Bài giảng Probability & Statistics - Lecture 8: Estimation - Bùi Dương Hải
8.1. Concept of Estimate Estimation is determining the approximate value of a unknown parameter on given data. Two types of estimate: Point estimate: single value Interval estimate: an interval that parameter falls into it with a determined probability level Ex. “The average height of VN’s male is 163 cm” “The average height of VN’s male is from 160 to 165 cm” 8.2. Point Estimate Point estimate of parameter is denoted by Estimator: a statistic calculated on random sample, an approximation to unknown parameter. Estimator is a random variable Estimate: specific value calculated on observed sample. Estimate is number.
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Lecture 8. ESTIMATION
[1] Chapter 8, Chapter 11
Concept of Estimate
Point Estimate
Maximum Likelihood Estimate
Interval Estimate
For Mean
For Proportion
For Variance (pp. 484 – 488)
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8.1. Concept of Estimate
Estimation is determining the approximate value of a
unknown parameter on given data.
Two types of estimate:
Point estimate: single value
Interval estimate: an interval that parameter falls into
it with a determined probability level
Ex. “The average height of VN’s male is 163 cm”
“The average height of VN’s male is from 160 to 165 cm”
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8.2. Point Estimate
Point estimate of parameter is denoted by
Estimator: a statistic calculated on random sample, an
approximation to unknown parameter. Estimator is a
random variable
Estimate: specific value calculated on observed
sample. Estimate is number.
Ex. Formula ̅ =
∑
is an estimator
Value ̅ =
= 5 is an estimate
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Point Estimate
Population parameters , , are unknown
Estimate by statistics from sample
Sample mean ̅ is point estimate for
Sample variance is point estimate for
Sample proportion ̅ is point estimate for
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Properties of Point Estimate
Estimate for by
Unbiased: is unbiased estimator of
=
Efficient: and are unbiased, is more efficient
than :
<
If is smallest in every unbiased estimator: most
efficient, best estimator
BUE (Best Unbiased Estimator)
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Example
Example.8.1. Population mean , sample ( , , , )
(a) Which of the followings are unbiased estimator?
(b) Which of the unbiased estimator is more efficient?
=
+
=
+
=
+
=
+
+
=
+
=
+ + ⋯ +
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Example
Example.8.2. Population mean of , there are two
sample. The first sample has size of 4, and mean of ̅ ;
the second sample has size of 8 and sample mean ̅ .
Which are the unbiased estimator, and which is more
efficient in the followings:
=
̅ +
̅ =
̅ +
̅
=
̅ +
̅ =
̅ +
̅
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8.3. Maximum Likelihood (ML) Estimate
Ex. Probability that a student pass an exam is = 0.6.
Which of the following sample is most likely to occurs
= ( , , ) = ( , , )
= ( , , ) = ( , , )
Likelihood function is Probability
= 0.6 ∗0.4 ∗0.4 =
= 0.4 ∗0.6 ∗0.4 =
= 0.4 ∗0.4 ∗0.4 =
= 0.6 ∗0.6 ∗0.6 =
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Maximum Likelihood (ML) Estimate
Ex. The sample is: = ( , , , )
Which of the following value of is most likely?
= 0.2; = 0.4; = 0.6; = 0.8
Likelihood function
= 0.2 = 0.2 ∗0.8 ∗0.2 ∗0.2 =
= 0.4 = 0.4 ∗0.6 ∗0.4 ∗0.4 =
= 0.6 = 0.6 ∗0.4 ∗0.6 ∗0.6 =
= 0.8 = 0.8 ∗0.2 ∗0.8 ∗0.8 =
Find the maximum likelihood estimate?
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8.4. Interval Estimate for Mean
The interval ( , )is interval estimate of , then
( < < ) is equal to (1 − )
( , )is confidence interval
(1 − ) is confidence level
= − is the width of confidence interval
: Lower Limit (LL)
: Upper Limit (UL)
The shorter (narrower) the is, the more accurate
the estimate is.
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Confidence Interval for Mean
Population is unknown, estimate for the mean
Sample size , mean of ̅, variance of
Confidence level (1 − ), confidence interval
− ( ) /
< < − ( ) /
or ± ( ) /
Value in the Table 2 (p. 976); ≈
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Confidence Interval for
Marignal Error = ( ) /
Confidence interval: ±
Narrower the Confidence Interval
Reduce standard deviation
Reduce confidence level
Increase sample size
=
/
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Example
Example 8.3. From the normality population, collect two
samples
Sample 1: 15, 17, 16, 20, 17
Sample 2: 18, 13, 14, 20, 15, 13
With confidence level of 95%, find confidence interval of
mean by using data in Sample 1, Sample 2, and Pooled
sample
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Example
Example 8.3 (cont.)
Sample 1:
̅ =
???
=
=
̅ ̅ ̅ ̅ ̅
???
=
= 5; 1 − = 0.95
Sample 2: ̅ = 15.5
= 8.3
Pooled sample: ̅ = 16.182
= 6.164
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Example
Example 8.4. To estimate the average score of students in
Maths, 40 students’ score are collected, and the sample
mean is 74.5, and sample variance is 64. Assumed that
score is normal distributed
(a) Find the 95% confidence interval of average score
(b) Find the 90% confidence interval of average score
(c) To reduce interval width to less than 4:
(c1) With level 95%, how many observations should be
surveyed
(c2) With above sample, how much confidence level
should be
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8.5. Interval Estimate for Proportion
Population proportion is unknown
Sample has size of , sample proportion ̅
Confidence interval level (1 − )
− /
( − )
< < + /
( − )
Or ± with = /
( )
Sample size: =
/
( )
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Example
Example 8.5. In 200 observed visitors, 50 of them buying
goods, and 40 using services. With confidence level of
95%
(a) Estimate of buyer proportion in the visitors
(b) To have interval that narrower than 10%, how many
observations should be surveyed?
(c) Estimate the proportion of visitors who do not use
services
(d) In 4000 visitors, estimate the total number of buyers
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8.6. Interval Estimate for Variance
Population variance is unknown
Sample has size of , sample proportion
Confidence interval level (1 − )
( − )
/
< <
( − )
/
Example 8.6. To estimate the variability of the time spent
on producing, manager random observed 40 times and
calculated the variance was 12.5 minutes2. Find the 95%
interval estimate of the producing time’s variance,
assumed that time is normal distributed.
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Key Concepts
Point Estimate
Unbiased, Efficient Estimator
Confidence Interval
Confidence Level, Marginal Error
Confidence Interval for Mean
Confidence Interval for Proportion
Confidence Interval for Sample
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Exercise
[1] Chapter 8
(349) 2, 6, 8, 10
(357) 13, 14, 17, 18, 21,
(261) 26, 30
(366) 32, 34, 36, 38, 40
Case Problem 1, 3
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