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Chap 8-1Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chapter 8
Estimation: Single Population
Statistics for Business and Economics
6th Edition
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-2
Chapter Goals
After completing this chapter, you should be able to:
Distinguish between a point estimate and a confidence interval estimate
Construct and interpret a confidence interval estimate for a single population mean using both the Z and t distributions
Form and interpret a confidence interval estimate for a single population proportion
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-3
Confidence Intervals
Content of this chapter Confidence Intervals for the Population
Mean, μ when Population Variance σ2 is Known when Population Variance σ2 is Unknown
Confidence Intervals for the Population Proportion, (large samples)p̂
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-4
Definitions
Một hàm ước lượng của một tham số tổng thể là: biến ngẫu nhiên phụ thuộc vào thông tin mẫu. . . mà giá trị của nó sẽ cho giá trị xấp xỉ của tham số
chưa biết
Một giá trị cụ thể của biến ngẫu nhiên này được gọi là một ước lượng
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-5
Point and Interval Estimates
Một điểm ước lượng là một con số, Một khoảng ước lượng cung cấp thêm thông
tin về độ biến thiên
Điểm ước lượng
Giới hạn tin cậy dưới
Giới hạn tin cậy trên
Độ rộng khoảng tin cậy
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-6
We can estimate a Population Parameter …
Ước lượng điểm
with a SampleStatistic
(a Point Estimate)
Trung bình
Tỷ lệ P
xμ
p̂
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-7
Ước lượng không chệch
được coi là hàm ước lượng điểm không
chệch của tham số nếu kỳ vọng, hay trung
bình của phân phối chọn mẫu của là
Examples: Trung bình mẫu là hàm ước lượng không chệch của μ Phương sai mẫu là hàm ước lượng không chệch củaσ2
Tỷ lệ mẫu là hàm ước lượng không chệch của P
θ̂
θ̂
θ)θE( ˆ
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-8
là hàm ước lượng không chệch, là hàm ước lượng chệch
1θ̂ 2θ̂
θ̂θ
1θ̂2θ̂
Unbiasedness(continued)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-9
Bias
Let be an estimator of
Độ chệch của (bias) được định nghĩa là hiệu giá trị trung bình của và
The bias of an unbiased estimator is 0
θ̂
θ̂
θ)θE()θBias( ˆˆ
θ̂
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-10
Ước lượng vững
Let be an estimator of
là hàm ước lượng vững của of nếu hiệu giá trị trung bình của nó và giảm dần khi tăng kích thước mẫu
Cần hàm ước lượng vững khi không thể chọn được hàm ước lượng không chệch
θ̂
θ̂θ̂
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-11
Hàm ước lượng hiệu quả nhất
Suppose there are several unbiased estimators of Hàm ước lượng hiệu quả nhất hay hàm ước lượng không
chệch phương sai cực tiểu của là hàm ước lượng với phương sai nhỏ nhất
Let and be two unbiased estimators of , based on the same number of sample observations. Then,
hữu hiệu hơn nếu Độ hữu hiệu tương đối của so với là:
)θVar()θVar( 21ˆˆ
)θVar(
)θVar( Efficiency Relative
1
2
ˆ
ˆ
1θ̂ 2θ̂
1θ̂ 2θ̂
1θ̂ 2θ̂
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-12
Khoảng tin cậy
How much uncertainty is associated with a point estimate of a population parameter?
An interval estimate provides more information about a population characteristic than does a point estimate
Such interval estimates are called confidence intervals
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-13
Ước lượng khoảng tin cậy
một khoảng giá trị : xét đến cả sự biến thiên của số thống kê
trong các mẫu khác nhau
dựa vào giá trị quan sát được từ 1 mẫu
cung cấp thông tin về mức độ sát với tham số tổng thể
mức độ tin cậy không bao giờ đạt 100%
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-14
Khoảng tin cậy và mức độ tin cậy
P(a < < b) = 1 - khoảng từ a đến b được gọi là khoảng tin cậy 100(1 - )% của
Định lượng (1 - ) is được gọi là mức độ tin cậy level của khoảng đó ( giữa 0 và 1)
In repeated samples of the population, the true value of the parameter would be contained in 100(1 - )% of intervals calculated this way.
