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Class

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 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 ID Age Sex ID Age Sex 1 30 M 11 38 F 21 74 F 31 32 M 41 46 M 2 48 M 12 24 M 22 21 M 32 28 F 42 34 F 3 52 M 13 48 F 23 29 F 33 35 M 43 44 F 4 25 F 14 45 M 24 48 M 34 49 M 44 68 M 5 63 F 15 28 F 25 37 M 35 18 M 45 24 F 6 50 F 16 39 M 26 52 F 36 56 F 46 34 F 7 18 F 17 37 F 27 25 F 37 48 F 47 55 F 8 28 F 18 63 F 28 44 M 38 38 F 48 39 M 9 72 M 19 20 M 29 29 F 39 52 F 49 40 F 10 25 F 20 81 F 30 66 M 40 33 F 50 30 F

8-5

Video Tutor

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Frequency

A Use your calculator to find the mean age and standard deviation for the population of the gym's first 50 members. Round to the nearest tenth.

=

; =

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

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 _x_and the standard deviation of the sample means _x .

_x_ =

; _x =

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E Now use your calculator's random number generator to choose a sample of 15 gym

__

members. Find the mean xfor 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 _x_and the standard deviation of the sample means _x .

_x_ =

; _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?

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 _x_. The standard deviation of the sampling distribution of the sample mean is denoted _xand is also called the standard error of the mean.

__

You may have discovered that _x_is close to xregardless of the sample size and that _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.

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Properties of the Sampling Distribution of the Sample Mean

If a random sample of size n is selected from a population with mean and standard deviation , then

(1) _x_= , (2) _x = _ __?n_, and (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.

=

=

n =

? Find the mean of the sampling distribution of the sample mean and the standard error of the mean.

_x_= =

_x_= _ _?n__= _ ______

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.

_x_- 2_x_ =

(- 2

) =

_x_+ 2_x_=

(+ 2

) =

So, 95% of the sample means fall between

g and

g.

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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.

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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 p^ for your sample.

p^ =

C Report your sample proportion to your teacher. As

10

other students report their sample proportions, create a

class histogram at right.

8

Frequency

6

D Calculate the mean of the sample proportions p^and

4

the standard deviation of the sample proportions p^.

Round to the nearest hundredth.

2

p^ =

; p^=

-0.05 0.15 0.35 0.55 0.75 0.95 0.05 0.25 0.45 0.65 0.85 1.05 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 p^ for your sample.

p^ =

10

F Report your sample proportion to your teacher. As other

8

students report their sample proportions, create a class

histogram at right.

6

Frequency

G Calculate the mean of the sample proportions p^and the standard deviation of the sample proportions p^.

Round to the nearest hundredth.

p^ =

; p^=

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 Sample Proportion

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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 p^ is the proportion of successes in a sample. The mean of the sampling distribution of the sample proportion is denoted p^. The standard deviation of the sampling distribution of the sample proportion is denoted p^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) p^= p,

(2) p^= ?p_ _(?1__n-?__p_?_) , and

(3) if both np and n(1 - p) are at least 10, then the sampling distribution of the sample proportion is approximately normal.

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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.

p^= p =

( ) ????????

p^= ?p__(?1__n-?__p_?_)= _ _______1__-__ _ __ _ _ __

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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.

p^- 2p^=

(- 2

) =

p^+ 2p^=

(+ 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

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.

A 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 p^ .

In this case, p = p^= p =

and n =

.

( ) and p^= ?p_ _(?_1_n-?__p?__) = ? _ __?__?__?_ _1_?-__?_ _ ?__? _ _ _?_

The reasonably likely values of p^ fall within 2 standard deviations of p^.

p^- 2p^= p^+ 2p^=

+

22((

) = ) =

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? Houghton Mifflin Harcourt Publishing Company Proportion of Successes in Population, p

B On the graph, draw a horizontal line segment at the level of 0.3 on the vertical axis to represent the interval of likely values of p^ 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 Draw a vertical line at 0.4 on the horizontal axis. This represents p^ = 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.

1 0.9 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, p^

REFLECT

5a. Is it possible that 30% of all students at the school have attended a football game? Is it likely? Explain.

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.

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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 p^- 2p^and p^+ 2 p^where p^ = 0.4. Since the bold line segments intersect at (0.4, 0.4), the vertical bold line segment has these same endpoints.

The above argument shows that you can find the endpoints of the confidence interval by finding the endpoints of the horizontal segment centered at p^ . 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.

Proportion of Successes in Population, p

1 0.9 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, p^

A Confidence Interval for a Population Proportion

A c% confidence interval for the proportion p of successes in

a population is given by

p^

-

zc

? p_^_(_? 1_n-_?__p^? __)

p

p^

+

zc

? p_^_(_? 1_n-_?__p^? __)

where p^ is the sample proportion, n is the sample size, and zcdepends upon the desired degree of confidence.

In order for this interval to describe the value of p reasonably accurately, three conditions must be met.

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. np^ and n(1 - p^) 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 z9 5%.)

Desired degree of confidence Value of z c

90% 1.645

95% 1.96

99% 2.576

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