Download Sampling Distributions

Name Class 8-5
Date Sampling Distributions
Extension: Confidence Intervals and Margins of Error
Essential question: How do you calculate a confidence interval and a margin
of error for a population proportion or mean?
CC.9–12.S.IC.4
1
Video Tutor
EXPLORE
Developing a Sampling Distribution
The table provides data about the first 50 people to join a new gym. For each person, the
table lists his or her member ID number, age, and sex.
ID
Age
Sex
ID
Age
Sex
ID
Age
Sex
1
30
M
11
38
F
21
74
F
2
48
M
12
24
M
22
21
M
3
52
M
13
48
F
23
29
F
4
25
F
14
45
M
24
48
M
5
63
F
15
28
F
25
37
6
50
F
16
39
M
26
52
7
18
F
17
37
F
27
8
28
F
18
63
F
28
9
72
M
19
20
M
29
10
25
F
20
81
F
30
ID
Age
Sex
ID
Age
Sex
31
32
M
32
28
F
41
46
M
42
34
F
33
35
M
43
44
F
34
49
M
44
68
M
M
35
18
M
45
24
F
F
36
56
F
46
34
F
25
F
37
48
F
47
55
F
44
M
38
38
F
48
39
M
29
F
39
52
F
49
40
F
66
M
40
33
F
50
30
F
A Use your calculator to find the mean age μ and standard deviation σ for the population
of the gym’s first 50 members. Round to the nearest tenth.
μ=
;σ=
B Use your calculator’s random number generator to choose a sample of 5 gym
__
members. Find the mean age x​
​  for your sample. Round to the nearest tenth.
__
​x​ =
C Report your sample mean to your teacher. As other students report their sample
means, create a class histogram below. To do so, shade a square above the appropriate
interval as each sample mean is reported. For sample means that lie on an interval
boundary (such as 39.5), shade a square on the interval to the right (39.5 to 40.5).
8
Frequency
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6
4
2
25.5 27.5 29.5 31.5 33.5 35.5 37.5 39.5 41.5 43.5 45.5 47.5 49.5 51.5 53.5 55.5
26.5 28.5 30.5 32.5 34.5 36.5 38.5 40.5 42.5 44.5 46.5 48.5 50.5 52.5 54.5
Sample Mean
D Calculate the mean of the sample means ​μ__x​
​​ ​  and the standard deviation of the sample
means ​σ_x​​​ ​  .
​μ​__x​​ ​  =
Chapter 8
; ​σ_x​​​ ​  =
459
Lesson 5
E Now use your calculator’s random number generator to choose a sample of 15 gym
__
members. Find the mean ​x​ for your sample. Round to the nearest tenth.
__
​x​ =
F Report your sample mean to your teacher and make a class histogram below.
Frequency
8
6
4
2
25.5 27.5 29.5 31.5 33.5 35.5 37.5 39.5 41.5 43.5 45.5 47.5 49.5 51.5 53.5 55.5
26.5 28.5 30.5 32.5 34.5 36.5 38.5 40.5 42.5 44.5 46.5 48.5 50.5 52.5 54.5
Sample Mean
G Calculate the mean of the sample means ​μ__x​
​​ ​  and the standard deviation of the sample
means ​σ_x​​​ ​  .
​μ​__x​​ ​  =
; ​σ_x​​​ ​  =
REFLECT
1a. In the class histograms, how does the mean of the sample means compare with the
population mean?
1b. What happens to the standard deviation of the sample means as the sample
size increases?
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1c. What happens to the shape of the histogram as the sample size increases?
The histograms that you made are sampling distributions. A sampling distribution
shows how a particular statistic varies across all samples of n individuals from the same
__
population. You have worked with the sampling distribution of the sample mean, ​x​ .
The mean of the sampling distribution of the sample mean is denoted ​μ__x​​​ ​  . The standard
deviation of the sampling distribution of the sample mean is denoted ​σ​_x​​ ​  and is also called
the standard error of the mean.
__
You may have discovered that ​μ​__x​​ ​  is close to ​x​ regardless of the sample size and that ​σ_x​​​ ​ 
decreases as the sample size n increases. These observations were based on simulations.
When you consider all possible samples of n individuals, you arrive at one of the major
theorems of statistics.
Chapter 8
460
Lesson 5
Properties of the Sampling Distribution of the Sample Mean
If a random sample of size n is selected from a population with mean μ and
standard deviation σ, then
(1) ​μ__x​​​  ​= μ,
(2) ​σ_x​​​  ​= ____
​ √σ   ​,  and
​ 
n ​ 
(3) the sampling distribution of the sample mean is normal if the population is