The confidence interval calculated in this manner is written as a < < b with 100(1 - )% confidence
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-15
Quy trình ước lượng
(trung bình, μ, là ẩn số)
Tổng thể
Mẫu ngẫu nhiên
X = 50
Mẫu
I am 95% confident that μ is between 40 & 60.
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-16
Mức độ tin cậy, (1-)
Suppose confidence level = 95% Also written (1 - ) = 0.95 A relative frequency interpretation:
From repeated samples, 95% of all the confidence intervals that can be constructed will contain the unknown true parameter
A specific interval either will contain or will not contain the true parameter No probability involved in a specific interval
(continued)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-17
Công thức chung
Công thức chung cho tất cả các khoảng tin cậy:
Giá trị của thừa số tin cậy phụ thuộc vào mức độ tin cậy cần đạt được
Điểm ước lượng (Thừa số tin cậy)(Sai số chuẩn)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-18
Khoảng tin cậy
trung bình tổng thể
không biết σ2
ConfidenceIntervals
tỷ lệ tổng thể
biết σ2
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-19
Khoảng tin cậy cho μ(biết σ2 )
Giả thiết Population variance σ2 is known Population is normally distributed If population is not normal, use large sample
Khoảng tin cậy ước lượng:
(where z/2 is the normal distribution value for a probability of /2 in each tail)
n
σzxμ
n
σzx α/2α/2
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-20
Sai số biên
The confidence interval,
Can also be written as
where ME is called the margin of error
Độ rộng của khoảng tin cậy bằng 2 lần sai số biên
n
σzxμ
n
σzx α/2α/2
MEx
n
σzME α/2
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-21
Giảm sai số biên
Có thể giảm sai số biên bằng cách:
the population standard deviation can be reduced (σ↓)
The sample size is increased (n↑)
The confidence level is decreased, (1 – ) ↓
n
σzME α/2
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-22
Tìm thừa số tin cậy z/2
Consider a 95% confidence interval:
z = -1.96 z = 1.96
.951
.0252
α .025
2
α
Point EstimateLower Confidence Limit
UpperConfidence Limit
Z units:
X units: Point Estimate
0
Find z.025 = 1.96 from the standard normal distribution table
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-23
Các mức độ tin cậy phổ biến
Commonly used confidence levels are 90%, 95%, and 99%
Confidence Level
Confidence Coefficient,
Z/2 value
1.28
1.645
1.96
2.33
2.58
3.08
3.27
.80
.90
.95
.98
.99
.998
.999
80%
90%
95%
98%
99%
99.8%
99.9%
1
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-24
μμx
Khoảng và mức độ tin cậy
Confidence Intervals
Intervals extend from
to
100(1-)%of intervals constructed contain μ;
100()% do not.
Sampling Distribution of the Mean
n
σzx
n
σzx
x
x1
x2
/2 /21
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-25
Example
A sample of 11 circuits from a large normal population has a mean resistance of 2.20 ohms. We know from past testing that the population standard deviation is 0.35 ohms.
Determine a 95% confidence interval for the true mean resistance of the population.
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-26
2.4068μ1.9932
.2068 2.20
)11(.35/ 1.96 2.20
n
σz x
Example
A sample of 11 circuits from a large normal population has a mean resistance of 2.20 ohms. We know from past testing that the population standard deviation is .35 ohms.
