USTPADPDD601

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UST/PAD/PDD601 Applied Quantitative Reasoning
Lecture 4. Parameter Estimation
Sugie Lee, Ph.D. Assistant Professor Urban
Planning, Design and Development Program Levin
College of Urban Affairs Cleveland State
University
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Part III. Inference

• Chapter 8. Parameter Estimation
• Point estimation vs. Interval estimation
• Interval estimate of a mean
• Students t distribution
• Interval estimate of other parameter
• Interval estimate of the mean difference between
  non-independent paired measures
• Interval estimate of the difference between two
  independent means
• Interval estimate of a proportion
• Interval estimate of the difference between two
  independent proportions
• Sampling size selection for limiting error

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Parameter Estimation

• Two Different Directions of Statistical Inference
  
• 1) Direct estimation of population parameter from
  the sample observations serving as approximations
  to the true population characteristic.
• 2) Hypotheses testing using the sample
  observations to support or discredit a
  priori hypotheses

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Direct Estimation of Population Parameter

• Point Estimation
• - To obtain a single value (the best possible
  approximation of the true value of the population
  mean or parameters) based upon a sample of
  observations However, point estimation does not
  provide us with information as to how close we
  might expect the approximation to be to the
  population parameter.
• Ex. The mean of a sample is an
  example of a point estimate of the population
  mean
• Ex. and The sample standard
  deviation and variance as respective estimates of
  the population is standard deviation and
  variance
• Unbiased estimate and efficient estimate

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Unbiased Estimate

• Unbiased Estimate
• Unbiased estimate means that in the long run the
  expected value of the estimate will equal the
  population parameter
• , rather than
  is an unbiased estimate of
• - A more extreme example than the population
  variance or standard deviation is the population
  range the range of a small sample of
  observations would tend to underestimate the
  range of the parent population

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Efficient Estimate

• Efficient Estimate
• Efficient estimate means that the estimate
  quickly becomes a more accurate estimate of the
  population parameter as the sample size increases
• Ex. While both mean and median are unbiased
  estimates of central tendency, the mean is a
  relatively more efficient estimate than the
  median
• Ex. The sample variance and standard
  deviation are more efficient estimates of
  variation than other statistics, such as the
  range or average deviation.
• - The mean , variance , standard
  deviation in the sampling distribution of a
  statistic are an unbiased and efficient estimates
  of a population parameters
  
  

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Interval Estimation

• Background
• Point estimate does not provide information
  regarding how close our sample statistic is to
  the population parameter
• Instead of a point estimate, we could state with
  a certain probability that it is within a
  particular distance of the parameter
• Ex. A point estimate of the population mean
  
• An interval estimation of the population
  mean
• - Confidence intervals and limits An interval
  estimate requires confidence intervals

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Interval Estimate of a Mean
or
or
The standard deviation of sampling errors,
the standard error of the mean,
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Distribution of Sampling Errors

• Shape of the Distribution of Sampling Errors
• The shape of the distribution of sampling errors
  will be identical to the shape of the sampling
  distribution of the mean because the constant
  value to sampling distribution of the mean (
  ) affects neither its standard deviation, nor
  the shape of the distribution

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Standard Normal Distribution (Z Distribution)
(a)
(c)
(b)
(d)
(e)
(f)
(g)
(h)
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Distribution of Sampling Errors,
1.65
-1.65
2.58
1.96
0
-2.58
-1.96
90 of the errors
95 of the errors
99 of the errors
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Confidence Intervals

• Determination of Confidence Intervals
• The shape and standard deviation of the
  distribution of sampling errors is the same as
  that of the sampling distribution of the mean
• In a normal distribution, 95 of the
  observations fall within
• standard deviations from the mean
• Since the standard deviation of the distribution
  of sampling errors is equal to the standard error
  of the mean, , it follows that

with 95 certainty
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Interval Estimation General Notation
(theta hat) is the sample estimate of
is the critical z value associated with a
given confidence interval
1.65 for a 90 confidence interval 1.96
for a 95 confidence interval
2.58 for a 99 confidence interval
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Interval Estimation of the Mean (Example)
Lets assume that we obtain a mean weight for the
sample of 100 ball bearings of
, and that we know from historical
records that the standard deviation of the
population of ball bearing weights is
. Calculate the standard error of the
mean Calculate the 95 confidence interval
Calculate the 99 confidence interval