normal; for all other populations, the sampling distribution of the
mean approaches a normal distribution as n increases.
The third property stated above is known as the Central Limit Theorem.
All normal distributions have the following properties, sometimes collectively called
the 68-95-99.7 rule:
• 68% of the data fall within 1 standard deviation of the mean.
• 95% of the data fall within 2 standard deviation of the mean.
• 99.7% of the data fall within 3 standard deviation of the mean.
You will learn more about the specific properties of normal distributions later in this
chapter.
CC.9–12.S.IC.4
2
EXAMPLE
Using the Sampling Distribution of the Sample Mean
Boxes of Cruncho cereal have a mean mass of 323 g with a standard deviation
of 20 g. You choose a random sample of 36 boxes of the cereal. What interval
captures 95% of the means for random samples of 36 boxes?
• Write the given information about the population and the sample.
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μ=
σ=
n=
• Find the mean of the sampling distribution of the sample mean and the standard
error of the mean.
​μ__x​​​ ​  = μ =
The sampling distribution of the sample mean is approximately normal. In a normal
distribution, 95% of the data fall within 2 standard deviations of the mean.
​μ​__x​​ ​  - 2​σ​__x​​ ​  =
​μ​__x​​ ​  + 2​σ​__x​​ ​  =
​σ​__x​​ ​  = ____
​ √σ   ​ = _______
 ​ 
≈
​ 
​ 
n ​ 
( 
+ 2​( 
- 2​
)
 )​=
  ​=
So, 95% of the sample means fall between
Chapter 8
g and
461
g.
Lesson 5
REFLECT
2a. When you choose a sample of 36 boxes, is it possible for the sample to have a mean
mass of 315 g? Is it likely? Explain.
CC.9–12.S.IC.4
3
explore
Developing Another Sampling Distribution
Use the table of data from the first Explore. This time you will develop a
sampling distribution based on a sample proportion rather than a sample mean.
A Find the proportion p of gym members in the population who are female.
p=
B Use your calculator’s random number generator to choose a sample of 5 gym
members. Find the proportion of female members p̂ for your sample.
p̂ =
C Report your sample proportion to your teacher. As
D Calculate the mean of the sample proportions ​μp̂​ ​and
the standard deviation of the sample proportions ​σp̂​ ​.
Round to the nearest hundredth.
8
6
4
2
-0.05 0.15 0.35 0.55 0.75 0.95
0.05 0.25 0.45 0.65 0.85 1.05
; ​σp̂​ ​=
Sample Proportion
E Now use your calculator’s random number generator
to choose a sample of 10 gym members. Find the proportion of female members p̂ for
your sample.
p̂ =
10
F Report your sample proportion to your teacher. As other
G Calculate the mean of the sample proportions ​μp̂​ ​and
the standard deviation of the sample proportions ​σp̂​ ​.
Round to the nearest hundredth.
8
Frequency
students report their sample proportions, create a class
histogram at right.
6
4
2
-0.05 0.15 0.35 0.55 0.75 0.95
0.05 0.25 0.45 0.65 0.85 1.05
​μ​p̂​=
Chapter 8
Sample Proportion
; ​σp̂​ ​=
462
Lesson 5
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​μ​p̂​=
10
Frequency
other students report their sample proportions, create a
class histogram at right.
REFLECT
3a. In the class histograms, how does the mean of the sample proportions compare
with the population proportion?
3b. What happens to the standard deviation of the sample proportions as the sample
size increases?
When you work with the sampling distribution of a sample proportion, p represents the
proportion of individuals in the population that have a particular characteristic (that
is, the proportion of “successes”) and p̂ is the proportion of successes in a sample. The
mean of the sampling distribution of the sample proportion is denoted ​μ​p̂​. The standard
deviation of the sampling distribution of the sample proportion is denoted ​σp̂​ ​and is also
called the standard error of the proportion.
Properties of the Sampling Distribution of the
Sample Proportion
If a random sample of size n is selected from a population with proportion of
successes p, then
(1) ​μp̂​ ​= p,
√