Solution:
(continued)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-27
Interpretation
We are 95% confident that the true mean resistance is between 1.9932 and 2.4068 ohms
Although the true mean may or may not be in this interval, 95% of intervals formed in this manner will contain the true mean
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-28
Confidence Intervals
Population Mean
ConfidenceIntervals
PopulationProportion
σ2 Unknown σ2 Known
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-29
Student’s t Distribution
Consider a random sample of n observations with mean x and standard deviation s from a normally distributed population with mean μ
Then the variable
follows the Student’s t distribution with (n - 1) degrees of freedom
ns/
μxt
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-30
If the population standard deviation σ is unknown, we can substitute the sample standard deviation, s
This introduces extra uncertainty, since s is variable from sample to sample
So we use the t distribution instead of the normal distribution
Confidence Interval for μ(σ2 Unknown)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-31
Assumptions Population standard deviation is unknown Population is normally distributed If population is not normal, use large sample
Use Student’s t Distribution Confidence Interval Estimate:
where tn-1,α/2 is the critical value of the t distribution with n-1 d.f. and an area of α/2 in each tail:
Confidence Interval for μ(σ Unknown)
n
Stxμ
n
Stx α/21,-nα/21,-n
(continued)
α/2)tP(t α/21,n1n
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-32
Student’s t Distribution
The t is a family of distributions
The t value depends on degrees of freedom (d.f.) Number of observations that are free to vary after
sample mean has been calculated
d.f. = n - 1
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-33
Student’s t Distribution
t0
t (df = 5)
t (df = 13)t-distributions are bell-shaped and symmetric, but have ‘fatter’ tails than the normal
Standard Normal
(t with df = ∞)
Note: t Z as n increases
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-34
Student’s t Table
Upper Tail Area
df .10 .025.05
1 12.706
2
3 3.182
t0 2.920The body of the table contains t values, not probabilities
Let: n = 3 df = n - 1 = 2 = .10 /2 =.05
/2 = .05
3.078
1.886
1.638
6.314
2.920
2.353
4.303
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-35
t distribution values
With comparison to the Z value
Confidence t t t Z Level (10 d.f.) (20 d.f.) (30 d.f.) ____
.80 1.372 1.325 1.310 1.282
.90 1.812 1.725 1.697 1.645
.95 2.228 2.086 2.042 1.960
.99 3.169 2.845 2.750 2.576
Note: t Z as n increases
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-36
Example
A random sample of n = 25 has x = 50 and s = 8. Form a 95% confidence interval for μ
d.f. = n – 1 = 24, so
The confidence interval is
2.0639tt 24,.025α/21,n
53.302μ46.69825
8(2.0639)50μ
25
8(2.0639)50
n
Stxμ
n
Stx α/21,-n α/21,-n
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-37
Confidence Intervals
Population Mean
σ Unknown
ConfidenceIntervals
PopulationProportion
σ Known
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-38
Confidence Intervals for the Population Proportion, p
An interval estimate for the population proportion ( P ) can be calculated by adding an allowance for uncertainty to the sample proportion ( ) p̂
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-39
Confidence Intervals for the Population Proportion, p
Recall that the distribution of the sample proportion is approximately normal if the sample size is large, with standard deviation
We will estimate this with sample data:
(continued)
n
)p(1p ˆˆ
n
P)P(1σP
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-40
Confidence Interval Endpoints
Upper and lower confidence limits for the population proportion are calculated with the formula
where z/2 is the standard normal value for the level of confidence desired
is the sample proportion n is the sample size
n
)p(1pzpP
n
)p(1pzp α/2α/2
ˆˆˆ
ˆˆˆ
p̂
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-41
Example
A random sample of 100 people
shows that 25 are left-handed.
Form a 95% confidence interval for
the true proportion of left-handers
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-42
Example
A random sample of 100 people shows that 25 are left-handed. Form a 95% confidence interval for the true proportion of left-handers.
(continued)
0.3349P0.1651
100
.25(.75)1.96
100
25P
100
.25(.75)1.96
100
25
n
)p(1pzpP
n
)p(1pzp α/2α/2
ˆˆˆ
ˆˆˆ
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-43
Interpretation
We are 95% confident that the true percentage of left-handers in the population is between
16.51% and 33.49%.
Although the interval from 0.1651 to 0.3349 may or may not contain the true proportion, 95% of intervals formed from samples of size 100 in this manner will contain the true proportion.
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-44
PHStat Interval Options
options
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-45
Using PHStat (for μ, σ unknown)
A random sample of n = 25 has X = 50 and S = 8. Form a 95% confidence interval for μ
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-46
Chapter Summary
Introduced the concept of confidence intervals
Discussed point estimates Developed confidence interval estimates Created confidence interval estimates for the
mean (σ2 known) Introduced the Student’s t distribution Determined confidence interval estimates for
the mean (σ2 unknown) Created confidence interval estimates for the
proportion