Uoc Luong Diem Va Khoang_2

Statistics for Business and Economics 6th Edition

Chapter 8 Estimation: Single Population

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chap 8-1

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 8-2

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, pˆ (large samples)

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chap 8-3

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 8-4

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 Giới hạn tin cậy dưới

Điểm ước lượng

Giới hạn tin cậy trên

Độ rộng khoảng tin cậy

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chap 8-5

Ước lượng điểm

We can estimate a Population Parameter …

Trung bình Tỷ lệ

with a Sample Statistic (a Point Estimate)

μ

x

P

pˆ

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chap 8-6

Ước lượng không chệch  θˆ được coi là hàm ước lượng điểm không chệch của tham số θ nếu kỳ vọng, hay trung bình của phân phối chọn mẫuθˆ của là θ

E(θˆ ) = θ

 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 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chap 8-7

Unbiasedness (continued)



θˆ 1là hàm ước lượng không chệch, θˆ 2 là hàm

ước lượng chệch

θˆ 1

θˆ 2

θ Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

θˆ Chap 8-8

Bias  Let θˆ be an estimator of θ  Độ chệch của θˆ (bias) được định nghĩa là hiệu giá trị trung bình của θˆ và θ ˆ ) = E( θ ˆ ) −θ Bias( θ

 The bias of an unbiased estimator is 0 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chap 8-9

Ước lượng vững  Let θˆ be an estimator of θ  θˆ là hàm ước lượng vững của of θ nếu hiệu giá trị trung bình của θˆ nó và θ giảm dần khi tăng kích thước mẫu  Cần hàm ước lượng vững khi không thể chọn được hàm ước lượng không chệch

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chap 8-10

Hàm ước lượng hiệu quả nhất  Suppose there are several unbiased estimators of θ  Hàm ước lượng hiệu quả nhất hay hàm ước lượng không chệch phương sai cực tiểu của θ là hàm ước lượng với phương sai nhỏ nhất  Let ˆ and ˆ be two unbiased estimators of θ , based on θ1 θ2 the same number of sample observations. Then, ˆθ hữu hiệu hơn θˆ 2 nếu Var(θˆ 1 ) < Var(θˆ 2 ) 1  Độ hữu hiệu tương đối của θˆ 1 so với θˆ 2 là: 

Var(θˆ 2 ) Relative Efficiency = Var(θˆ ) 1

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chap 8-11

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 8-12

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 8-13

Khoảng tin cậy và mức độ tin cậy  P(a < θ < b) = 1 - α  khoảng từ a đến b được gọi là khoảng tin cậy 100(1 - α )% của θ  Định lượng (1 - α ) is được gọi là mức độ tin cậy level của khoảng đó (α giữa 0 và 1)  In repeated samples of the population, the true value of the parameter θ would be contained in 100(1 α )% of intervals calculated this way.  The confidence interval calculated in this manner is written as a < θ < b with 100(1 - α )% confidence Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chap 8-14

Quy trình ước lượng Mẫu ngẫu nhiên Tổng thể (trung bình, μ, là ẩn số)

X = 50

I am 95% confident that μ is between 40 & 60.

Mẫu

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chap 8-15

Mức độ tin cậy, (1-α ) (continued)

 Suppose confidence level = 95%  Also written (1 - α ) = 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

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chap 8-16

Công thức chung  Công thức chung cho tất cả các khoảng tin cậy: Điểm ước lượng ± (Thừa số tin cậy)(Sai số chuẩn)

 Giá trị của thừa số tin cậy phụ thuộc vào mức độ tin cậy cần đạt được

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chap 8-17

Khoảng tin cậy Confidence Intervals trung bình tổng thể

biết σ2

tỷ lệ tổng thể

không biết σ2

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chap 8-18

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:

x − z α/2

σ σ < μ < x + z α/2 n n

(where zα /2 is the normal distribution value for a probability of α /2 in each tail) Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chap 8-19

Sai số biên  The confidence interval,

x − z α/2

σ σ < μ < x + z α/2 n n

 Can also be written as x ± ME where ME is called the margin of error ME = z α/2

σ n

 Độ rộng của khoảng tin cậy bằng 2 lần sai số biên

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chap 8-20

Giảm sai số biên ME = z α/2

σ n

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 – α ) ↓

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chap 8-21

Tìm thừa số tin cậy zα /2  Consider a 95% confidence interval:

1 − α = .95

α = .025 2 Z units: X units:

α = .025 2

z = -1.96 Lower Confidence Limit

0 Point Estimate

z = 1.96 Upper Confidence Limit

 Find z.025 = ± 1.96 from the standard normal distribution table Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chap 8-22

Các mức độ tin cậy phổ biến  Commonly used confidence levels are 90%, 95%, and 99% Confidence Level 80% 90% 95% 98% 99% 99.8% 99.9%

Confidence Coefficient,

Zα /2 value

.80 .90 .95 .98 .99 .998 .999

1.28 1.645 1.96 2.33 2.58 3.08 3.27

1− α

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chap 8-23

Khoảng và mức độ tin cậy Sampling Distribution of the Mean α/2

Intervals extend from σ x−z n

1− α

α/2

x

μx = μ

x1 x2

to

σ x+z n Confidence Intervals Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

100(1-α )% of intervals constructed contain μ; 100(α )% do not. Chap 8-24

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 8-25

Example (continued)

 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:

σ x±z n = 2.20 ± 1.96 (.35/ 11) = 2.20 ± .2068 1.9932 < μ < 2.4068

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chap 8-26

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 8-27

Confidence Intervals Confidence Intervals Population Mean

σ2 Known

Population Proportion

σ2 Unknown

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chap 8-28

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

x −μ t= s/ n follows the Student’s t distribution with (n - 1) degrees of freedom

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chap 8-29

Confidence Interval for μ (σ2 Unknown)  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 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chap 8-30

Confidence Interval for μ (σ Unknown) (continued)

 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:

x − t n-1,α/2

S S < μ < x + t n-1,α/2 n n

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:

P(t n−1 > t n−1,α/2 ) = α/2 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chap 8-31

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 8-32

Student’s t Distribution Note: t

Z as n increases

Standard Normal (t with df = ∞) t (df = 13)

t-distributions are bellshaped and symmetric, but have ‘fatter’ tails than the normal

t (df = 5)

0 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

t Chap 8-33

Student’s t Table Upper Tail Area df

.10

.05

.025

1 3.078 6.314 12.706

Let: n = 3 df = n - 1 = 2 α = .10 α /2 =.05

2 1.886 2.920 4.303 α /2 = .

3 1.638 2.353 3.182 The body of the table contains t values, not probabilities Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

05

0

2.920 t Chap 8-34

t distribution values With comparison to the Z value Confidence t Level (10 d.f.)

t (20 d.f.)

t (30 d.f.)

Z ____

.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 8-35

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

t n−1,α/2 = t 24,.025 = 2.0639

The confidence interval is S S x − t n-1,α/2 < μ < x + t n-1,α/2 n n 8 8 50 − (2.0639) < μ < 50 + (2.0639) 25 25 46.698 < μ < 53.302 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chap 8-36

Confidence Intervals Confidence Intervals Population Mean

σ Known

Population Proportion

σ Unknown

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chap 8-37

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 ( pˆ )

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chap 8-38

Confidence Intervals for the Population Proportion, p (continued)

 Recall that the distribution of the sample proportion is approximately normal if the sample size is large, with standard deviation

P(1− P) σP = n  We will estimate this with sample data:

pˆ (1− pˆ ) n Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chap 8-39

Confidence Interval Endpoints  Upper and lower confidence limits for the population proportion are calculated with the formula

pˆ − z α/2

ˆ (1− pˆ ) pˆ (1− pˆ ) p < P < pˆ + z α/2 n n

 where  p zˆ α /2 is the standard normal value for the level of confidence desired  is the sample proportion  n is the sample size Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chap 8-40

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 8-41

Example (continued)

 A random sample of 100 people shows that 25 are left-handed. Form a 95% confidence interval for the true proportion of left-handers. pˆ − z α/2

ˆ (1− pˆ ) pˆ (1− pˆ ) p < P < pˆ + z α/2 n n

25 .25(.75) 25 .25(.75) − 1.96 < P < + 1.96 100 100 100 100 0.1651 < P < 0.3349 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chap 8-42

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 8-43

PHStat Interval Options

options

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chap 8-44

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 8-45

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 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chap 8-46