions
BPS chapter 11
© 2010 W.H. Freeman and Company
Parameters and statistics
The average fuel tank capacity of all cars made by Ford is 14.7 gallons.
This value represents a
a)
b)
c)
d)
Parameter because it is an average from all possible cars.
Parameter because it is an average from all Ford cars.
Statistic because it is an average from a sample of all cars.
Statistic because it is an average from a sample of American cars.
Parameters and statistics (answer)
The average fuel tank capacity of all cars made by Ford is 14.7 gallons.
This value represents a
a)
b)
c)
d)
Parameter because it is an average from all possible cars.
Parameter because it is an average from all Ford cars.
Statistic because it is an average from a sample of all cars.
Statistic because it is an average from a sample of American cars.
Parameters and statistics
The fraction of all American adults who received at least one speeding
ticket last year can be represented by
a)
b)
c)
d)
.
.
.
.
Parameters and statistics (answer)
The fraction of all American adults who received at least one speeding
ticket last year can be represented by
a)
b)
c)
d)
.
.
.
.
Parameters and statistics
The mean distance traveled in a year by a sample of truck drivers can
be represented by
a)
b)
c)
d)
.
.
.
.
Parameters and statistics (answer)
The mean distance traveled in a year by a sample of truck drivers can
be represented by
a)
b)
c)
d)
.
.
.
.
Sample size
We wish to estimate the mean price, , of all hotel rooms in Las Vegas.
The Convention Bureau of Las Vegas did this in 1999 and used a
sample of n = 112 rooms. In order to get a better estimate of  than
the 1999 survey, we should
a)
b)
c)
Take a larger sample because the sample mean will be closer to .
Take a smaller sample since we will be less likely to get outliers.
Take a different sample of the same size since it does not matter
what n is.
Sample size (answer)
We wish to estimate the mean price, , of all hotel rooms in Las Vegas.
The Convention Bureau of Las Vegas did this in 1999 and used a
sample of n = 112 rooms. In order to get a better estimate of  than
the 1999 survey, we should
a)
b)
c)
Take a larger sample because the sample mean will be closer
to .
Take a smaller sample since we will be less likely to get outliers.
Take a different sample of the same size since it does not matter
what n is.
Law of Large Numbers
The Law of Large Numbers says that
a)
b)
c)
d)
If the population is large, the estimate for  would be better than if
the population was small.
is a good estimate for  as long as the size of the population was
large.
If we increase our sample size, will be a better estimate for .
As long as the values for x are large, our estimate for  will be good.
Law of Large Numbers (answer)
The Law of Large Numbers says that
a)
b)
c)
d)
If the population is large, the estimate for  would be better than if
the population was small.
is a good estimate for  as long as the size of the population was
large.
If we increase our sample size, will be a better estimate for .
As long as the values for x are large, our estimate for  will be good.
Sampling distribution
If the first graph shows the population, which plot could be the sampling
distribution of if all samples of size n = 50 are drawn?
a)
b)
c)
Plot A
Plot B
Plot C
Sampling distribution (answer)
If the first graph shows the population, which plot could be the sampling
distribution of if all samples of size n = 50 are drawn?
a)
b)
c)
Plot A
Plot B
Plot C
Sampling distribution
Which of the following is true?
a)
b)
c)
d)
The shape of the sampling distribution of is always bell-shaped.
The shape of the sampling distribution of gets closer to the shape
of the population distribution as n gets large.
The shape of the sampling distribution of
gets approximately
normal as n gets large.
The mean of the sampling distribution of
gets closer to µ as n
gets large.
Sampling distribution (answer)
Which of the following is true?
a)
b)
c)
d)
The shape of the sampling distribution of is always bell-shaped.
The shape of the sampling distribution of gets closer to the shape
of the population distribution as n gets large.
The shape of the sampling distribution of
gets
approximately normal as n gets large.
The mean of the sampling distribution of
gets closer to µ as n
gets large.