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Interval Estimation of the Mean (Example)
From a random sample of graduate students (n100)
in the urban college, the mean GRE score is 1800
and the population standard deviation is 150.
Assuming that GRE scores are normally
distributed, construct the 95 confidence
interval around this mean
Answer 1770.6lt lt1829.4
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Interval Estimation of the Mean (Example)
From a random sample of graduate students (n50)
in the urban college, the mean GRE score is 1800
and the population standard deviation is 150.
Assuming that GRE scores are normally
distributed, construct the 95 confidence
interval around this mean. Compare the results.
What is the difference? What is the lesson?
Previous questions answer 1770.6lt lt1829.4
Answer 1758.41lt lt1841.59
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Interval Estimation of the Mean (Example)
From a random sample of graduate students (n100)
in the urban college, the mean GRE score is 1800
and the population standard deviation is 50.
Assuming that GRE scores are normally
distributed, construct the 95 Confidence
Interval around this mean. Compare the results.
What is the difference? What is the lesson?
From a random sample of graduate students (n50)
in the urban college, the mean GRE score is 1800
and the population standard deviation is 150.
Assuming that GRE scores are normally
distributed, construct the 99 Confidence
Interval around this mean. Compare the results.
What is the difference? What is the lesson?
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Interval Estimate of the Mean (Notation)
Normal population
Non-normal population Via the C.L.T, normal
distribution (ngt30)
When ngt30 unknown known
When nlt30 (student t-distribution) unknown
known
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Approximate Interval Estimation of the Mean
(Example)
Lets assume that we obtain a mean weight for the
sample of 100 ball bearings of
, and that we do not know standard
deviation of the population of ball bearing
weights. However we know the sample standard
deviation s as an estimate of . Lets say
. Calculate the standard
error of the mean
instead of Calculate the 95 confidence
interval Calculate the 99 confidence
interval
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Interval Estimation of the Mean (Students
t-distribution)
When nlt30 (student t-distribution) unknown
known
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Interval Estimation ( t-distribution)
(Page. 565)
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Interval Estimation of the Mean ( t-distribution
Example)
Interviews are conducted with 15 randomly
selected home-owners in a city, seeking
information about their housing costs. They
reported an average of 915 in maintenance and
repair in the last year, with a standard
deviation of 115. Historically, repair costs in
your city have been normally distributed. What is
the 99 confidence interval around this sample
mean?
Degree of freedomn-115-114 For 14 df, find
the critical value of t associated with 99
confidence interval.
Answer
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Interval Estimation of the Mean ( t-distribution
Example)
Consider a governmental agency report that a
particular model automobile gets 24 miles per
gallon of gasoline.
n9 cars

s4 mpg Calculate the standard error of the
mean and 95 interval
Degree of freedomn-19-18 For df8, find the
critical value of t associated with 95
confidence interval.
Answer
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Interval Estimate Various Parameter Estimation
Problem
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Interval Estimate of the Mean Difference Between
Non-Independent Paired Measures (Notation)
Normal population
Non-normal population Via the C.L.T, normal
distribution (ngt30)
When ngt30 unknown known
When nlt30 (student t-distribution) unknown
known
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Interval Estimate of the Mean Difference Between
Non-Independent Paired Measures (Example)
A sample (n150) of high school seniors are shown
two different CSU commercials (A and B) and asked
to rate each from 0-100 based on their
effectiveness. The mean difference between
commercial ratings (A-B) is 7, and the standard
deviation is 4. Construct the 95 confidence
interval around this mean difference.
Answer
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Interval Estimation of the Difference Between Two
Independent Means (Notation)
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Interval Estimation of the Difference Between Two
Independent Means (Example)
Two independent random samples of adults are
drawn sample one is of urban residents (n75),
and sample two is of suburban residents (n60).
Average years of education is calculated, and
they are 17.5(urban) and 23.4(suburban). The
population standard deviations are 5(urban) and
6(suburban). Find the 95 confidence interval the
difference between the two population means.
0.98
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Interval Estimate of a Proportion (Notation)
Standard deviation of the binary population
Standard error of a sample proportion
mean of the sampling distribution
Standard error of a sample proportion (number of
successes)
mean of the sampling distribution (number of
successes)
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Interval Estimate of a Proportion (Example)
From a sample of citizen (n150), it was
determined that the probability of support for an
upcoming school levy was 0.28. What is the 99
confidence interval around this population
probability, p?
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Interval Estimate of the Difference Between Two
Independent Proportions (Example)
Independent random samples were taken of parents
at two elementary schools. For sample 1(n100),
the probability of levy support 0.35. For sample
2(n65), the probability of levy support is 0.45.
What is the 95 confidence interval for the
difference between these two independent
proportions?
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Sample Size Selection for Limiting Error
Suppose that we are interested in estimating the
mean diameter of the trees in a large stand of
timber. Assume we do not want to be in error by
more than .5 inches, with 95 assurance. How many
trees should we sample and measure for this
degree of precision? Since we do not know the
value of , the standard deviation of the
tree diameters, we perform a pilot study and
obtain of s4.7 inches.
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