ESTIMATION
Prepared by:
Paolo Lorenzo Bautista
Estimation
 We
wish to estimate a characteristic of
the population, by using information from
the sample
 Types of Estimates


Point Estimate – single value
Interval Estimate – interval or range
Est : PLBautista
ESTIMATION OF PARAMETERS
One Population
 One


Mean
Case 1
Case 2
 One
Proportion
 One Variance
Two Populations
 Difference
Means


of Two
Case 1 to 3
Paired Observations
 Difference
of Two
Proportions
 Ratio of Two
Variances Est : PLBautista
Estimation for One Mean
 Best
Point Estimate:
 Interval Estimation for μ

CASE 1: σ is known, OR n ≥ 30
 The
(1-α)100% confidence interval for μ is
given by
MARGIN OF ERROR
Est : PLBautista
Example 1
 The
mean and standard deviation for the
sample point average of a random
sample of 36 college seniors are
calculated to be 2.6 and 0.3 respectively.
Find a 95% confidence interval for the
grade point average of the entire senior
class.
Est : PLBautista
Determining the Sample Size
 If
the sample mean is used as an estimate
of µ, we can be (1-α)100% confident that
the error will not exceed E when the
sample size is
 How
large a sample do we need if we do
not wish our error to exceed 0.05?
Est : PLBautista
Case 2: σ is unknown, AND n < 30
A
(1-α)100% confidence interval for μ is
given by
MARGIN OF ERROR
 The
t-value has n-1 degrees of freedom
Est : PLBautista
Example 2
A
random sample of 16 mid-sized cars
tested for fuel consumption gave a mean
of 26.4 miles per gallon with a standard
deviation of 2.3 miles per gallon. Assuming
that the miles per gallon given by all midsized cars have a normal distribution, find
a 99% confidence interval for the
population mean.
Est : PLBautista
Exercise
 In
an effort to estimate the mean amount
spent per customer for dinner at a major
Atlantis restaurant, data were collected
for a sample of 49 customers.


Assume a population standard deviation of
$5, and a 95% confidence level. What is the
margin of error?
If the sample mean is $34.80, what is the
95% confidence interval for the population
mean?
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Exercise
 In
the testing of a new production
method, 18 employees were selected
randomly and asked to try the new
method. The sample mean production
rate was 80 pph, and the sample s.d. was
10 pph. Provide 90% and 95% confidence
intervals for the population mean
production rate for the new method,
assuming the population has a normal
probability distribution.
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Estimation for the Proportion of
Success
 Assume
a binomial experiment is being
conducted
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Estimation for the Proportion of
Success
 Best
point estimate:
 A (1-α)100% confidence interval for p is
given by
MARGIN OF ERROR
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Example 3
 In
a random sample of n = 500 families
owning television sets in Quezon City, it
was found that x = 340 subscribed to
Destiny. Find a 95% confidence interval for
the actual proportion of families in this city
who subscribe to Destiny.
Est : PLBautista
Determining the Sample Size
 If
we use the sample proportion as an
estimate of p, we can be (1 – α)100%
confident that the error will not exceed E
when the sample size is approximately
 How
large a sample would be needed if
we do not wish our error to exceed 0.02 at
95% confidence?
Est : PLBautista
Exercise

A random sample of 75 college students is
selected and 16 are found to have cars on
campus.



Use a 95% confidence interval to estimate the
fraction of students who have cars on campus.
How large a sample is needed if we do not wish
our error to exceed 0.075?
In a random sample of 1000 homes in a
certain city, it is found that 628 are heated by
natural gas. Find the 98% confidence interval
for the fraction of homes in this city that are
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heated by natural gas.
Estimation for the Variance
 Recall:
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Estimation for the Variance
 Best
Point Estimate:
 A (1-α)100% confidence interval for σ2 is
given by
 The
chi-squared values have n-1 degrees
of freedom
Est : PLBautista
Example 4
 The
following are the volumes, in
deciliters, of 10 cans of peaches
distributed by a certain company: 46.4,
46.1, 45.8, 47.0, 46.1, 45.9, 46.9, 45.8, 45.2,
and 46.0. Find a 95% confidence interval
for the variance of all such cans assuming
that volume is normally distributed.
Est : PLBautista
ESTIMATION OF PARAMETERS
One Population
 One


