Download Chapter 1 Introduction to Data

Chapter 9
Inferring
Population
Means
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Slide 1
Chapter 9 Topics
• Apply the Central Limit Theorem for sample
means
• Make inferences about population means
using confidence intervals and hypothesis
tests
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Slide 2
Section 9.1
Studio 8. Pearson Education Ltd
SAMPLE MEANS OF RANDOM
SAMPLES
• Discuss the Accuracy and Precision of the Sample
Mean as an Estimate of the Population Mean
• Find the Standard Error of the Sampling
Distribution of the Sample Mean
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Slide 3
Three Characteristics of the
Sample Mean
1. Accuracy
2. Precision
3. Probability distribution
Notation Review:
μ = population mean
x = sample mean
σ = population standard deviation
s = sample standard deviation
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Slide 44
Accuracy and Precision of the Sample
Mean
The sample mean is unbiased when estimating
the population mean – on average, the sample
mean is the same as the population mean.
The precision of the sample mean depends on
the variability in the population. The more
observations we collect, the more precise the
sample mean becomes.
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Sampling Distribution of the Sample
Mean
The sampling distribution can be thought of as
the distribution of all possible sample means
that would result from drawing repeated
samples of a certain size from the population.
Reminder: The standard deviation of the
sampling distribution is called the standard
error.
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Effect of Sample Size on the Sampling
Distribution
As the sample size increases the standard error decreases but
the mean remains the same.
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Accuracy and Bias
of the Sample Mean
For all populations, the sample mean, if based
on a random sample, is unbiased when
estimating the population mean. The standard
error of the sample mean is s .
n
As the sample size increases, the sample mean
becomes more precise.
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Example: Playlists
A student has a large digital music library. The
mean length of the songs is 258 seconds with a
standard deviation of 87 seconds. The student
creates a playlist that consists of 30 randomly
selected songs.
a. Is the mean value of 258 seconds a
parameter or a statistic? Explain.
b. What should the student expect the average
song length to be for his playlist?
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Example: Playlists
c. What is the standard error for the mean song
length of 30 randomly selected songs?
d. Would a playlist of 50 songs have a standard
error that is greater than or less than your
answer to part c? Explain.
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10
Example: Playlists
a. The mean value of 258 seconds is a
parameter because it is the mean of the
population (all songs in the digital music
library)
b. The student should expect that the average
song length of the playlist is 258 seconds
(sample mean is typically the same as the
population mean)
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11
Example: Playlists
c. Standard error =
s
87
=
= 15.88 seconds
n
30
d. If the playlist had 50 songs, the standard
error would be smaller because the standard
error decreases as the sample size increases.
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12
Section 9.2
sonya etchison. Shutterstock
CENTRAL LIMIT THEOREM FOR
SAMPLE MEANS
• Apply the Central Limit Theorem for Sample Means
to Calculate Probabilities
• Describe the Features of the t-distribution
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Slide 13
Central Limit Theorem
for Sample Means
If certain conditions are met, the Central Limit
Theorem assures us that the distribution of
sample means follows an approximately Normal
distribution no matter what the shape of the
population distribution.
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14
Conditions to Check
When determining whether you can apply the Central
Limit Theorem to analyze data, consider these three
conditions:
1. Random Sample and Independence. Each
observation is collected randomly from the
population and observations are independent of
each other.
2. Large sample. Either the population distribution is
Normal or the sample size is large (usually 25 is large
enough).
3. Big population. If the sample is collected without
replacement then the population must be at least
10 times larger than the sample size.
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15
Central Limit Theorem
If the three conditions are met, then the sampling
distribution of x
is approximately
æ s ö
N ç m,
÷
è
nø
The larger the sample size the better the
approximation. If the population is normal to begin
with then the sampling distribution is exactly a Normal
distribution, regardless of the sample size.
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16
Visualizing Distributions of the Sample
Mean
The figure below shows the distribution of in-state
tuition and fees for all two-year colleges in the United
States for the 2012-2013 academic year. Note that it is
skewed and multimodal.
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17
Visualizing Distributions of the Sample
Mean
The figures below show the distribution of the sample mean for
samples of size 30 and size 90 drawn from the skewed
multimodal distribution on the previous slide. Note the shape of
each is approximately Normal and the standard error decreases
as the sample size increases.