p(1 - p)
(2) ​σp̂​ ​= ​ ​ ________
 
 
, and
n ​ ​ 
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(3) if both np and n(1 - p) are at least 10, then the sampling distribution of
the sample proportion is approximately normal.
CC.9–12.S.IC.4
4
EXAMPLE
Using the Sampling Distribution of the Sample
Proportion
About 40% of the students at a university live off campus. You choose a random
sample of 50 students. What interval captures 95% of the proportions for
random samples of 50 students?
A Write the given information about the population and the sample, where a success is a
student who lives off campus.
p=
n=
B Find the mean of the sampling distribution of the sample proportion and the
standard error of the proportion.
​μp̂​ ​= p =
Chapter 8
√
( 
)

1
p(1
p)
​
 ​
___________________
 ​ ​
​σ​p̂​= ​ ​ ________
 
 
 
​ 
  
  
   ≈
 ​ ​
=
​
n
√
463
Lesson 5
C Check that np and n(1 - p) are both at least 10.
np =
·
=
n(1 - p) =
·
=
Since np and n(1 - p) are both greater than 10, the sampling distribution is
approximately normal.
D In a normal distribution, 95% of the data fall within 2 standard deviations of the mean.
​μ​p̂​- 2​σ​p̂​=
​μ​p̂​+ 2​σ​p̂​=
( 
+ 2​( 
)
 )​=
- 2​
  ​=
So, 95% of the sample proportions fall between
and
.
REFLECT
4a. How likely is it that a random sample of 50 students includes 31 students who live
off campus? Explain.
Previously, you investigated sampling from a population whose parameter of interest
(mean or proportion) is known. In many real-world situations, you collect sample data
from a population whose parameter of interest is not known. Now you will learn how to
use sample statistics to make inferences about population parameters.
CC.9–12.S.IC.4
5
explore
Analyzing Likely Population Proportions
A
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You survey a random sample of 50 students at a large high school and find that
30% of the students have attended a school football game. You cannot survey
the entire population of students, but you would like to know what population
proportions are reasonably likely in this situation.
Suppose the proportion p of the population that has attended a school football game is
30%. Find the reasonably likely values of the sample proportion p̂ .
In this case, p =
​μ​p̂ ​= p =
and n =
.
√
( 
)

1
p(1
- p)
​
 ​
________
and ​σp̂ 
​ ​= ​ ​  n ​ ​ 
 = ​ ​ ____________________
  
  
 ​
    ​≈
√
The reasonably likely values of p̂ fall within 2 standard deviations of ​μp̂ ​ ​.
​μ​p̂ ​- 2​σp̂ ​ ​ =
​μ​p̂ ​+ ​2σ​p̂ ​ =
Chapter 8
( 
( 
- 2​
​
+2
)
 )​=
  ​=
464
Lesson 5
segment at the level of 0.3 on the vertical
axis to represent the interval of likely
values of p̂ that you found above.
C
Now repeat the process for p = 0.35, 0.4,
0.45, and so on to complete the graph.
You may wish to divide up the work with
other students and pool your findings.
D
1
On the graph, draw a horizontal line
Draw a vertical line at 0.4 on the
horizontal axis. This represents p̂ = 0.4.
The line segments that this vertical line
intersects are the population proportions
for which a sample proportion of 0.4 is
reasonably likely.
0.9
Proportion of Successes in Population, p
B
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Proportion of Successes in Sample, p̂
REFLECT
5a. Is it possible that 30% of all students at the school have attended a football game?
Is it likely? Explain.
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5b. Is it possible that 60% of all students at the school have attended a football game? Is
it likely? Explain.
5c. Based on your graph, which population proportions do you think are reasonably
likely? Why?
A confidence interval is an approximate range of values that is likely to include
an unknown population parameter. The level or degree of a confidence interval, such as
95%, gives the probability that the interval includes the true value of the parameter.
Recall that when data are normally distributed, 95% of the values fall within
2 standard deviations of the mean. Using this idea in the Explore, you found
a 95% confidence interval for the proportion of all students who have attended
a school football game.
Chapter 8
465
Lesson 5
The above argument shows that you can find
the endpoints of the confidence interval by
finding the endpoints of the horizontal segment
centered at p̂ . You know how to do this using
the formula for the standard error of the
sampling distribution of the sample proportion
from earlier in this lesson. Putting these ideas
together gives the following result.
1
0.9
Proportion of Successes in Population, p
To develop a formula for a confidence interval,
notice that the vertical bold line segment in the
figure, which represents the 95% confidence
interval you found in the Explore, is about
the same length as the horizontal bold line
segment. The horizontal bold line segment
has endpoints ​μ​p̂ ​- 2​σp̂ ​ ​and ​μ​p̂ ​+ 2​σp̂ ​ ​where
p̂ = 0.4. Since the bold line segments intersect
at (0.4, 0.4), the vertical bold line segment has
these same endpoints.
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Proportion of Successes in Sample, p̂
A Confidence Interval for a Population Proportion
A c% confidence interval for the proportion p of successes in
a population is given by
  