Sampling distribution
True or false: The shape of the sampling distribution of
more normal the larger your sample size is.
a)
b)
True
False
becomes
Sampling distribution (answer)
True or false: The shape of the sampling distribution of
more normal the larger your samples size is.
a)
b)
True
False
becomes
Sampling distribution
True or false: The standard deviation of the sampling distribution of
is always less than the standard deviation of the population when
the sample size is at least 2.
a)
b)
True
False
Sampling distribution (answer)
True or false: The standard deviation of the sampling distribution of
is always less than the standard deviation of the population when
the sample size is at least 2.
a)
b)
True
False
Sampling distribution
The theoretical sampling distribution of
a)
b)
c)
d)
Gives the values of 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 values in a random sample of
size n.
Sampling distribution (answer)
The theoretical sampling distribution of
a)
b)
c)
d)
Gives the values of 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 values in a random sample of
size n.
Standard error
What does
a)
b)
c)

n
measure?
The spread of the population.
The spread of the ‘s.
Different values
could be.
Standard error (answer)
What does
a)
b)
c)

n
measure?
The spread of the population.
The spread of the ‘s.
Different values
could be.
Sample size
True or false: If we want the mean of all possible
we must use a large sample size.
a)
b)
True
False
‘s to equal , then
Sample size (answer)
True or false: If we want the mean of all possible
we must use a large sample size.
a)
b)
True
False
‘s to equal , then
Sampling distributions
The following density curve represents waiting times at a customer
service counter at a national department store. The mean waiting
time is 5 minutes with standard deviation 5 minutes. If we took all
possible samples of size n = 100, how would you describe the
sampling distribution of the ‘s?
a)
b)
c)
d)
Shape = right skewed, center = 5, spread = 5
Shape = less right skewed, center = 5, spread = 0.5
Shape = approx. normal, center = 5, spread = 5
Shape = approx. normal, center = 5, spread = 0.5
Sampling distributions (answer)
The following density curve represents waiting times at a customer
service counter at a national department store. The mean waiting
time is 5 minutes with standard deviation 5 minutes. If we took all
possible samples of size n = 100, how would you describe the
sampling distribution of the ‘s?
a)
b)
c)
d)
Shape = right skewed, center = 5, spread = 5
Shape = less right skewed, center = 5, spread = 0.5
Shape = approx. normal, center = 5, spread = 5
Shape = approx. normal, center = 5, spread = 0.5
Central Limit Theorem
Which is a true statement about the Central Limit Theorem?
a)
b)
c)
d)
We need to take repeated samples in order to estimate .
It only applies to populations that are Normally distributed.
It says that the distribution of ‘s will have the same shape as the
population.
It requires the condition that the sample size, n, is large and that the
samples were drawn randomly.
Central Limit Theorem (answer)
Which is a true statement about the Central Limit Theorem?
a)
b)
c)
d)
We need to take repeated samples in order to estimate .
It only applies to populations that are Normally distributed.
It says that the distribution of ‘s will have the same shape as the
population.
It requires the condition that the sample size, n, is large and
that the samples were drawn randomly.
Central Limit Theorem
True or false: The Central Limit Theorem allows us to compute
probabilities on when the conditions are met.
a)
b)
True
False
Central Limit Theorem (answer)
True or false: The Central Limit Theorem allows us to compute
probabilities on when the conditions are met.
a)
b)
True
False
Sampling distributions
The following density curve shows a sampling distribution of
created
by taking all possible samples of size n = 6 from a population that
was very left-skewed. Which of the following would result in a
decrease of the area to the left of 0.5 (denoted by the vertical line)?
a)
b)
c)
d)
Increasing the number of samples taken.
Taking a different sample.
Decreasing n.
Increasing the number of observations in each sample.
Sampling distributions (answer)
The following density curve shows a sampling distribution of
created
by taking all possible samples of size n = 6 from a population that
was very left-skewed. Which of the following would result in a
decrease of the area to the left of 0.5 (denoted by the vertical line)?
a)
b)
c)
d)
Increasing the number of samples taken.
Taking a different sample.
Decreasing n.
Increasing the number of observations in each sample.
Sampling distributions
What effect does increasing the sample size, n, have on the center of
the sampling distribution of ?
a)
b)
c)
d)
The mean of the sampling distribution gets closer to the mean of the
population.
The mean of the sampling distribution gets closer to 0.
