数理统计——假设检验
合集下载
相关主题
- 1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。
Chapter 9
Fundamentals of Hypothesis Testing: One-Sample Tests
Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall
Chap 9-1
Learning Objectives
Chap 9-12
Level of Significance and the Rejection Region
H0: μ = 30 H1: μ ≠ 30
a /2
Level of significance = a a /2
30 Critical values Rejection Region
This is a two-tail test because there is a rejection region in both tails
In this chapter, you learn:
The basic principles of hypothesis testing How to use hypothesis testing to test a mean The assumptions of each hypothesis-testing
Decision Rule: If the test statistic falls in the rejection region, reject H0 ; otherwise do not reject H0
procedure, how to evaluate them, and the consequences if they are seriously violated How to avoid the pitfalls involved in hypothesis testing The ethical issues involved in hypothesis testing
Chap 9-8
The Hypothesis Testing
Process
(continued)
Sampling Distribution of X
20
If it is unlikely that you would get a sample mean of this value ...
μ = 50 If H0 is true
Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall
Chap 9-14
Z Test of Hypothesis for the Mean (σ Known)
Convert sample statistic ( X ) to a ZSTAT test statistic
Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall
Chap 9-2
What is a Hypothesis?
A hypothesis is a claim (assertion) about a population parameter:
Refers to the status quo or historical value Null always contains “=“sign May or may not be rejected
Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall
Challenges the status quo
Alternative never contains the “=”sign
May or may not be proven Is generally the hypothesis that the
researcher is trying to prove
H0 : μ 30
Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall
H0 : X 30
Chap 9-4
The Null Hypothesis, H0
Begin with the assumption that the null hypothesis is true Similar to the notion of innocent until proven guilty
The confidence level of a hypothesis test is (1-α)*100%.
Level of significance = Alpha= α
Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall
population mean
Example: The mean monthly cell phone bill in this city is μ = $42
Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall
Chap 9-3
The Null Hypothesis, H0
States the claim or assertion to be tested
Example: The average diameter of a
manufactured bolt is 30mm ( H0 : μ 30 )
Is always about a population parameter, not about a sample statistic
Generally, If the sample mean is far from the stated population mean, the null hypothesis is rejected.
How far is “far enough” to reject H0?
The critical value of a test statistic creates a “line in the sand” for decision making -- it answers the question of how far is far enough.
... When in fact this were the population mean…
Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall
X ... then you reject the null hypothesis
H1: μ ≠ 50
Sample the population and find sample mean.
Population
Sample
Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall
Chap 9-7
The Hypothesis Testing
Chap 9-5
The Alternative Hypothesis, H1
Is the opposite of the null hypothesis
e.g., The average diameter of a manufactured bolt is not equal to 30mm ( H1: μ ≠ 30 )
In other words, getting a sample mean of 20 is so unlikely if the population mean was 50, you conclude that the population mean must not be 50.
Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall
Region of Rejection
Region of Non-Rejection
Critical Values
Region of Rejection
“Too Far Away” From Mean of Sampling Distribution
Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall
Chap 9-15
Critical Value Approach to Testing
For a two-tail test for the mean, σ known:
Convert sample statistic ( X ) to test statistic (ZSTAT)
Determine the critical Z values for a specified level of significance a from a table or computer
Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall
Chap 9-10
The Test Statistic and Critical Values
Sampling Distribution of the test statistic
that 百度文库 = 50.
Chap 9-9
The Test Statistic and Critical Values
Generally, If the sample mean is close to the stated population mean, the null hypothesis is not rejected.
Hypothesis Tests for
σKKnnoowwnn (Z test)
The test statistic is:
Xμ
Z
STAT
σ
n
Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall
σUUnnkknnoowwnn (t test)
Process
(continued)
Suppose the sample mean age was X = 20.
This is significantly lower than the claimed mean population age of 50.
If the null hypothesis were true, the probability of getting such a different sample mean would be very small, so you reject the null hypothesis.
Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall
Chap 9-13
Hypothesis Tests for the Mean
Hypothesis Tests for
Known (Z test)
Unknown (t test)
Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall
Chap 9-6
The Hypothesis Testing Process
Claim: The population mean age is 50.
H0: μ = 50,
Chap 9-11
Possible Errors in Hypothesis Test Decision Making
(continued)
The confidence coefficient (1-α) is the probability of not rejecting H0 when it is true.
