Chapter 9 PowerPoint 2014

Report
Estimating the Value of a Parameter
Using Confidence Intervals
Chapter 9
9.1: Confidence Interval for a
Population Mean – When population
standard deviation is known
• Confidence Interval: Interval of numbers for
an unknown parameter.
• Level of Confidence: The expected proportion
of intervals that will contain the parameter.
Denoted as a percent
Example
• 95% level of confidence
▫ Implies that if 100 different intervals are
constructed based on obtained samples, we will
expect 95 of the intervals to contain the true
parameter.
▫ i.e.: we are 95% confident that the obtained
interval contains the true population parameter.
▫ For example, if we constructed a 99% confidence
interval with a lower bound of 52 and an upper bound
of 71, we would interpret the interval as follows: “We
are 99% confident that the population mean, μ, is
between 52 and 71.
Requirements
• Population from which the
sample was drawn must be
normally distributed or the
sample size must be greater than
or equal to 30
Margin of Error
• Confidence interval estimates for the population
mean are in the form of
▫ Point estimate + margin of error
The Margin of error is a measure of how accurate the
point estimate is.
For the previous example, if we constructed a 99% confidence
interval with a lower bound of 52 and an upper bound of 71
then the point estimate would be in the middle [61.5], and
the margin of error would be 9.5
3 Factors on Margin of Error
• Level of Confidence: as the level of
confidence increases, the margin of error also
increases.
• Sample Sizes: As the size of the random
sample increases, the margin of error decreases.
• Standard deviation of the Population: The
more spread there is in the population, the wider
our interval will be for a given level of
confidence.
The Role of Sample Szie
Sample
Size
Margin of
Error
Confidence
Interval
17
0.012
(2.452, 2.476)
35
0.009
(2.455, 2.473)
Note: The larger sample size reduces the margin of error
The Role of Level of Confidence
Confidence
Level
Margin of
Error
Confidence
Interval
90%
0.008
(2.456, 2.472)
99%
0.012
(2.452, 2.476)
Note: As we increase the level of confidence, the
margin of error increased
Summaries
• If we want to be ‘more confident’ we
need to increase the sample size
• If we want to reduce the margin of
error, we would need to decrease our
level of confidence or increase our
sample size
The Z-interval
• If we know the population standard deviation,
then the confidence interval is found using a
formula called the Z-interval
• Lower:
x  z 2 

n
Upper:
x  z 2 

n
• Use Calculator:
▫ Menu: 6:statistics 6: Confidence Intervals


1: Z-interval
#23: A simple random sample of size n is drawn from a
population that is normally distributed with population
standard deviation known to be 13. The sample mean is found
to be 108
a) Compute the 96% confidence interval for μ if the
sample size n is 25.
a) Compute the 88% confidence interval for μ if the
sample size is 10. How does decreasing the sample size
affect the margin of error?
#23: A simple random sample of size n is drawn from a population that is normally
distributed with population standard deviation known to be 13. The sample mean is
found to be 108
c) Compute the 88% confidence interval if the sample size is
25. compare to part a, how does decreasing the confidence
interval affect the margin of error?
d) Could we have computed confidence intervals in parts a-c if
the population had not been normally distributed? Why?
a) If an analysis revealed three outliers greater than the mean,
how would this affect the confidence interval?
9.2: Confidence Interval for a
Population Mean – When population
standard deviation is unknown
• Since population standard deviation is unknown, we
need to use sample standard deviation: t-interval
Determine a t-value
• Find the t-value such that the area under the tdistribution to the right of the t-value is 0.2 assuming
10 degrees of freedom. That is, find t0.20 with 10
degrees of freedom.
• The unknown value of t is labeled, and the area under
the curve to the right of t is shaded. The value of t0.20
with 10 degrees of freedom is invt(.80,10) = 0.8791.
• **Remember Inverse only does area to the LEFT!
Example:
• The pasteurization process reduces the amount of bacteria
found in dairy products, such as milk. The following data
represent the counts of bacteria in pasteurized milk (in
CFU/mL) for a random sample of 12 pasteurized glasses of
milk. Data courtesy of Dr. Michael Lee, Professor, Joliet
Junior College.
x  6.41
s  4.55
• Construct a 95% confidence interval for the bacteria count.
A Gallup poll conducted May 20-22, 2005 asked 1006 Americans “During the
past year, about how many books, either hardcover or paperback, did you
read either all or part of the way through?” Results of the survey indicated
that the mean was 13.4 books and s = 16.6 books. Construct a 99%
confidence interval for the mean number of books that Americans read either
all or part of during the preceding year. Interpret the interval.
9.3: Confidence Intervals for a
Population Proportion
• A point estimate is an unbiased estimator of
the parameter. The point estimate for the
population proportion is pˆ 
x
n
where x is the
number of individuals in the sample with the
specified characteristic and n is the sample
size.

Example
In July of 2008, a Quinnipiac University Poll asked 1783
registered voters nationwide whether they favored or
opposed the death penalty for persons convicted of
murder. 1123 were in favor.
Obtain a point estimate for the proportion of registered
voters nationwide who are in favor of the death penalty
for persons convicted of murder.
Confidence Interval
• 1-Porportion Z-interval
▫ Menu: 6: Statistics, 6: Confidence Intervals,
5: 1-prop z Interval
• Required:
▫ np(1-p) > 10
▫ n < 0.05N
Example
In July of 2008, a Quinnipiac University Poll asked 1783 registered
voters nationwide whether they favored or opposed the death
penalty for persons convicted of murder. 1123 were in favor.
Obtain a 90% confidence interval for the proportion of registered voters
nationwide who are in favor of the death penalty for persons
convicted of murder
• P=0.63
• np(1-p) =
• For Calculator: x = 1123, n = 1783, CI = 0.90
#21: A Zogby Interactive survey conducted Feb. 20-21,
2008 found that 1,322 of 1,979 randomly selected adult
Americans believe that traditional journalism is out of
touch with Americans want from their news.
a) Obtain a point estimate for the proportion of
adult Americans who believe that traditional
journalism is out of touch.
b) Verify that the requirements for constructing a
confidence interval for p are satisfied.
#21: A Zogby Interactive survey conducted Feb. 20-21,
2008 found that 1,322 of 1,979 randomly selected adult
Americans believe that traditional journalism is out of
touch with Americans want from their news.
c) Construct and interpret a 96% Confidence
interval
d) Is possible that the proportion of adult
Americans is below 60? Is this likely?

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