The variability of the population mean is decreased.
It has no effect. The mean of the sampling distribution always
equals the mean of the population.
Sampling distributions (answer)
What effect does increasing the sample size, n, have on the center of
the sampling distribution of ?
a)
b)
c)
d)
The mean of the sampling distribution gets closer to the mean of the
population.
The mean of the sampling distribution gets closer to 0.
The variability of the population mean is decreased.
It has no effect. The mean of the sampling distribution always
equals the mean of the population.
Sampling distributions
What effect does increasing the sample size, n, have on the spread of
the sampling distribution of ?
a)
b)
c)
d)
The spread of the sampling distribution gets closer to the spread of
the population.
The spread of the sampling distribution gets larger.
The spread of the sampling distribution gets smaller.
It has no effect. The spread of the sampling distribution always
equals the spread of the population.
Sampling distributions (answer)
What effect does increasing the sample size, n, have on the spread of
the sampling distribution of ?
a)
b)
c)
d)
The spread of the sampling distribution gets closer to the spread of
the population.
The spread of the sampling distribution gets larger.
The spread of the sampling distribution gets smaller.
It has no effect. The spread of the sampling distribution always
equals the spread of the population.
Sampling distributions
What effect does increasing the sample size, n, have on the shape of
the sampling distribution of ?
a)
b)
c)
The shape of the sampling distribution gets closer to the shape of
the population.
The shape of the sampling distribution gets more bell-shaped.
It has no effect. The shape of the sampling distribution always
equals the shape of the population.
Sampling distributions (answer)
What effect does increasing the sample size, n, have on the shape of
the sampling distribution of ?
a)
b)
c)
The shape of the sampling distribution gets closer to the shape of
the population.
The shape of the sampling distribution gets more bell-shaped.
It has no effect. The shape of the sampling distribution always
equals the shape of the population.
Sampling distributions
Which of the following would result in a decrease in the spread of the
approximate sampling distribution of ?
a)
b)
c)
d)
Increasing the sample size.
Increasing the number of samples taken.
Increasing the population standard deviation
Decreasing the value of the population mean.
Sampling distributions (answer)
Which of the following would result in a decrease in the spread of the
approximate sampling distribution of ?
a)
b)
c)
d)
Increasing the sample size.
Increasing the number of samples taken.
Increasing the population standard deviation
Decreasing the value of the population mean.
Sampling distributions
Time spent working out at a local gym is normally distributed with mean 
= 43 minutes and standard deviation  = 6 minutes. The gym took a
sample of size n = 24 from its patrons. What is the distribution of ?
a)
b)
c)
Normal with mean  = 43 minutes and standard deviation  = 6
minutes.
6
Normal with mean  = 43 minutes and standard deviation  =
24
minutes.
Cannot be determined because the sample size is too small.
Sampling distributions (answer)
Time spent working out at a local gym is normally distributed with mean 
= 43 minutes and standard deviation  = 6 minutes. The gym took a
sample of size n = 24 from its patrons. What is the distribution of ?
a)
b)
c)
Normal with mean  = 43 minutes and standard deviation  = 6
minutes.
6
Normal with mean  = 43 minutes and standard deviation  =
24
minutes.
Cannot be determined because the sample size is too small.
Control charts
On a control chart, what are on the x and y axes?
a)
b)
c)
Time is on the x-axis and the upper control limit is on the y-axis.
The lower control limit is on the x-axis and the upper control limit is on
the y-axis.
Time is on the x-axis and values of the means are on the y-axis.
Control charts (answer)
On a control chart, what are on the x and y axes?
a)
b)
c)
Time is on the x-axis and the upper control limit is on the y-axis.
The lower control limit is on the x-axis and the upper control limit is on
the y-axis.
Time is on the x-axis and values of the means are on the y-axis.
Control charts
Look at the following plots. Which show(s) a process that is out-ofcontrol?
a)
b)
c)
d)
Plot A
Plot B
Both plots
Neither plot
Control charts (answer)
Look at the following plots. Which show(s) a process that is out-ofcontrol?
a)
b)
c)
d)
Plot A
Plot B
Both plots
Neither plot