Mean
Case 1
Case 2
 One
Proportion
 One Variance
Two Populations
 Difference
Means


of Two
Case 1 to 3
Paired Observations
 Difference
of Two
Proportions
 Ratio of Two
Variances Est : PLBautista
Estimation for the Difference of
Two Means
 Best
Point Estimate:
 Interval Estimation:
 Case 1:

A (1-α)100% confidence interval for μ1-μ2 is
given by
Est : PLBautista
MARGIN OF ERROR
Example 5
A
standardized chemistry test was given
to 50 girls and 75 boys. The girls made an
average grade of 76 with a standard
deviation of 6, while the boys made an
average grade of 82 with a standard
deviation of 8. Find a 96% confidence
interval for the difference of the two
population means.
Est : PLBautista
Case 2: σ1=σ2 unknown, AND
n1<30 and n2<30

A (1-α)100% confidence interval for μ1-μ2 is
given by

The t-value has
Est : PLBautista
Example 6

A course in statistics is taught to 12 students by the
transmissive teaching method. A second group of 10
students was given the same course by means of the
transformative teaching method. At the end of the
trimester the same examination was given to each
group. The 12 students who used the transmissive
method got an average grade of 85 with a standard
deviation of 4, while the 10 students under the
transformative method had an average grade of 81
with a standard deviation of 5. Find a 90%
confidence interval for the difference between
population means, assuming the populations are
approximately normally distributed with equal
Est : PLBautista
variances.
Case 3: σ1≠σ2 unknown, AND
n1<30 and n2<30

A (1-α)100% confidence interval for μ1-μ2 is
given by

The t-value has degrees of freedom given by:
Est : PLBautista
Example 7

Records for the past 15 years have shown the
average rainfall in a certain region of the
country for the month of May to be 4.93
centimeters, with a standard deviation of 1.14
centimeters. A second region of the country
has had an average rainfall in May of 2.64
centimeters with a standard deviation of 0.66
centimeters during the past 10 years. Find a
95% confidence interval for the difference of
the true average rainfalls in these two regions,
assuming that the observations come from
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normal populations with different variances.
Paired Observations
 Used
when the samples are not selected
independently
 The
t-value has n-1 degrees of freedom
 n represents the no. of pairs
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Example 8

Twenty college freshmen were divided into 10
pairs, each member of the pair having
approximately the same IQ. One of each pair
was selected at random and assigned to a
statistics section using a transformative
method of teaching. The other member of
each pair was assigned to a section in which
the professor lectured. At the end of the
trimester, each group was given the same
examination and the following grades were
recorded:
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Example 8
Pair
Transformative
Lecture
1
76
81
2
60
52
3
85
87
4
58
70
5
91
86
6
75
77
7
82
90
8
64
63
9
79
85
10
88
83
Find a 98%
confidence
interval for the
true difference
in the two
learning
procedures.
Est : PLBautista
Estimation for the Difference of
Two Proportions
 Best
point estimate:
 A (1-α)100% confidence interval for p1-p2
is given by
Est : PLBautista
Example 9
A
poll is taken among the residents of a
city and the surrounding county to
determine the feasibility of a proposal to
construct a civic center. If 2400 of 5000
city residents favor the proposal and 1200
of 2000 county residents favor it, find a
90% confidence interval for the true
difference in the fractions favoring the
proposal to construct the civic center .
Est : PLBautista
Estimation for the Ratio of Two
Variances
A
(1-α)100% confidence interval for the
ratio of two variances is given by
Est : PLBautista
Example 10

A standardized placement test in
mathematics was given to 25 boys and 16
girls. The boys made an average grade of 82,
with a standard deviation of 8, while the girls
made an average grade of 78 with a
standard deviation of 7. Find a 98%
confidence interval for σ12/σ22, where σ12 and
σ22 are the variances of the populations of
grades for all boys and girls, respectively, who
at some time have taken or will take this
exam. Assume the populations to be normally
distributed.
Est : PLBautista