Statistical Estimation
Chapter 3
McGraw-Hill/Irwin ©The McGraw-Hill Companies, Inc. 2008
GOALS
Define a point estimate.
Define level of confidence.
Construct a confidence interval for the population
mean when the population standard deviation is
known.
Construct a confidence interval for a population mean
when the population standard deviation is unknown.
Construct a confidence interval for a population
proportion.
Determine the sample size for attribute and variable
sampling.
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3.1. Basic concepts
In statistical inference, one estimates about the
population based on the result obtained from the
sample selected from that population.
Thus, estimation is a process by which we estimate
various unknown population parameters from sample
statistics.
Any sample statistic that is used to estimate a
population parameter is called an estimator.
and an estimate is a numerical value of an estimator.
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3.1. Basic concepts
The sample mean is often used as an estimator of
the population mean. Suppose that we calculate
the mean daily revenue of a store for a random
sample of 6 days and find it to be 1110 birr. If
we use this value to estimate the daily revenue for
the whole year, then the value 1110 birr would
be an estimate.
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3.1. Basic concepts
Point estimate – A single number computed from a
sample and used to estimate a population parameter.
Interval estimate – The interval, within which a
population parameter probably lies, based on sample
information.
Sampling error – The difference between a sample
statistic and its corresponding population parameter.
Confidence interval – An interval estimate which is
associated with degree of confidence of containing the
population parameter is called Confidence Interval.
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3.2. Point and Interval Estimates
A point estimate is the statistic, computed from
sample information, which is used to estimate
the population parameter.
– Sample mean for population mean.
A confidence interval estimate is a range of
values constructed from sample data so that
the population parameter is likely to occur
within that range at a specified probability.
– The specified probability is called the level of
6 confidence.
3.3. Point Estimation
Point estimation is a statistical procedure in
which we use a single value to estimate a
population parameter.
A point estimate is a single number that is
used as an estimate of a population
parameter, and is derived from a random
sample taken from the population of interest.
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3.3. Point Estimation
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3.4 Interval Estimation
Interval estimation is a statistical procedure in
which we find a random interval with a specified
probability of containing the parameter being
estimated.
An interval estimate is an interval that provides
an upper bound and a lower bound for a
specific population parameter whose value is
unknown.
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3.4 Interval Estimation
This interval estimate has an associated degree of confidence of
containing the population parameter.
Such interval estimates are also called Confidence intervals and
are calculated from random samples.
The interval estimate is an interval that includes the point
estimate.
Standard Error is A value that measures the spread of the
sample means around the population mean.
The standard error is reduced when the sample size is increased.
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3.5. Confidence Interval for the
Population Mean ()
To compute a confidence interval for a population mean,
we will consider two situations:
Case 1: We use sample data to estimate μ with x and the
population standard deviation (σ)
is known.
Case 2: We use sample data to estimate μ with x and the
population standard deviation is unknown.
In this case, we substitute the sample standard deviation
(s) for the population standard deviation (σ).
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Factors Affecting Confidence Interval Estimates
The factors that determine the width of a confidence
interval are:
1.The sample size, n.
2.The variability in the population, usually σ
estimated by s.
3.The desired level of confidence.
Confidence Level
The percentage of all possible confidence
intervals that will contain the true population
parameter.
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Confidence Interval for a Mean (σ Known)
Thus, a (1 - ) 100% confidence interval for the population
mean is given by:
𝒁𝜶 𝟐 𝝈/ 𝒏 is called Margin of Error
MoE is The amount that is added and subtracted to the point
estimate to determine the endpoints of the confidence interval.
MoE Also, is a measure of how close we expect the point
estimate to be to the population parameter with the specified
13 level of confidence.
Example
In a certain small city, to estimate the mean monthly
expenditure for food, a random sample of 25
households was randomly selected yielding a mean of
200 birr. From experience, it is known that such
expenditures are normally distributed with a standard
deviation of 50 Birr.
A. What is the point estimate of the mean monthly
expenditures for food of all households in the city?
B. Find a 95 percent confidence interval for the mean
monthly expenditures for food of all households in the
city.
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Example
Solution: -
Given
𝑋 = 200 Birr
= 50 Birr
n = 25
A. A point estimate of the population mean is the sample
mean Thus, = 200 Birr.
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Example
B) For 95 % confidence interval, let us find confidence
coefficient .
