Chapter 9
Estimating the Value of a Parameter
Chapter 9.1 Estimating a Population Proportion
Objective A :Point Estimate
A point estimate is the value of a statistic that estimates the value of a parameter.
x
The best point estimate of the population proportion is a sample proportion ( pˆ  ).
n
The best point estimate of the population mean is a sample mean ( x 
 x ).
n
Since p̂ varies from sample to sample, we use an interval based on p̂ to capture the unknown
population proportion with a level of confidence.
Objective B : Confidence Interval
A confidence interval for an unknown parameter consists of an interval of numbers based on
a point estimate.
The level of confidence represents the expected proportion of intervals that will contain the
parameter if a large number of different samples is obtained.
The level of confidence is denoted as 1    100% . The level of confidence controls the width
of the interval.
Confidence interval estimates for a parameter are of the form:
Point estimate  margin of error.
Confidence interval for p :
pˆ  Z  / 2   pˆ where  pˆ 
pˆ (1  pˆ )
provided that n pˆ (1  pˆ )  10 .
n
The value of Z / 2 is called the critical value of the distribution.
The margin of error, E , in a 1    100% confidence interval for a population proportion is
given by E  Z  / 2
pˆ (1  pˆ )
. The width of the interval is determined by the margin of error.
n
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Example 1:Use StatCrunch to determine the critical value Z / 2 that corresponds to the given level
of confidence.
(a) 90%
(b) 95%
(c) 98%
(d) 92%
Example 2: Determine the margin of error for p with x  540 and n  900 at a 99% level
of confidence.
Example 3: A Rasmussen Reports national survey of 1000 adult Americans found that 18% dreaded
Valentine's Day. Construct a 95% confidence interval for the population proportion of adult
Americans who dread Valentine's Day. Explain what does the interval mean.
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Example 4: Construct a confidence interval of the population proportion at the given level
of confidence.
x  80, n  200, 96% confidence
Example 5: In a study of 1228 randomly selected medical malpractice lawsuits, it is found that 856
if them were later dropped or dismissed.
(a) What is the best point of estimate of the proportion of medical malpractice lawsuits
that are dropped or dismissed?
(b) Use StatCrunch to construct a 99% confidence interval for the population proportion
of medical malpractice lawsuits that are dropped or dismissed?
(c) Interpret the interval.
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Objective C :Sample Size Needed for Estimating the Population Proportion p
The sample size required to obtain a 1    100% confidence interval for p with a
margin of error E is given by
Z

n  pˆ (1  pˆ )  / 2 
 E 
2
Round up to the next integer
p̂ is a prior estimate of p
If a prior estimate of p is unavailable, the sample size required is
Z 
n  0.25   /2 
 E 
2
Round up to the next integer
Example 1 : An urban economist wishes to estimate the proportion of Americans who own
their homes. What size sample should be obtained if he wishes the estimate to be
within 0.02 with 90% confidence if
(a) he uses a 2010 estimate of 0.669 obtained from the U.S Census Bureau?
(b) he does not use any prior estimates?
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Example 2: In a Gallup poll conducted in October 2010, 64% of the people polled answered
"more strict" to the following question: "Do you feel that the laws covering the sale
of firearms should be made more strict as they are now?" Suppose the margin of
error in the poll was 3.5% and the estimate was made with 95% confidence. At
least how many people were surveyed?
Example 3: A Gallup poll conducted in November 2010 found that 493 of 1050 adult Americans
believe it is the responsibility of the federal government to make sure all Americans
have healthcare coverage.
(a) Obtain a point estimate for the proportion of adult Americans who believe it is the
responsibility of the federal government to make sure all Americans have healthcare
coverage.
(b) Verify the requirements for constructing a confidence interval for p are satisfied.
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(c) Use StatCrunch to construct a 95% confidence interval for the proportion of adult
Americans who believe it is the responsibility of the federal government to make
sure all Americans healthcare coverage. Interpret the interval.
(d) You wish to conduct your own study for the proportion of adult Americans who
believe it is the responsibility of the federal government to make sure all Americans
have healthcare coverage. What sample size would be needed for the estimate to be
within 3 percentage points with 90% confidence if you use the estimate obtained in
part (a).
(e) You wish to conduct your own study for the proportion of adult Americans who
believe it is the responsibility of the federal government to make sure all Americans
have healthcare coverage. What sample size would be needed for the estimate to be
within 3 percentage points with 90% confidence if you do not have a prior estimate?
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Chapter 9.2 Estimating a Population Mean
Objective A : Point Estimate
The best point estimate of the population mean,  , is the sample mean, x .
Objective B :Student's t - distribution
Properties of the t - distribution
1. The t - distribution is different for different degrees of freedom ( df  n  1 ).
2. The t - distribution has the same general symmetric bell shape as the standard normal
distribution but its area in the tails is a little greater than the area in the tails of the standard
normal distribution due to the greater variability that is expected with small samples.