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18
Example:
Weights of 10-Year-Old Boys
According to data from the National Health and
Nutrition Exam Survey, the mean weight of 10-year-old
boys is 88.3 pounds with a standard deviation of 2.06
pounds. Assume the distribution of weights is Normal.
a. Suppose we take a random sample of 30 boys from
this population. Can we find the approximate
probability that the average weight of this sample
will be above 89 pounds? If so, find it. If not,
explain why.
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19
Example:
Weights of 10-Year-Old Boys
b. Suppose we take a random sample of 10 boys from
this population. Can we find the approximate
probability that the average weight of this sample
will be above 89 pounds? If so, find it. If not,
explain why not.
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20
Example:
Weights of 10-Year-Old Boys
a. Suppose we take a random sample of 30 boys from this
population. Can we find the approximate probability that
the average weight of this sample will be above 89 pounds?
If so, find it. If not, explain why.
We have a random sample, the population is at least 10 times
larger than the sample, and the population is Normal so we can
apply the Central Limit Theorem. The distribution of sample
means is
æ
2.06 ö
N ç88.3,
÷
è
30 ø
We can use technology to find the probability that the sample
mean is greater than 89 pounds.
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21
The standard error =
s
2.06
=
= 0.3761
n
30
The probability that the
mean of the sample is
greater than 89 pounds
is 0.03.
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Slide 22
Example:
Weights of 10-Year-Old Boys
b. Suppose we take a random sample of 10 boys from this
population. Can we find the approximate probability that
the average weight of this sample will be above 89 pounds?
If so, find it. If not, explain why not.
We have a random sample, the population is at least 10 times
larger than the sample, and the population is Normal. So, we can
apply the Central Limit Theorem even though the sample size is
less than 25. The distribution of sample means is

2.06 
N  88.3,
.


10 
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23
The standard error is

2.06

n
 0.6514.
10
The probability the
sample mean is greater
than 89 pounds is 0.1413.
Note that this probability
is different than our
answer to part (a) due to
the larger standard error
in the distribution of
samples of size 10.
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Slide 24
Example: Home Prices
Home prices in a certain community have a distribution that is
skewed right. The mean of the home prices is $498,000 with a
standard deviation of $25,200.
a. Suppose we take a random sample of 30 homes in this
community. What is the probability that the mean of this
sample is between $500,000 and $510,000?
b. Suppose we take a random sample of 10 homes in this
community. Can we find the approximate probability that
the mean of the sample is more than $510,000? If so, find it.
If not, explain why not.
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25
Example: Home Prices
a. Suppose we take a random sample of 30 homes in this
community. What is the probability that the mean of this
sample is between $500,000 and $510,000?
We have a random sample. The sample is large (greater than 25)
and the population is large (we assume there are more than
10x30 = 300 homes in the community) so according to the
Central Limit Theorem the distribution of sample means is

25200 
N  498, 000,
.


30 
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26
The standard error is

25200

 4600.87.
n
30
The probability the
sample mean is
between $500,000
and $510,000 is
0.3273.
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Slide 27
Example: Home Prices
b. Suppose we take a random sample of 10 homes in this
community. Can we find the approximate probability that
the mean of the sample is more than $510,000? If so, find it.
If not, explain why not.
Because the population of home prices is skewed we would need
a sample size of at least 25 to apply the Central Limit Theorem.
Since the sample size is 10, we cannot find this probability.
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28
Identify the Distribution
The next slide shows three distributions. One of
these distributions is a population. The other
two distributions are approximate sample
distributions of the sample means randomly
sampled from the population, one of sample
size 10 and the other of sample size 25. Match
the graph with the correct distribution.
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29
Identify the Distribution
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30
Identify the Distribution
The Central Limit Theorem tells us that the sampling distribution
for sample means are approximately Normal. Since this graph is
not approximately Normal it is the graph of the population.
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31
Identify the Distribution
As the sample size increases, the standard error (spread) of the
distribution decreases. Since Figure (a) has more spread than
Figure (c), Figure (a) is the distribution for samples of size 25 and
Figure (c) is the distribution for samples of size 10.
Figure (a)
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Figure (c)
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32
The t-Statistic
Hypothesis tests and confidence intervals for
estimating and testing the mean are based on a statistic
called the t-statistic:
x -m
t=
s
n
Since the population standard deviation is almost
always unknown, we divide by an estimate of the
standard error, using the sample standard deviation s
instead of s .