  
p̂ (1 - p̂ )
p̂ (1 - p̂ )
p̂ - z​ c​ ​ ​   ​ _________
 
 
 
 
≤ p ≤ p̂ + ​zc​ ​​   ​ _________
n ​ ​ 
n ​ ​ 
√
√
where p̂ is the sample proportion, n is the sample size, and ​
z​c​depends upon the desired degree of confidence.
In order for this interval to describe the value of p reasonably accurately, three conditions
must be met.
© Houghton Mifflin Harcourt Publishing Company
1.There are only two possible outcomes associated with the parameter of interest.
The population proportion for one outcome is p, and the proportion for the other
outcome is 1 - p.
2. np̂ and n(1 - p̂ ) must both be at least 10.
3.The size of the population must be at least 10 times the size of the sample, and the
sample must be random.
Use the values in the table below for ​zc​ ​. (Note that for greater accuracy you should use
1.96 rather than 2 for ​z95%
​ ​.)
Desired degree of confidence
90%
95%
99%
Value of z​ ​c​
1.645
1.96
2.576
Chapter 8
466
Lesson 5
CC.9–12.S.IC.4
6
EXAMPLE
Finding a Confidence Interval for a Proportion
In a random sample of 100 four-year-old children in the United States, 76 were
able to write their name. Find a 95% confidence interval for the proportion p of
four-year-olds in the United States who can write their name.
A
Determine the sample size n, the proportion p̂ of four-year-olds in the sample who can
write their name, and the value of ​zc​ ​for a 95% confidence interval.
n =
B
p̂ =
​z​c​=
Substitute the values of n, p̂, and ​z​c​into the formulas for the endpoints of the
confidence interval. Then simplify and round to two decimal places.
√
√
√
( 
)
( 
)

1​
 ​
​ ​ ____________________
  
  
 ​
    ​≈

p̂(1 - p̂)
p̂ - ​z​c​​ ​ ________
  
 ​=
n ​ 

p̂(1 - p̂)
p̂ + ​z​c​​ ​ ________
  
 ​=
n ​ 
So, you can state with 95% confidence that the proportion of all four-year-olds in the
United States who can write their name lies between
√
-
+