3. The t - distribution has a mean of t  0 at the center of the distribution.
4. As the sample size n gets larger, the t - distribution gets closer to the standard normal
distribution.
Example 1: Use StatCrunch to determine the t -value.
(a) Find the t -value such that the area in the right tail is 0.05 with 19 degrees of freedom.
(b) Find the t -value such that the area left of the t -value is 0.02 with 6 degrees of freedom.
(c) Find the critical t -value that corresponds to 90% confidence. Assume 12degrees of freedom.
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In general, the population standard deviation is unknown for estimating a population mean based
on a sample mean. The t -distribution is used to off-set the additional variability introduced by using
s in place of  .
Objective C :Confidence Interval for a Population Mean
Constructing a 1    100% Confidence Interval for 
Point estimate  margin of error
s
s
where E  t  / 2 
.
x  t /2 
n
n
provided the data come from a population that is normally distributed, or the sample size is large.
Example 1: A simple random sample of size n  30 has been obtained. From the normal probability
plot and boxplot, judge whether a t -interval should be constructed.
(a)
(b)
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Example 2: A simple random sample of size n is drawn from a population that is normally distributed.
The sample mean, x , is found to be 50, and the sample standard deviation, s , is found to be 8.
(a) Use StatCrunch to construct a 98% confidence interval for  if the sample size, n , is 20.
(b) Use StatCrunch to construct a 98% confidence interval for  if the sample size, n , is 15.
How does decreasing the sample size affect the margin of error, E ?
(c) Construct a 95% confidence interval for  if the sample size, n , is 20.
Compare the results to those obtained in part (a). How does decreasing the level of
confidence affect the margin of error, E ?
(d) Could we have computed the confidence intervals in parts (a) to (c) if the population
had not been normally distributed? Why?
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Example 3: Determine the point estimate of the population mean and margin of error for the
following confidence interval.
Lower bound: 5
Upper bound: 23
Example 4 : How much time do Americans spend eating or drinking? Suppose for a random
sample of 1001 Americans age 15 or older, the mean amount of time spent eating
or drinking per day is 1.22 hours with a standard deviation of 0.65 hour.
(a) A histogram of time spent eating and drinking each day is skewed right. Use this
result to explain why a large sample size is needed to construct a confidence
interval for the mean time spent eating and drinking each day.
(b) Use StatCrunch to determine and a 95% confidence interval for the mean amount of
time Americans age 15 or older spend eating and drinking each day. Interpret the interval.
(c) Could the interval be used to estimate the mean amount of time a 9-year-old American
spends eating and drinking each day? Explain.
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Objective D : Determining the Sample Size n
The sample size required to estimate the population mean,  , with a level of confidence
1    100% within a specified margin of error,
E , is given by
 Z s 
n    /2 
 E 
where n is rounded up to the nearest whole number.
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Note: The t -distribution approaches the standard normal z - distribution as the samplesize increases.
Example 1: A researcher wanted to determine the mean number of hours per week(Sunday through
Saturday) the typical person watches television. Results from the Sullivan Statistics Survey
indicate that s  7.5 hours.
(a) How many people are needed to estimate the number of hours people watch television
per week within 2 hours with 95% confidence?
(b) How many people are needed to estimate the number of hours people watch television
per week within 1 hour with 95% confidence?
(c) What effect does doubling the required accuracy have on the sample size?
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Chapter 9 Estimating a Population Standard Deviation (Supplementary Materials)
Objective A : Point Estimate
The best point estimate of the population variance,  2 , is the sample variance, s 2 .
Objective B : Chi-Square Distribution
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Example 1: Use StatCrunch to find the critical values  12 / 2 and  2 / 2 for the given level of confidence
and sample size.
(a) 90% confidence, n  23
(b) 99% confidence, n  15
Objective C : Confidence Interval for a Population Variance or Standard Deviation
(1   ) 100% of the values of  2 will lie between  12 / 2 and  2 / 2 .
( Recall:  2 
(n  1) s 2
2
)
To find a (1   ) 100% confidence interval about  , take the square root of the lower bound and upper
bound.
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Example 1: A simple random sample of size n is drawn from a population that is known to be normally
distributed. The sample variance, s 2 , is determined to be 19.8.
(a) Use StatCrunch to construct a 95% confidence interval for  2 if the sample size, n , is 10.
(b) Use StatCrunch to construct a 95% confidence interval for  2 if the sample size, n , is 25.
How does increasing the sample size affect the width of the interval?
(c) Use StatCrunch to construct a 99% confidence interval for  2 if the sample size, n , is 10.
Compare the results with those obtained in part (a). How does increasing the level of
confidence affect the width of the confidence interval?
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Example 2: Travelers per taxes for flying, car rentals, and hotels. The following data represent the total
travel tax for a 3-day business trip in eight randomly selected cities. It was verified that the
data are normally distributed. Use StatCrunch to construct a 90% confidence interval for the
standard deviation travel tax for a 3-day business trip. Interpret the interval.
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