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33
The t-Distribution
The t-statistic does not follow the Normal distribution,
because the denominator changes with every sample
size.
The t-statistic is more variable than the z-statistic,
whose denominator is always the same.
If the three conditions for using the Central Limit
Theorem hold, the t-statistic follows a distribution
called the t-distribution.
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34
The t-Distribution
•
•
•
•
Symmetric and “bell-shaped”
Has thicker tails than the Normal distribution
Shape depends on the degrees of freedom (df)
If df is small, the tails are thick; as df increases, tails get
thinner.
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35
Section 9.3
Imagemore Co., Ltd
ANSWERING QUESTIONS ABOUT
THE MEAN OF A POPULATION
• Construct and Interpret a Confidence Interval for a
Population Mean
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Slide 36
Answering Questions about the Mean
of a Population
There are two approaches for answering questions
about a population mean:
1. Confidence intervals – Used for estimating
parameter values
2. Hypothesis tests – Used for deciding whether a
parameter’s value is one thing or another
These are the same methods as introduced in previous
chapters for population proportions but modified to
work with population means.
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37
Confidence Interval
• Provides a range of plausible values for the
population mean along with a measure of the
uncertainty in our estimate
• Is a measure of the uncertainty in our
estimate – the higher the level of confidence,
the better our estimate is
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38
When to Use Confidence Intervals
Use confidence intervals whenever you are
estimating the value of a population parameter
on the basis of a random sample.
Do NOT use a confidence interval if there is no
uncertainty in your estimate. If you have data
for the entire population you need to find a
confidence interval since the population
parameter is known – there is no need to
estimate it.
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39
Confidence Interval for a Population
Mean: Conditions
Before constructing a confidence interval for a
population mean, check these three conditions:
1. Random, independent sample
2. Large sample – Either the population is Normally
distributed or the sample size is at least 25.
3. Big population – If the sample is collected without
replacement the population must be at least 10
times larger than the sample size.
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40
Example: Car Prices
A used car website wanted to estimate the mean price of a
2012 Nissan Altima. The site gathered data on a random
sample of 30 such cars and found a sample mean of
$16,610 and a sample standard deviation of $2736. The
95% confidence interval for the mean cost of this model car
based on this data is
(15588, 17632)
a. Describe the population. Is the number $16,610 an
example of a parameter or a statistic?
b. Verify that the conditions for a valid confidence interval
are met.
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41
Example: Car Prices
a. Describe the population. Is the number $16610 an example
of a parameter or a statistic?
The population is all 2012 Nissan Altimas that are for sale. The
number $16,610 is a statistic because it is a measure of a
sample, not a population.
b. Verify that the conditions for a valid confidence interval are
met.
The sample is random and independent. The sample size (30) is
large (at least 25) and the population is big (we can assume there
are at least 10 x 40 = 400 2012 Altimas for sale).
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42
Interpretation of the
Confidence Level
The confidence level is a measure of how well
the method used to produce the confidence
interval performs. For example, a 95%
confidence interval means that if we were to
take many random samples of the same size
from the same population, we expect 95 of
them would “work” (contain the population
parameter) and five of them would be “wrong”
(not contain the population parameter).
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43
Calculating the Confidence Interval
General structure:
Estimator ± margin of error
where
margin of error = (multiplier) x SE
and
SE =
s
n
Because we usually do not know the population standard
deviation, we replace SE with its estimate that uses the
sample standard deviation.
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44
One-Sample t-Interval
The one-sample t-interval is a confidence interval for a
population mean.
where
and
x ±m
m = t *SEEST
SEEST =
s
n
The multiplier t* is found using the t-distribution with
n–1 degrees of freedom.
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45
One-Sample t-Interval
To construct a one-sample t-interval you need
four pieces of information:
1. The sample mean, x , that you calculate from
the data.
2. The sample standard deviation, s, which you
calculate from the data.
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46
One-Sample t-Interval
3. The sample size, n.
4. The multiplier, t*, which you can look up in a
table or use technology. t* is determined by
the confidence level and the sample size. The
value of t* tells us how wise the margin of
error is, in terms of standard errors. For
example, if t*= 2 then our margin of error is 2
standard errors wide.
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47
Finding t*
t* can be found using a t-distribution table, but
because most tables stop at 35 or 40 degrees of
freedom, it it best to use technology to
construct confidence intervals.