1​
 ​
​ ​ ____________________
  
  
    ​≈
 ​
and
.
REFLECT
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6a. Find the 99% confidence interval for p and describe how increasing the degree of
confidence affects the range of values. Why does this make sense?
You can use reasoning similar to the argument in the Explore to develop a formula for a
confidence interval for a population mean.
A Confidence Interval for a Population Mean
A c% confidence interval for the mean μ in a normally distributed
population is given by
__
σ   ​ ≤ μ ≤ __x​
σ   ​ 
x​
​  - z​ c​ ​​ ____
​  + ​zc​ ​​ ____
√
​ 
n ​ 
√
​ 
n ​ 
__
where x​
​  is the sample mean, n is the sample size, σ is the population
­standard deviation, and ​zc​ ​depends upon the desired degree of confidence.
Note that it is assumed that the population is normally distributed and that you know the
population standard deviation σ. In a more advanced statistics course, you can develop a
confidence interval formula that does not depend upon a normally distributed population
or knowing the population standard deviation.
Chapter 8
467
Lesson 5
CC.9–12.S.IC.4
7
EXAMPLE
Finding a Confidence Interval for a Mean
In a random sample of 20 students at a large high school, the mean score on
a standardized test is 610. Given that the standard deviation of all scores at
the school is 120, find a 99% confidence interval for the mean score among all
students at the school.
__
A Determine the sample size n, the sample mean ​x​ , the population standard deviation σ,
and the value of ​zc​ ​for a 99% confidence interval.
__
n=
σ=
​z​c​=
B
​x​ =
__
Substitute the values of n, x​
​  , σ, and ​zc​ ​into the formulas for the endpoints of the
confidence interval. Then simplify and round to the nearest whole number.
__
σ    ​=
​x​ - ​z​c​ ​ ____
√
​ 
n ​ 
__
σ    ​=
​x​ + ​z​c​ ​ ____
√
​ 
n ​ 
-
__________
​ 
 
 ​ 
≈
√ ​ 
​
+
__________
​ 
 
 ​ 
≈
√ ​ 
​
So, you can state with 99% confidence that the mean score among all students
at the school lies between
and
.
REFLECT
7a. What is the 99% confidence interval when the sample size increases to 50? Describe
how increasing the sample size affects the confidence interval.
© Houghton Mifflin Harcourt Publishing Company
7b. What do you assume about the test scores of all students at the school in order to
use the formula for the confidence interval?
In the previous example, you found the 99% confidence interval 541 ≤ µ ≤ 679,
which is a range of values centered at µ = 610. You can write the confidence interval
as 610 ± 69, where 69 is the margin of error. The margin of error is half the length of
a confidence interval.
Chapter 8
468
Lesson 5
Margin of Error for a Population Proportion
The margin of error E for the proportion of successes in a population with
sample proportion p̂ and sample size n is given by
√
  