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Example
Then Z/2 = Z0.05/2 = Z0.025 = 1.96 (from the
table of standard normal)
Thus, a 95 % confidence interval for the mean is
= 200 (1.96)
= 200 19.6
= (180.40 Birr, 219.60 Birr)
I.e. we are 95 percent confident that the true mean
monthly expenditure for food () is between 180.40
Birr and 219.60 Birr.
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Example 2
The Ethiopian Management Association (EMA) is
studying the income of store managers in the retail
industry. A random sample of 49 managers reveals a
sample mean of Br. 45,420. The standard deviation of
this population is Br. 2,050.
The association would like answers to the following
questions:
1. What is the population mean?
2. What is a reasonable range of values for the
population mean?
18 3. How do we interpret these results?
Confidence Interval for a Mean (σ Unknown)
• The column on the left margin
is identified as “df.”
• This refers to the number of
degrees of freedom.
• The number of degrees of
freedom is the number of
observations in the sample
minus the number of samples,
written n - 1.
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Example
A tire manufacturer wishes to investigate the tread life
of its tires. A sample of 10 tires driven 50,000 K.M
revealed a sample mean of 0.32 C.M of tread
remaining with a standard deviation of 0.09 C.M.
Construct a 95% confidence interval for the population
mean.
Solution
The manufacturer can be reasonably sure (95% confident)
that the mean remaining tread depth is between 0.256
and 0.384 C.M.
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Example 2
A manufacturer claims that its tire lasts 20,000 miles on
average. A consumer organization tests a random sample of 64
tires and reported an average of 19,200 miles with a standard
deviation of 2,000 miles. Does a 99 % confidence interval for
the mean life of all tires produced by the manufacturer supports
the claim?
Solution
.
.
Hence, we are 99 percent confident that the true mean mileage
is at most 19,845 Which is less than the claimed mean 20,000
miles. Therefore, the claim is not true.
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A CONFIDENCE INTERVAL FOR A POPULATION
PROPORTION
PROPORTION The fraction, ratio, or percent
indicating the part of the sample or the population
having a particular trait of interest.
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Example
A national survey of 900 women footballers was conducted to
learn how women golfers view their treatment at golf courses in
the Ethiopia. The survey found that 396 of the women
footballers were satisfied with the availability of tee
times.
Required
A. the point estimate of the proportion of the population of
women footballers who are satisfied with the availability of
tee times.
B. Construct the confidence interval at 95% level of confidence.
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Example
Thus, the margin of error is .0324 and the 95% confidence
interval estimate of the population proportion is .4076 to
.4724.
Using percentages, the survey results enable us to state
with 95% confidence that between 40.76% and 47.24% of all
women footballers are satisfied with the availability of tee
times.
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Example 2
The Kuriftu resort is thinking of starting a new promotion. When a
customer checks out of the resort after spending 5 or more days, the
customer would be given a voucher that is good for 2 free nights on the
next stay of 5 or more nights at the resort.
The marketing manager is interested in estimating the proportion of
customers who return after getting a voucher. From a simple random
sample of 100 customers, 62 returned within 1 year after receiving the
voucher.
Construct A confidence interval estimate for the true population
proportion
Using the sample of 100 customers and a 95% confidence interval, the
manager estimates that the true percentage of customers who will take
advantage of the two-free night option will be between 52.5% and
71.5%.
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Sample size to estimate a population mean
• Whenever we take a sample for inferential purposes, there is
always a sampling error. This sampling error is controlled by
selecting a sample that is adequate in size. If the sample size is
small, then we may fail to achieve the objective of our analysis,
and if it is too large, then we waste the resources when we gather
the sample.
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Example
A student in public administration wants to estimate the mean
monthly earnings of city council members in large cities. She can
tolerate a margin of error of Br.100 in estimating the mean. She
would also prefer to report the interval estimate with a
95% level of confidence. The student found a report by the
Department of Labor that reported a standard deviation of
Br.1,000. What is the required sample size?
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Sample Size to Estimate a Population Proportion
where:
n is the size of the sample.
z is the standard normal z value corresponding to the desired
level of confidence.
π is the population proportion.
E is the maximum allowable error.
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Example
The student in the previous example also wants to estimate
the proportion of cities that have private refuse collectors.
The student wants to estimate the population proportion
with a margin of error of .10, prefers a level of
confidence of 90%, and has no estimate for the population
proportion.
Required
What is the sample size?
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