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48
Example: Highway Speeds
Data on the speed (in mph) for random sample of 30
cars travelling on a highway was collected. The mean
speed was 63.3 mph with a standard deviation of 5.23
mph.
a. Find the 95% confidence interval for the mean
speed of all cars travelling on the highway. Verify
that the necessary conditions hold.
b. Interpret the interval.
c. Is it plausible that the mean speed of cars on the
highway is 67 mph? Why or why not?
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49
Example: Highway Speeds
a. Find the 95% confidence interval for the mean speed of
all cars travelling on the highway. Verify that the
necessary conditions hold.
The sample is random and independent. We can assume
that the population is large (at least
10 x 30 = 300 cars travel on the highway). The sample is
large (at least 25) so the necessary conditions hold. Use
technology to create the interval.
StatCrunch: Stats > T stats > One sample > with summary
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50
Confidence
Interval:
(61.35, 65.25)
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Slide 51
Example: Highway Speeds
b. Interpret the interval.
c. Is it plausible that the mean speed of cars on the
highway is 67 mph? Why or why not?
We are 95% confidence that the mean speed of cars on
the highway is between 61.35 and 65.25 mph. Since 67
is not in our interval, it is not a plausible value for the
mean speed of cars on the highway.
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52
Using the TI-84 Calculator
To construct a confidence interval for a population
mean on the TI-84 calculator:
1. Push STAT > TESTS then select T-INTERVAL.
2. If you have data in a list, select DATA. Enter the name of
the list where the data is stored, leave Freq: 1, select your
C-level and press Calculate.
If you do not have data in a list, select STATS. Enter the
sample mean, standard deviation, sample size. Select your Clevel and press Calculate.
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53
Example: Movies
A random sample of 35 college students was asked how
many movies they had seen in the previous month. The
sample mean was 4.14 movies with a standard deviation of
10.02.
a. Construct a 90% confidence interval for the mean
number of movies college students see per month.
b. If we were to use the data to construct the 95%
confidence interval, would the interval be wider or
narrower than the 90% confidence interval?
c. What would be the effect of taking a larger sample on
the width of the interval?
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54
The 90%
confidence
interval is
(1.28, 7.00).
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Slide 55
Example: Movies
a. We are 90% confident that the mean number of movies
seen by college students in a month is between 1.28 and
7.00.
b. A 95% confidence interval would be wider than a 90%
confidence interval because a 95% confidence interval
would have a larger t* multiplier than a 90% confidence
interval.
c. If we take a larger sample, the standard error would be
smaller. This means the margin of error would be
smaller and so the interval would be narrower.
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56
Section 9.4
Monkey Business Images. Shutterstock
HYPOTHESIS TESTING FOR MEANS
• Conduct a Hypothesis Test for a Population Mean
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Slide 57
Four Steps for Hypothesis Testing
1. Hypothesize State your hypotheses about
the population parameter.
2. Prepare Choose a significance level and test
statistic, check conditions and assumptions.
3. Compute to Compare Compute a test
statistic and p-value.
4. Interpret Do you reject the null hypothesis
or not? What does this mean?
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58
Hypothesis Test: Claim about a
Population Mean
Test Statistic for a One-Sample t-Test:
x - m0
t=
SEEST
where SEEST
s
=
n
If conditions hold, the test statistic follows a t-distribution with
df = n – 1.
Two Conditions:
1. Random, independent sample
2. Large sample: The population must be Normal or the sample
size must be at least 25.
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59
Example: Nursing
In 2010, the mean years of experience among a
nursing staff was 14.3 years. A nurse manager
took a survey of a random sample of 35 nurses
at the hospital and found a sample mean of
18.37 years with a standard deviation of 11.12
years. Do we have evidence that the mean
years of experience among the nursing staff at
the hospital has increased? Use a significance
level of 0.05.
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60
Example: Nursing
1. Hypotheses H0: μ = 14.3
Ha: μ > 14.3
2. Prepare Random, independent sample; sample size is large
(35 > 25)
3. Compute to Compare
SEEST
11.12
=
= 1.88
35
t=
18.37 -14.3
= 2.16
1.88
Use technology to find the p-value that corresponds with
t = 2.16
df = 34.
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61
4. Interpret
With a p-value of 0.019
and a significance level
of 0.05, we would reject
H0.
Evidence suggests that
mean experience
among the nursing staff
has increased.
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Slide 62
One- and Two-sided Alternative
Hypotheses
Just as with hypothesis tests for proportions, hypothesis tests
can be one-sided or two-sided depending on the research
question. The choice of the alternative hypothesis determines
how the p-value is calculated.
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63
Using Technology
Both the test-statistic and p-value can be
computed using technology.
StatCrunch:
Stats > t Stats > One Sample > with summary
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64
Using the TI-84 Calculator
To conduct a hypothesis test for a mean with one
population on the TI-84 calculator:
1. Push STAT > TESTS then select the T-test option.
2. If your data is in a list, select DATA, enter the number in
H0, the list the data is in. Leave Freq at 1, and highlight the
sign in your Ha. If you have summary statistics, select
STATS and enter the required statistics.
3. Press Calculate or Draw. The test statistic and p-value will
be displayed.
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Example: Hockey Attendance
In the 2010 season, average home attendance
for NHL hockey games was 17,072. Suppose a
sports statistician took a random sample of 30
home hockey games during the 2014 season and
found a sample mean of 18,104 with a standard
deviation of 1203.5. Can we conclude that
mean attendance at NHL games has changed
since the 2010 season?
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Example: Hockey Attendance
1. Hypothesize H0: μ = 17072 Ha: μ ≠ 17072
2. Prepare Random independent sample,
sample size is large, assume population is
large.
3. Use technology to find the test statistic and pvalue.
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3. Compute
to
Compare
Test statistic:
t = 4.70
p-value:
p < 0.0001
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Example: Hockey Attendance
Since the p-value is less than our significance
level, we reject H0.
Yes, mean attendance at hockey games has
changed since 2010.
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Section 9.5
Robert Kneschke. Shutterstock
COMPARING TWO POPULATION
MEANS
• Conduct Hypothesis Tests and Construct
Confidence Intervals About a Population Mean for
Independent and Dependent Samples
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Independent vs.
Dependent Samples
When comparing two populations, it is
important to note whether the data are two
independent samples or are paired (dependent)
samples.
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Paired/Dependent Samples
Each observation in one group is coupled or
paired with one particular observation in the
other group.
Examples:
• “Before and after” comparisons
• Related objects/people (twins, siblings,
spouses)
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Example:
Dependent or Independent?
People chosen in a random sample are asked
how many minutes they spend reading and how
many minutes they spent exercising during a
certain day. Researchers wanted to know how
different the mean amounts of time were for
each activity.
Is this a dependent or an independent sample?
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Example:
Dependent or Independent?
A sample of men and women each had their
hearing tested. Researchers wanted to know
whether, typically, mean and women differed in
their hearing ability.
Is this a dependent or an independent sample?
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Example:
Dependent or Independent?
A random sample of married couples are asked
how many minutes per day they spent
exercising. Means were compared to see if the
mean exercise times for husbands and wives
differed.
Is this a dependent or an independent sample?
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Example:
Dependent or Independent?
1. In the reading and exercise example, the samples are
dependent. The study was based on one sample of
people who were measured twice (once for reading and
once for exercise).
2. In the hearing example, the samples are independent
because two different populations were used: one of
men and the other of women. The people were not
related.
3. In the study involving exercise of married couples, the
sample is dependent because each husband is paired
with his wife.
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Confidence Intervals:
Independent Samples
To construct the confidence interval for the difference
in population means given independent samples, check
three conditions:
1. Random samples and independence – Both samples
are randomly taken from their populations and each
observation is independent of any other.
2. Independent samples – The samples are not paired.
3. Large samples – The populations are approximately
Normal OR the sample sizes are each 25 or more.
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Confidence Intervals:
Independent Samples
General structure:
Estimate ± margin of error
Specifically:
(estimate of difference) ± t* (SEestimate of difference)
The estimate of difference = Meanfirst sample – Meansecond sample
SEestimate of difference =
s12 s22
+
n1 n2
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Confidence Intervals:
Independent Samples
Two-Sample t-Interval
( x1 - x2 ) ± t
*
s12 s22
+
n1 n2
t* is based on an approximate t-distribution with
df as the smaller of n1 – 1 and n2 – 1. For more
accuracy, use technology.
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Interpreting Confidence Intervals:
Independent Samples μ1 – μ2
Interpreting confidence intervals for the difference of
population means given independent samples is the
same as interpreting confidence intervals for the
difference of population proportions.
1. If 0 is in the interval, there is no significant
difference between μ1 and μ2.
2. If both values in the confidence interval are positive,
then μ1>μ2.
3. If both values in the confidence interval are
negative, then μ1<μ2
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Example: Units
A college randomly surveyed day students and evening
students to determine the number of units students
were enrolled in. The data is shown in the table below.
Use the data to find a 95% confidence interval for the
difference in the mean difference in number of units
for the two groups. Based on your confidence interval,
is there a difference in mean number of units taken by
day and evening students at this college?
Day
Evening
n
72
68
mean
8.7
5.9
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3.7
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Checking Conditions:
Using Technology
Checking conditions:
1. Random sample and independence
2. Samples are independent
3. Large sample size (at least 25)
StatCrunch:
STAT > T-stats > Two sample > with summary
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95% Confidence
Interval:
(1.66, 3.94)
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Example: Units
95% Confidence Interval: (1.66, 3.94)
This is a confidence interval for μday – μevening
Since both values are positive, we know μday > μevening
Day students enroll, on the average, in more units than
evening students at this college. The difference in the
mean number of units enrolled is estimated between
1.66 and 3.94.
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Using the TI-84 Calculator
To construct a confidence interval for the difference of
population means (independent samples) on the
TI-84 calculator:
1. Push STAT > TESTS then select option 2-SAMPTInt.
2. Select DATA (if you have data in a list) or STATS (if you have
summary stats).
3. Enter the sample means, sample standard deviations, and
sample sizes as prompted by the calculator, enter the
confidence level and calculate.
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Example:
Home Prices in Albuquerque
Data on the selling price (in hundreds of $) was obtained for a
sample of homes in Albuquerque, New Mexico. Homes were
classified as located in the northeast (NE) section of the city or in
another location in the city. The data are shown in the table
below. Construct a 95% confidence interval for the difference in
mean home price between the NE section and other sections in
the city. Is there a difference in home prices?
NE
other
n
39
78
mean
972.8
1107.4
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s
320.4
401.5
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Example:
Home Prices in Albuquerque
Check conditions:
1. Random sample and independence
2. Independent samples
3. Sample size is at least 25
Use appropriate technology to construct the
interval.
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Confidence Interval
(-270.73, 1.53)
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Example:
Home Prices in Albuquerque
Confidence Interval (–270.73, 1.53)
The interval is for μNE - μother
Since the interval contains 0, there is no
significant difference in the average home prices
between the two areas.
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Hypothesis Testing: Two Means
1. Hypothesize
H0: μ1 = μ2
Ha: μ1 ≠ μ2 or μ1 < μ2 or μ1 ≠=> μ2
2. Prepare
1. Random samples, independent observations
2. Independent samples
3. Both populations are approximately Normal
OR both sample sizes are at least 25 or more.
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Hypothesis Testing: Two Means
3. Compute to Compare t =
difference in sample means – what H0 says the difference is
SEEST
H0 usually says the difference is 0, and
so
SEEST
s12 s22
=
+
n1 n2
x1 - x2
t=
SEEST
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Hypothesis Testing: Two Means
4. Interpret
Compare the p-value to the significance level. If
the p-value is less than or equal to α, we reject the null
hypothesis.
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Note about Pooling
When using technology for a two-sample t-test,
use the unpooled version. It is more accurate in
most situations than the pooled version.
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Example: Units – Hypothesis Test
Use the units data to test the claim that the mean number of
units taken by day and evening students at the college are
different.
Day
Evening
n
72
68
mean
8.7
5.9
s
3.1
3.7
1. Hypothesize
H0: μday = μevening
Ha: μday ≠ μevening
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Example: Units – Hypothesis Test
2. Prepare
1. Random samples, independent observations
2. Independent samples
3. Both sample sizes are at least 25 or more
3. Compute to Compare Use technology to compute
the test statistic and p-value.
StatCrunch: STAT > T Stats > Two Sample > with
summary
TI-84: 2-SampTTest
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95
Test statistic:
t = 4.84
p-value: < 0.0001
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Example: Units – Hypothesis Test
4. Interpret
Since our p-value is close to 0, we reject H0. There is a
difference in mean units between day and night
students at this college.
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Using the TI-84 Calculator
To run a hypothesis test for the difference of
population means (independent samples) on the TI-84
calculator:
1. Push STAT > TESTS then select option 2-SAMPTTest.
2. Select DATA (if you have data in a list) or STATS (if you have
summary stats)
3. Enter the sample means, sample standard deviations, and
sample sizes as prompted by the calculator, enter the sign
in Ha and calculate.
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Example: Home Prices in Albuquerque
– Hypothesis Test
Use the data on home prices in Albuquerque to test the
claim that the mean price of homes in the NE differs
from other areas of the city. Use a significance level of
0.05.
n
mean
s
NE
39
972.8
320.4
other
78
1107.4
401.5
1. Hypothesize
H0: μNE = μother
Ha: μNE ≠ μother
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Example: Home Prices in Albuquerque
– Hypothesis Test
2. Prepare
1. Random samples, independent observations
2. Independent samples
3. Both sample sizes are at least 25 or more
3. Compute to Compare
Use technology to compute the test statistic and
p-value.
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Test statistic:
t = -1.96
p-value: 0.0526
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Example: Home Prices in Albuquerque
– Hypothesis Test
4. Interpret
Because the p-value (0.0526) is not less than the
significance level, we cannot reject H0. We cannot
conclude that prices in the NE neighborhood differ
from the rest of the city.
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Dependent Samples
• Transform the original data from two variables
into a single variable that contains the
difference between the scores in Group 1 and
Group 2.
• After the differences have been computed, we
can apply either a confidence interval
approach or a hypothesis test approach to the
differences.
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Example: Reading Intervention
Suppose 4th grade children are exposed to a reading
intervention program designed to improve scores on a
reading assessment. Children were given a pre-test and
post-test before and after the reading intervention and the
scores are shown in the table below. Assume that all
conditions needed to construct the confidence interval are
met. Construct a 95% confidence interval for the mean
difference in reading score after participating in the
program. Based on your confidence interval, do you believe
the reading program is effective?
Pre 10 14 21 18 15 16 18 19 20 25 16
Post 12 14 23 19 21 20 19 21 20 24 16
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Example: Reading Intervention
Pre 10
Post 12
14
14
21
23
18
19
15
21
16
20
18
19
19
21
20
20
25
24
16
16
Diff
0
-2
-1
-6
-4
-1
-2
0
1
0
-2
The differences are pre – post. Use technology to create a 95%
confidence interval using the difference data.
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Example: Reading Intervention
The 95% confidence interval for the difference between
the pre-test and post-test scores is (–2.90, –0.19).
Since the confidence interval does not include 0, there
is a significant difference in the mean pre- and post-test
scores; however, the difference is not large in size (only
between 0.19 and 2.9 points).
Since the differences were for pre-test minus post-test,
having negative values means the the post-test score
was greater than the pre-test score, so there was
improvement after the reading intervention.
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Example: Reading Intervention –
Hypothesis Test
Use the reading pre/post data to conduct a hypothesis
test to see if there was a difference in mean pre/post
test scores after the reading intervention. Assume the
conditions for conducting a hypothesis test are met.
Use a 0.05 significance level.
1. Hypothesize
H0: μdifference = 0 (there is no difference in scores)
Ha: μdifference ≠ 0
2. Prepare We were told to assume the conditions
for conducting a hypothesis test are met.
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Example: Reading Intervention –
Hypothesis Test
3. Compute to Compare
Test statistic: t = -2.54
p-value: 0.03
Since the p-value is less than 0.05, reject H0.
4. Interpret
Yes, there is a difference in mean pre- and posttest scores.
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Section 9.6
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OVERVIEW OF ANALYZING MEANS
• Compare the Methods of the Chapter Used to
Analyze Means
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A Lot of Repetition…
• The hypothesis test for two means is very
similar to the test for one mean.
• The hypothesis test for paired data is really a
special case of the one-sample t-test.
• Hypothesis tests use almost the same
calculations as confidence intervals and they
impose the same conditions.
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Don’t Accept H0
If the p-value is larger than the significance level,
we do not reject the null hypothesis.
This is different from “accepting” the null
hypothesis. Just because we do not reject the
null hypothesis does not mean that we now
believe the null hypothesis is true.
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Confidence Intervals and Hypothesis
Tests
If the alternative hypothesis is two-sided, a
confidence interval can be used instead of a
hypothesis test. Both approaches will always
reach the same conclusion.
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