p̂ (1 - p̂ )
E = z​ c​ ​​    ​ ________
 
 
n ​ ​ 
where ​zc​ ​depends on the degree of the confidence interval.
Margin of Error for a Population Mean
The margin of error E for the mean in a normally distributed population
__
with standard deviation σ, sample mean x​
​  , and sample size n is given by
σ   ​ 
E = ​z​c​​ ____
√
​ 
n ​ 
where ​zc​ ​depends on the degree of the confidence interval.
From the above formulas, it is clear that the margin of error decreases as the sample size
n increases. This suggests using a sample that is as large as possible; however, it is often
more practical to determine a margin of error that is acceptable and then calculate the
required sample size.
pra c t i c e
Bags of SnackTime Popcorn have a mean mass of 15 ounces with a standard
deviation of 1.5 ounces. A quality control inspector selects a random sample of
40 bags of popcorn at the factory. Find each of the following.
  1. What is the population mean?
© Houghton Mifflin Harcourt Publishing Company
  2. What is the sample mean?
  3. What is the sample standard deviation?
  4. The interval that captures 95% of the proportions for random samples
of 40 bags of popcorn
5. In a random sample of 100 U.S. households, 37 had a pet dog.
a. Do the data satisfy the three conditions for the confidence interval formula for
a population proportion? Why or why not?
b. Find a 90% confidence interval for the proportion p of U.S. households that
have a pet dog.
Chapter 8
469
Lesson 5
c. Find a 95% confidence interval for the proportion p of U.S. households that
have a pet dog.
6. In a quality control study, 200 cars made by a particular company were randomly
selected and 13 were found to have defects in the electrical system.
a. Give a range for the percent p of all cars made by the company that have defective
electrical systems, assuming you want to have a 90% degree of confidence.
b. What is the margin or error?
c. How does the margin of error change if you want to report the range
of percents p with a 95% degree of confidence?
7. The mean annual salary for a random sample of 300 kindergarten through 12th
grade teachers in a particular state is $50,500. The standard deviation among the
state’s entire population of teachers is $3,700. Find a 95% confidence interval for the
mean annual salary μ for all kindergarten through 12th grade teachers in the state.
8. You survey a random sample of 90 students at a university whose students’ grade-
point averages (GPAs) have a standard deviation of 0.4. The surveyed students
have a mean GPA of 3.1.
© Houghton Mifflin Harcourt Publishing Company
a. How likely is it that the mean GPA among all students at the university is 3.25?
Explain.
b. What are you assuming about the GPAs of all students at the university? Why?
9. The margin of error E for the proportion of successes in a population may be
estimated by ____
​ √1   ​ where n is the sample size. Explain where this estimate comes
​ 
n ​ 
from. (Hint: Assume a 95% confidence interval.)
Chapter 8
470
Lesson 5
8-5
Name ________________________________________
Class __________________
Name Class Date __________________
Date Chapter
Practice
Additional
Practice
Sampling Distributions
2
For Exercises 1–3, use the following information: The mean height of
the population of male high school students in a school district is
67.2 inches with a standard deviation of 2.8 inches.
1. Random samples are drawn from the
population. What is the mean of the
sampling distribution of the sample
mean?
2. What is the standard error of the mean
for a random sample of 25 students?
for a random sample of 64 students?
____________________________
_____________________________
3. What interval captures 95% of the sample means for a sample size of 64 students?
________________________________________________________________________
For Exercises 4 and 5, use the following information: From a random
sample of 30 customers who bought a soft-serve frozen yogurt cone,
26% chose chocolate-vanilla swirl.
4. Find npˆ and n ( 1 − pˆ ) where n is the sample size and p̂ is the sample proportion.
________________________________________________________________________
© Houghton Mifflin Harcourt Publishing Company
5. Is it reasonable to use this sample to construct a confidence interval for the population
proportion of customers who would choose chocolate-vanilla swirl? Explain.
________________________________________________________________________
6. In a random sample of 44 students from a school of 1290 students, 60% chose red
and blue for the new school colors over red and white. Find a 90% confidence
interval for the proportion p of students at the school who prefer red and blue. Can
you conclude with 90% confidence that the majority of students prefer red and blue?
________________________________________________________________________
7. In a random sample of 550 students in a state, the mean math score on a
standardized test was 22.6. Find a 95% confidence interval for the population mean
score μ for the state given that the standard deviation is 4.8.
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Find the margin of error E for the statistic at the given confidence level.
8. population proportion, if sample proportion p̂ = 0.11 and sample size n = 124; 99%
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9. population mean, if standard deviation σ = 6.8 and sample size n = 1000; 95%
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Chapter 8
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Lesson 5
Holt McDougal Algebra 1
Name ________________________________________ Date __________________ Class __________________
Problem
Solving
Problem
Solving
Chapter
Sampling Distributions
2
Out of a random sample of n = 40 students from a high school with
more than 2000 students, p̂ = 58% say that their morning travel time
to school is less than 30 minutes.
1. Can you use the data to construct a reasonably accurate confidence interval for the
proportion of students with a morning travel time of less than 30 minutes? Explain.
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2. Can you conclude that the morning travel time for the majority of the students is less
than 30 minutes at the 90%, 95%, or 99% confidence level? Explain.
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3. If you increase the sample size to n = 125 students and obtain the same sample
proportion of p̂ = 0.58, does this change your answer to Exercise 2? Explain.
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For Exercises 4 and 5, a random sample is drawn from a population with
a normally distributed statistic with mean µ and standard deviation σ = 20.
4. What is the margin of error E for estimating μ from the sample mean at
the 95% confidence level for a sample size of n = 100?
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5. How can you use the margin of error formula to find the minimum sample size so
that E ≤ 2.0 at the 95% confidence level? What is this sample size?
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6. What is the relationship between the margin of error E for a population proportion
when the sample proportion is p̂ = 0.2 and when the sample proportion is p̂ = 0.5 for
a given confidence level and sample size?
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Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
Chapter 8
472
50
Lesson 5
Holt McDougal Algebra 1
© Houghton Mifflin Harcourt Publishing Company
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