Fundamentals of Hypothesis Testing: One-Sample Tests
Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall
Chap 9-1
Learning Objectives
Chap 9-12
Level of Significance and the Rejection Region
H0: μ = 30 H1: μ ≠ 30
a /2
Level of significance = a a /2
30 Critical values Rejection Region
This is a two-tail test because there is a rejection region in both tails
In this chapter, you learn:
The basic principles of hypothesis testing How to use hypothesis testing to test a mean The assumptions of each hypothesis-testing
Decision Rule: If the test statistic falls in the rejection region, reject H0 ; otherwise do not reject H0
procedure, how to evaluate them, and the consequences if they are seriously violated How to avoid the pitfalls involved in hypothesis testing The ethical issues involved in hypothesis testing
Chap 9-8
The Hypothesis Testing
Process
(continued)
Sampling Distribution of X
20
If it is unlikely that you would get a sample mean of this value ...
μ = 50 If H0 is true
Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall
Chap 9-14
Z Test of Hypothesis for the Mean (σ Known)
Convert sample statistic ( X ) to a ZSTAT test statistic
Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall
Chap 9-2
What is a Hypothesis?
A hypothesis is a claim (assertion) about a population parameter:
Refers to the status quo or historical value Null always contains “=“sign May or may not be rejected
Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall
Challenges the status quo
Alternative never contains the “=”sign
May or may not be proven Is generally the hypothesis that the
researcher is trying to prove
H0 : μ 30
Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall
H0 : X 30
Chap 9-4
The Null Hypothesis, H0
Begin with the assumption that the null hypothesis is true Similar to the notion of innocent until proven guilty
The confidence level of a hypothesis test is (1-α)*100%.
Level of significance = Alpha= α
Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall
population mean
Example: The mean monthly cell phone bill in this city is μ = $42
Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall
Chap 9-3
The Null Hypothesis, H0
States the claim or assertion to be tested
Example: The average diameter of a
manufactured bolt is 30mm ( H0 : μ 30 )
Is always about a population parameter, not about a sample statistic
Generally, If the sample mean is far from the stated population mean, the null hypothesis is rejected.
How far is “far enough” to reject H0?
The critical value of a test statistic creates a “line in the sand” for decision making -- it answers the question of how far is far enough.
... When in fact this were the population mean…
Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall
X ... then you reject the null hypothesis
H1: μ ≠ 50
Sample the population and find sample mean.
Population
Sample
Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall
Chap 9-7
The Hypothesis Testing
Chap 9-5
The Alternative Hypothesis, H1
Is the opposite of the null hypothesis
e.g., The average diameter of a manufactured bolt is not equal to 30mm ( H1: μ ≠ 30 )
In other words, getting a sample mean of 20 is so unlikely if the population mean was 50, you conclude that the population mean must not be 50.
Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall
Region of Rejection
Region of Non-Rejection
Critical Values
Region of Rejection
“Too Far Away” From Mean of Sampling Distribution
Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall
Chap 9-15
Critical Value Approach to Testing
For a two-tail test for the mean, σ known:
Convert sample statistic ( X ) to test statistic (ZSTAT)
Determine the critical Z values for a specified level of significance a from a table or computer
Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall
Chap 9-10
The Test Statistic and Critical Values
Sampling Distribution of the test statistic
that 百度文库 = 50.
Chap 9-9
The Test Statistic and Critical Values
Generally, If the sample mean is close to the stated population mean, the null hypothesis is not rejected.
Hypothesis Tests for
σKKnnoowwnn (Z test)
The test statistic is:
Xμ
Z
STAT
σ
n
Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall
σUUnnkknnoowwnn (t test)
Process
(continued)
Suppose the sample mean age was X = 20.
This is significantly lower than the claimed mean population age of 50.
If the null hypothesis were true, the probability of getting such a different sample mean would be very small, so you reject the null hypothesis.
Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall
Chap 9-13
Hypothesis Tests for the Mean
Hypothesis Tests for
Known (Z test)
Unknown (t test)
Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall
Chap 9-6
The Hypothesis Testing Process
Claim: The population mean age is 50.
H0: μ = 50,
Chap 9-11
Possible Errors in Hypothesis Test Decision Making
(continued)
The confidence coefficient (1-α) is the probability of not rejecting H0 when it is true.