Download 8.2 Estimating Population Means

8.2 Estimating Population
Means
LEARNING GOAL
Learn to estimate population means and compute the
associated margins of error and confidence intervals.
Review: Central Limit Theorem
Suppose we take many random samples of
size n for a variable with any distribution
and record the means of each sample.
Then:
1. The distribution of means will be
approximately a normal distribution.
2.
3.
Notation
Inferential Statistics
Inferential statistics uses
sample statistics to estimate
population parameters.
Calories per day
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Suppose a sample of 100 college students are
asked to record the number of calories they
eat in a day. The mean of the sample is 2560
calories with a standard deviation of 320
calories.
Does that indicate that all college students on
average eat 2560 calories per day?
Would all samples of 100 students have a
mean of 2560 calories?
2560 calories is a single number estimate of
the population mean, but we want to find an
interval estimate or confidence interval.
95% Confidence Interval for
Population Mean
Back to calories per day
Suppose a sample of 100 college students
are asked to record the number of
calories they eat in a day. The mean of
the sample is 2560 calories with a
standard deviation of 320 calories.
 Compute the 95% confidence interval for
the number of calories consumed per day
for all college students.
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What does a 95% Confidence
Interval Mean?
If we repeat the process of
obtaining samples and
constructing confidence intervals,
in the long run 95% of the
confidence intervals will contain
the true population mean.
Time to earn a bachelor’s degree
In a study of the length of time that students
require to earn bachelor’s degrees, 80 students
are randomly selected and they are found to
have a mean of 4.8 years and a standard
deviation of 2.2 years (based on data from the
National Center for Education Statistics).
 What is a single value estimate of the time it
takes for the average student to complete a
bachelor’s degree?
 Find a 95% confidence interval for the time it
takes for the average student to complete a
bachelor’s degree.
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Sample Size Needed
•Many times we know what margin
of error we want, but need to
know the sample size.
•For a 95% confidence interval:
•Use a past study, similar situation,
or preliminary study to estimate .
Sample Size for Weights of Quarters
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The Tyco Video Game Corporation finds that it is
losing income because of slugs used in its video
games. The machines must be adjusted to accept
coins only if they fall within set limits. In order to
set those limits, the mean weight of quarters in
circulation must be estimated. How many
quarters must we randomly select if we want to
be 95% confident that the sample mean is within
0.025 g of the true population mean for all
quarters? Based on results from a sample of
quarters we can estimate the standard deviation
as 0.068 g.
8.3 Estimating Population
Proportions
LEARNING GOAL
Learn to estimate population proportions and compute
the associated margins of error and confidence intervals.
Internet shopping
In a Gallup poll, 1025 randomly selected
adults were surveyed and 297 of them
said they used the Internet for shopping
at least a few times a year.
 What is a single number estimate for the
proportion of adults who use the Internet
for shopping?
 Express the proportion as a percent.
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95% Confidence Interval for a
Population Proportion
Internet shopping
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In a Gallup poll, 1025 randomly selected
adults were surveyed and 297 of them said
they used the Internet for shopping at least a
few times a year.
Find a 95% confidence interval for the
proportion of all adults that use the Internet
for shopping.
If a traditional retail store wants to estimate
the percentage of adult Internet shoppers in
order to determine the maximum impact of
Internet shoppers on its sales, what
percentage should be used?
Survey Responses
In a survey of 1002 people, 701 said that
they voted in a recent presidential election
(based on data from ICR Research Group).
Voting records show that 61% of eligible
voters actually vote.
 Find a 95% confidence interval estimate of
the proportion of people who say that they
voted.
 Are the survey results consistent with the
actual voter turnout of 61%? Why or why
not?
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Choosing the Correct Sample Size
To estimate a population
proportion with a 95% degree of
confidence, the sample size should
be at least:
Downloaded Songs
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The music industry must adjust to the growing
practice of consumers downloading songs instead
of buying CDs. It therefore becomes important
to estimate the proportion of songs that are
currently downloaded. How many randomly
selected individuals that have purchased music
must be surveyed to determine the percentage
that were obtained by downloading if we want to
have 95% confidence with a margin of error of no
more than one percentage point? Of no more
than two percentage points?