Download Lesson 15-7 Sample and Population

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Algebra 2: Lesson 15-7 Sample and Population Proportions
Learning
1.
What is sampling variability?
2.
How is standard deviation used in sample proportions?
3.
How do you estimate a population proportion?
4.
What happens to the standard deviation of a sample proportion as you increase the number of data values?
DEFINITIONS

A population proportion is the ratio of members of a population with a particular characteristic to the
total members of the population.

A sample proportion is the ratio of members of a sample of the population with a particular characteristic
to the total members of the sample.

Not every random sample results in the same estimate of a population characteristic; there will be some
sampling variability. Larger sample sizes, however, tend to produce more accurate estimates.

A dotplot is a type of graphic display used to compare frequency counts within categories or groups.
 Each dot represents a specific number of observations from a set of data.
 The dots are stacked in a column over a category, so that the height of the column represents the
relative of absolute frequency of observations in the category.
 The pattern of data in a dotplot can be described in terms of symmetry and skewness.
Example: A group of eleventh graders wanted to estimate the population proportion of students in their high
school who drink at least one soda per day. Each student selected a different random sample of
students and
calculated the proportion that drink at least one soda per day. The dot plot below shows the sampling
distribution.
 Describe the shape of the distribution.
Estimating a Sample Proportion
Example: A recent poll stated that
of Americans pay “a great deal” or a “fair amount” of attention to the
nutritional information that restaurants provide. This poll was based on a random sample of
adults living in
the United States. The
corresponds to a proportion of
, and
is called a sample proportion. It is an
estimate of the proportion of all adults who would say they pay “a great deal” or a “fair amount” of attention to
the nutritional information that restaurants provide.
If you were to take a random sample of
attention to nutritional information?
Americans, how many would you predict would say that they pay
Estimating a Population Proportion
Example: A teacher gave a class of students a container of dried beans. Some of the beans in the container are
black. The teacher wanted the students to see what happens when you take a sample of beans from the
container and use the proportion of black beans in the sample to estimate the proportion of black beans in the
container (a population proportion)
Each person in the class randomly selected a sample of 20 beans from the container by carefully mixing all the
beans and then selecting one bean and recording its color. They would then replace the bean, mix the bag, and
continue to select one bean at a time until 20 beans have been selected. The results were recorded in the
sampling distribution below.
a) Describe the shape of the distribution.
b) What was the smallest sample proportion observed?
c) What was the largest sample proportion observed?
d) What sample proportion occurred most often?
e) Using technology, find the mean and standard deviation of the sample proportions used to construct the
sampling distribution created by the class.
f)
How does the mean of the sampling distribution compare with the population proportion of 0.40?
What do you think would happen to the sampling distribution if everyone in class took a random sample of 40
beans from the container? The class decided to repeat the process described above, but this time you will draw a
random sample of 40 beans instead of 20.
a) Describe the shape of the distribution.
b) What was the smallest sample proportion observed?
c) What was the largest sample proportion observed?
d) What sample proportion occurred most often?
e) Using technology, find the mean and standard deviation of the sample proportions used to construct the
sampling distribution created by the class.
f)
How does the mean of the sampling distribution compare with the population proportion of 0.40?
g) How does the mean of the sample distribution based on random samples of size 20 compare to the mean
of the sampling distribution based on random samples of size 40?
h) As the sample size increased from 20 to 40, what happened to the standard deviation?
i)
What do you think would happen to the variability (standard deviation) of the distribution of the sample
proportions if the sample size for each samples was 80 instead of 40? Explain.
Summary
The sampling distribution of the sample proportion can be approximated by a graph of the sample proportions for
many different random samples. The mean of the sampling distribution of the sample proportions will be
approximately equal to the value of the population proportion.
As the sample size increases, the sampling variability in the sample proportion decreases; in other words, the
standard deviation of the sampling distribution of the sample proportions decreases.
PRACTICE
1.
A class of
eleventh graders wanted to estimate the proportion of all juniors and seniors at their high school
with part-time jobs after school. Each eleventh grader took a random sample of
juniors and seniors and
then calculated the proportion with part-time jobs. Following are the
sample proportions.
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
a.
Construct a dot plot of the sample proportions.
b.
Describe the shape of the distribution.
c.
Using technology, find the mean and standard deviation of the sample proportions.
d.
Do you think that the proportion of all juniors and seniors at the school with part-time jobs could be
Do you think it could be
? Justify your answers based on your dot plot.
e.
Suppose the eleventh graders had taken random samples of size . How would the distribution of
sample proportions based on samples of size
differ from the distribution for samples of size ?
?
2. Below are three dot plots of the proportion of tails in , , or
simulated flips of a coin. The mean and
standard deviation of the sample proportions are also shown for each of the three dot plots. Match each dot
plot with the appropriate number of flips. Clearly explain how you matched the plots with the number of
simulated flips.
Dot Plot 1
Mean:
Standard deviation:
Sample Size: _________
Explain:
Dot Plot 2
Mean:
Standard deviation:
Sample Size: _________
Explain:
Dot Plot 3
Mean:
Standard deviation:
Sample Size: _________
Explain: