Suche senden
Hochladen
Chap#9 hypothesis testing (3)
•
90 gefällt mir
•
14,227 views
shafi khan
Folgen
Melden
Teilen
Melden
Teilen
1 von 77
Jetzt herunterladen
Downloaden Sie, um offline zu lesen
Empfohlen
Rejection Region.ppt.ppt
Rejection Region.ppt.ppt
AlonaNayAgcaoili
STATISTICS: Hypothesis Testing
STATISTICS: Hypothesis Testing
jundumaug1
Chapter 6-THEORETICAL & CONCEPTUAL FRAMEWORK
Chapter 6-THEORETICAL & CONCEPTUAL FRAMEWORK
Ludy Mae Nalzaro,BSM,BSN,MN
Advanced statistics
Advanced statistics
Romel Villarubia
Stat3 central tendency & dispersion
Stat3 central tendency & dispersion
Forensic Pathology
How to write a statement problem
How to write a statement problem
businesscollege_plmar
Statistical Estimation and Testing Lecture Notes.pdf
Statistical Estimation and Testing Lecture Notes.pdf
Dr. Tushar J Bhatt
Bartlett's test
Bartlett's test
Irfan Hussain
Weitere ähnliche Inhalte
Was ist angesagt?
Intro to probability
Intro to probability
getyourcheaton
Two Means, Independent Samples
Two Means, Independent Samples
Long Beach City College
Determining the Sample Size
Determining the Sample Size
University of Rizal System-Morong
Samplels & Sampling Techniques
Samplels & Sampling Techniques
Jo Balucanag - Bitonio
Statistics and probability lesson6&7
Statistics and probability lesson6&7
MARIA CHRISTITA POLINAG
Inferences about Two Proportions
Inferences about Two Proportions
Long Beach City College
Resume
Resume
kclyn
Basics of Hypothesis Testing
Basics of Hypothesis Testing
Long Beach City College
6. point and interval estimation
6. point and interval estimation
ONE Virtual Services
Statistical treatment of data
Statistical treatment of data
senseiDelfin
Solution to the Practice Test 3A, Chapter 6 Normal Probability Distribution
Solution to the Practice Test 3A, Chapter 6 Normal Probability Distribution
Long Beach City College
Other Measures of Location
Other Measures of Location
Samuel John Parreño
Hypergeometric probability distribution
Hypergeometric probability distribution
Nadeem Uddin
Testing a claim about a standard deviation or variance
Testing a claim about a standard deviation or variance
Long Beach City College
The Interpretation Of Quartiles And Percentiles July 2009
The Interpretation Of Quartiles And Percentiles July 2009
Maggie Verster
Z-Test with Examples
Z-Test with Examples
Anas Bin Madni
Advanced statistics Lesson 1
Advanced statistics Lesson 1
Cliffed Echavez
Parametric Statistics
Parametric Statistics
jennytuazon01630
Experimental research
Experimental research
dhinnar
Complements and Conditional Probability, and Bayes' Theorem
Complements and Conditional Probability, and Bayes' Theorem
Long Beach City College
Was ist angesagt?
(20)
Intro to probability
Intro to probability
Two Means, Independent Samples
Two Means, Independent Samples
Determining the Sample Size
Determining the Sample Size
Samplels & Sampling Techniques
Samplels & Sampling Techniques
Statistics and probability lesson6&7
Statistics and probability lesson6&7
Inferences about Two Proportions
Inferences about Two Proportions
Resume
Resume
Basics of Hypothesis Testing
Basics of Hypothesis Testing
6. point and interval estimation
6. point and interval estimation
Statistical treatment of data
Statistical treatment of data
Solution to the Practice Test 3A, Chapter 6 Normal Probability Distribution
Solution to the Practice Test 3A, Chapter 6 Normal Probability Distribution
Other Measures of Location
Other Measures of Location
Hypergeometric probability distribution
Hypergeometric probability distribution
Testing a claim about a standard deviation or variance
Testing a claim about a standard deviation or variance
The Interpretation Of Quartiles And Percentiles July 2009
The Interpretation Of Quartiles And Percentiles July 2009
Z-Test with Examples
Z-Test with Examples
Advanced statistics Lesson 1
Advanced statistics Lesson 1
Parametric Statistics
Parametric Statistics
Experimental research
Experimental research
Complements and Conditional Probability, and Bayes' Theorem
Complements and Conditional Probability, and Bayes' Theorem
Andere mochten auch
ESPA 4123 - Statistika Ekonomi Modul 7 : Uji Hipotesis
ESPA 4123 - Statistika Ekonomi Modul 7 : Uji Hipotesis
Ancilla Kustedjo
Four steps to hypothesis testing
Four steps to hypothesis testing
Hasnain Baber
Data processing and analysis final
Data processing and analysis final
Akul10
Data editing ( In research methodology )
Data editing ( In research methodology )
Np Shakeel
research methodology data processing EDITING
research methodology data processing EDITING
Suvin Lal
Hypothesis
Hypothesis
17somya
Literature review in research methodology
Literature review in research methodology
raison sam raju
Data Preparation and Processing
Data Preparation and Processing
Mehul Gondaliya
Hypothesis Testing
Hypothesis Testing
Harish Lunani
Hypothesis testing; z test, t-test. f-test
Hypothesis testing; z test, t-test. f-test
Shakehand with Life
Literature Review (Review of Related Literature - Research Methodology)
Literature Review (Review of Related Literature - Research Methodology)
Dilip Barad
Research Design
Research Design
gaurav22
Types of Research
Types of Research
Vaisali Krishnakumar
Andere mochten auch
(13)
ESPA 4123 - Statistika Ekonomi Modul 7 : Uji Hipotesis
ESPA 4123 - Statistika Ekonomi Modul 7 : Uji Hipotesis
Four steps to hypothesis testing
Four steps to hypothesis testing
Data processing and analysis final
Data processing and analysis final
Data editing ( In research methodology )
Data editing ( In research methodology )
research methodology data processing EDITING
research methodology data processing EDITING
Hypothesis
Hypothesis
Literature review in research methodology
Literature review in research methodology
Data Preparation and Processing
Data Preparation and Processing
Hypothesis Testing
Hypothesis Testing
Hypothesis testing; z test, t-test. f-test
Hypothesis testing; z test, t-test. f-test
Literature Review (Review of Related Literature - Research Methodology)
Literature Review (Review of Related Literature - Research Methodology)
Research Design
Research Design
Types of Research
Types of Research
Ähnlich wie Chap#9 hypothesis testing (3)
Biomath12
Biomath12
guruppkn11
1Copyright © 2015 The McGraw-Hill Companies, Inc. Permission r.docx
1Copyright © 2015 The McGraw-Hill Companies, Inc. Permission r.docx
felicidaddinwoodie
Lecture 15 - Hypothesis Testing (1).pdf
Lecture 15 - Hypothesis Testing (1).pdf
RufaidahKassem1
Chapter 10
Chapter 10
University of Dhaka
MTH120_Chapter11
MTH120_Chapter11
Sida Say
8. testing of hypothesis for variable & attribute data
8. testing of hypothesis for variable & attribute data
Hakeem-Ur- Rehman
1667390753_Lind Chapter 10-14.pdf
1667390753_Lind Chapter 10-14.pdf
AriniputriLestari
Chapter 10 One sample test of hypothesis.ppt
Chapter 10 One sample test of hypothesis.ppt
rhanik1596
Telesidang 4 bab_8_9_10stst
Telesidang 4 bab_8_9_10stst
Nor Ihsan
Chapter 10
Chapter 10
Tuul Tuul
Estimation and hypothesis testing 1 (graduate statistics2)
Estimation and hypothesis testing 1 (graduate statistics2)
Harve Abella
Data Analysis Stats Problems Worksheet.docx
Data Analysis Stats Problems Worksheet.docx
studywriters
Javier Garcia - Verdugo Sanchez - Six Sigma Training - W2 Hypothesis Test
Javier Garcia - Verdugo Sanchez - Six Sigma Training - W2 Hypothesis Test
J. García - Verdugo
Statistical inference concept, procedure of hypothesis testing
Statistical inference concept, procedure of hypothesis testing
AmitaChaudhary19
Data analysis
Data analysis
Shameem Ali
Ch09(1)
Ch09(1)
Marouane Zouzhi
Hypothesis Testing: Central Tendency – Non-Normal (Compare 2+ Factors)
Hypothesis Testing: Central Tendency – Non-Normal (Compare 2+ Factors)
Matt Hansen
Testing of hypothesis
Testing of hypothesis
Jags Jagdish
49697462-bman12.pptx
49697462-bman12.pptx
ManoloTaquire
Hypothesis Testing techniques in social research.ppt
Hypothesis Testing techniques in social research.ppt
Solomonkiplimo
Ähnlich wie Chap#9 hypothesis testing (3)
(20)
Biomath12
Biomath12
1Copyright © 2015 The McGraw-Hill Companies, Inc. Permission r.docx
1Copyright © 2015 The McGraw-Hill Companies, Inc. Permission r.docx
Lecture 15 - Hypothesis Testing (1).pdf
Lecture 15 - Hypothesis Testing (1).pdf
Chapter 10
Chapter 10
MTH120_Chapter11
MTH120_Chapter11
8. testing of hypothesis for variable & attribute data
8. testing of hypothesis for variable & attribute data
1667390753_Lind Chapter 10-14.pdf
1667390753_Lind Chapter 10-14.pdf
Chapter 10 One sample test of hypothesis.ppt
Chapter 10 One sample test of hypothesis.ppt
Telesidang 4 bab_8_9_10stst
Telesidang 4 bab_8_9_10stst
Chapter 10
Chapter 10
Estimation and hypothesis testing 1 (graduate statistics2)
Estimation and hypothesis testing 1 (graduate statistics2)
Data Analysis Stats Problems Worksheet.docx
Data Analysis Stats Problems Worksheet.docx
Javier Garcia - Verdugo Sanchez - Six Sigma Training - W2 Hypothesis Test
Javier Garcia - Verdugo Sanchez - Six Sigma Training - W2 Hypothesis Test
Statistical inference concept, procedure of hypothesis testing
Statistical inference concept, procedure of hypothesis testing
Data analysis
Data analysis
Ch09(1)
Ch09(1)
Hypothesis Testing: Central Tendency – Non-Normal (Compare 2+ Factors)
Hypothesis Testing: Central Tendency – Non-Normal (Compare 2+ Factors)
Testing of hypothesis
Testing of hypothesis
49697462-bman12.pptx
49697462-bman12.pptx
Hypothesis Testing techniques in social research.ppt
Hypothesis Testing techniques in social research.ppt
Chap#9 hypothesis testing (3)
1.
9-1
Chapter 9 Hypothesis Testing © The McGraw-Hill Companies, Inc., 2000
2.
9-2
Outline 9-1 Introduction 9-2 Steps in Hypothesis Testing 9-3 Large Sample Mean Test 9-4 Small Sample Mean Test 9-5 Proportion Test © The McGraw-Hill Companies, Inc., 2000
3.
9-3
Outline 9-6 Variance or Standard Deviation Test 9-7 Confidence Intervals and Hypothesis Testing © The McGraw-Hill Companies, Inc., 2000
4.
9-4
Objectives Understand the definitions used in hypothesis testing. State the null and alternative hypotheses. Find critical values for the z test. © The McGraw-Hill Companies, Inc., 2000
5.
9-5
Objectives State the five steps used in hypothesis testing. Test means for large samples using the z test. Test means for small samples using the t test. © The McGraw-Hill Companies, Inc., 2000
6.
9-6
Objectives Test proportions using the z test. Test variances or standard deviation using the chi-square test. Test hypotheses using confidence intervals. © The McGraw-Hill Companies, Inc., 2000
7.
9-7
9-2 Steps in Hypothesis Testing A Statistical hypothesis is a conjecture about a population parameter. This conjecture may or may not be true. The null hypothesis, symbolized by H0, hypothesis is a statistical hypothesis that states that there is no difference between a parameter and a specific value or that there is no difference between two parameters. © The McGraw-Hill Companies, Inc., 2000
8.
9-8
9-2 Steps in Hypothesis Testing The alternative hypothesis, hypothesis symbolized by H1, is a statistical hypothesis that states a specific difference between a parameter and a specific value or states that there is a difference between two parameters. © The McGraw-Hill Companies, Inc., 2000
9.
9-9
9-2 Steps in Hypothesis Testing - Example A medical researcher is interested in finding out whether a new medication will have any undesirable side effects. The researcher is particularly concerned with the pulse rate of the patients who take the medication. © The McGraw-Hill Companies, Inc., 2000
10.
9-10
9-2 Steps in Hypothesis Testing - Example What are the hypotheses to test whether the pulse rate will be different from the mean pulse rate of 82 beats per minute? H0: µ = 82 H1: µ ≠ 82 This is a two-tailed test. © The McGraw-Hill Companies, Inc., 2000
11.
9-11
9-2 Steps in Hypothesis Testing - Example A chemist invents an additive to increase the life of an automobile battery. If the mean lifetime of the battery is 36 months, then his hypotheses are H0: µ ≤ 36 H1: µ > 36 This is a right-tailed test. © The McGraw-Hill Companies, Inc., 2000
12.
9-12
9-2 Steps in Hypothesis Testing - Example A contractor wishes to lower heating bills by using a special type of insulation in houses. If the average of the monthly heating bills is $78, her hypotheses about heating costs will be H0: µ ≥ $78 H0: µ < $78 This is a left-tailed test. © The McGraw-Hill Companies, Inc., 2000
13.
9-13
9-2 Steps in Hypothesis Testing A statistical test uses the data obtained from a sample to make a decision about whether or not the null hypothesis should be rejected. © The McGraw-Hill Companies, Inc., 2000
14.
9-14
9-2 Steps in Hypothesis Testing The numerical value obtained from a statistical test is called the test value. value In the hypothesis-testing situation, there are four possible outcomes. © The McGraw-Hill Companies, Inc., 2000
15.
9-15
9-2 Steps in Hypothesis Testing In reality, the null hypothesis may or may not be true, and a decision is made to reject or not to reject it on the basis of the data obtained from a sample. © The McGraw-Hill Companies, Inc., 2000
16.
9-16
9-2 Steps in Hypothesis Testing H0 True H0 False Error Error Correct Correct Reject Type II Type decision decision H0 Correct Correct Error Error Do not decision decision Type II Type II reject H0 © The McGraw-Hill Companies, Inc., 2000
17.
9-17
9-2 Steps in Hypothesis Testing A type I error occurs if one rejects the null hypothesis when it is true. A type II error occurs if one does not reject the null hypothesis when it is false. © The McGraw-Hill Companies, Inc., 2000
18.
9-18
9-2 Steps in Hypothesis Testing The level of significance is the maximum probability of committing a type I error. This probability is symbolized by α (Greek letter alpha). That is, P(type I error)=α . P(type II error) = β (Greek letter beta). © The McGraw-Hill Companies, Inc., 2000
19.
9-19
9-2 Steps in Hypothesis Testing Typical significance levels are: 0.10, 0.05, and 0.01. For example, when α = 0.10, there is a 10% chance of rejecting a true null hypothesis. © The McGraw-Hill Companies, Inc., 2000
20.
9-20
9-2 Steps in Hypothesis Testing The critical value(s) separates the critical region from the noncritical region. The symbol for critical value is C.V. © The McGraw-Hill Companies, Inc., 2000
21.
9-21
9-2 Steps in Hypothesis Testing The critical or rejection region is the range of values of the test value that indicates that there is a significant difference and that the null hypothesis should be rejected. © The McGraw-Hill Companies, Inc., 2000
22.
9-22
9-2 Steps in Hypothesis Testing The noncritical or nonrejection region is the range of values of the test value that indicates that the difference was probably due to chance and that the null hypothesis should not be rejected. © The McGraw-Hill Companies, Inc., 2000
23.
9-23
9-2 Steps in Hypothesis Testing A one-tailed test (right or left) indicates that the null hypothesis should be rejected when the test value is in the critical region on one side of the mean. © The McGraw-Hill Companies, Inc., 2000
24.
9-24
9-2 Finding the Critical Value for α = 0.01 (Right-Tailed Test) Critical 0.9900 region Noncritical region α = 0.01 0.4900 0 z = 2.33 © The McGraw-Hill Companies, Inc., 2000
25.
9-25
9-2 Finding the Critical Value for α = 0.01 (Left-Tailed Test) For a left-tailed test when α = 0.01, the critical value will be –2.33 and the critical region will be to the left of –2.33. © The McGraw-Hill Companies, Inc., 2000
26.
9-26
9-2 Finding the Critical Value for α = 0.01 (Two-Tailed Test) In a two-tailed test, the null hypothesis should be rejected when the test value is in either of the two critical regions. © The McGraw-Hill Companies, Inc., 2000
27.
9-27
9-2 Finding the Critical Value for α = 0.01 (Two-Tailed Test) α /2 = 0.005 Critical Critical 0.9900 region region Noncritical region α /2 = 0.005 z = –2.58 0 z = 2.58 © The McGraw-Hill Companies, Inc., 2000
28.
9-28
9-3 Large Sample Mean Test The z test is a statistical test for the mean of a population. It can be used when n ≥ 30, or when the population is normally distributed and σ is known. The formula for the z test is given on the next slide. © The McGraw-Hill Companies, Inc., 2000
29.
9-29
9-3 Large Sample Mean Test X −µ z= σ √n where X = sample mean µ = hypothesized population mean σ = population deviation n = sample size © The McGraw-Hill Companies, Inc., 2000
30.
9-30
9-3 Large Sample Mean Test - Example A researcher reports that the average salary of assistant professors is more than $42,000. A sample of 30 assistant professors has a mean salary of $43,260. At α = 0.05, test the claim that assistant professors earn more than $42,000 a year. The standard deviation of the population is $5230. © The McGraw-Hill Companies, Inc., 2000
31.
9-31
9-3 Large Sample Mean Test - Example Step 1: State the hypotheses and identify the claim. H0: µ ≤ $42,000 H1: µ > $42,000 (claim) Step 2: Find the critical value. Since α = 0.05 and the test is a right-tailed test, the critical value is z = +1.65. Step 3: Compute the test value. © The McGraw-Hill Companies, Inc., 2000
32.
9-32
9-3 Large Sample Mean Test - Example Step 3: z = [43,260 – 42,000]/[5230/√30] = 1.32. Step 4: Make the decision. Since the test value, +1.32, is less than the critical value, +1.65, and not in the critical region, the decision is “Do not reject the null hypothesis.” © The McGraw-Hill Companies, Inc., 2000
33.
9-33
9-3 Large Sample Mean Test - Example Step 5: Summarize the results. There is not enough evidence to support the claim that assistant professors earn more on average than $42,000 a year. See the next slide for the figure. © The McGraw-Hill Companies, Inc., 2000
34.
9-34
9-3 Large Sample Mean Test - Example 1.65 Reject α = 0.05 1.32 © The McGraw-Hill Companies, Inc., 2000
35.
9-35
9-3 Large Sample Mean Test - Example A national magazine claims that the average college student watches less television than the general public. The national average is 29.4 hours per week, with a standard deviation of 2 hours. A sample of 30 college students has a mean of 27 hours. Is there enough evidence to support the claim at α = 0.01? © The McGraw-Hill Companies, Inc., 2000
36.
9-36
9-3 Large Sample Mean Test - Example Step 1: State the hypotheses and identify the claim. H0: µ ≥ 29.4 H1: µ < 29.4 (claim) Step 2: Find the critical value. Since α = 0.01 and the test is a left-tailed test, the critical value is z = –2.33. Step 3: Compute the test value. © The McGraw-Hill Companies, Inc., 2000
37.
9-37
9-3 Large Sample Mean Test - Example Step 3: z = [27– 29.4]/[2/√30] = – 6.57. Step 4: Make the decision. Since the test value, – 6.57, falls in the critical region, the decision is to reject the null hypothesis. © The McGraw-Hill Companies, Inc., 2000
38.
9-38
9-3 Large Sample Mean Test - Example Step 5: Summarize the results. There is enough evidence to support the claim that college students watch less television than the general public. See the next slide for the figure. © The McGraw-Hill Companies, Inc., 2000
39.
9-39
9-3 Large Sample Mean Test - Example -6.57 Reject -2.33 © The McGraw-Hill Companies, Inc., 2000
40.
9-40
9-3 Large Sample Mean Test - Example The Medical Rehabilitation Education Foundation reports that the average cost of rehabilitation for stroke victims is $24,672. To see if the average cost of rehabilitation is different at a large hospital, a researcher selected a random sample of 35 stroke victims and found that the average cost of their rehabilitation is $25,226. © The McGraw-Hill Companies, Inc., 2000
41.
9-41
9-3 Large Sample Mean Test - Example The standard deviation of the population is $3,251. At α = 0.01, can it be concluded that the average cost at a large hospital is different from $24,672? Step 1: State the hypotheses and identify the claim. H0: µ = $24,672 H1: µ ≠ $24,672 (claim) © The McGraw-Hill Companies, Inc., 2000
42.
9-42
9-3 Large Sample Mean Test - Example Step 2: Find the critical values. Since α = 0.01 and the test is a two-tailed test, the critical values are z = –2.58 and +2.58. Step 3: Compute the test value. Step 3: z = [25,226 – 24,672]/[3,251/√35] = 1.01. © The McGraw-Hill Companies, Inc., 2000
43.
9-43
9-3 Large Sample Mean Test - Example Step 4: Make the decision. Do not reject the null hypothesis, since the test value falls in the noncritical region. Step 5: Summarize the results. There is not enough evidence to support the claim that the average cost of rehabilitation at the large hospital is different from $24,672. © The McGraw-Hill Companies, Inc., 2000
44.
9-44
9-3 Large Sample Mean Test - Example 1.01 Reject Reject -2.58 2.58 © The McGraw-Hill Companies, Inc., 2000
45.
9-45
9-3 P-Values Besides listing an α value, many computer statistical packages give a P- value for hypothesis tests. The P-value is the actual probability of getting the sample mean value or a more extreme sample mean value in the direction of the alternative hypothesis (> or <) if the null hypothesis is true. © The McGraw-Hill Companies, Inc., 2000
46.
9-46
9-3 P-Values The P-value is the actual area under the standard normal distribution curve (or other curve, depending on what statistical test is being used) representing the probability of a particular sample mean or a more extreme sample mean occurring if the null hypothesis is true. © The McGraw-Hill Companies, Inc., 2000
47.
9-47
9-3 P-Values - Example A researcher wishes to test the claim that the average age of lifeguards in Ocean City is greater than 24 years. She selects a sample of 36 guards and finds the mean of the sample to be 24.7 years, with a standard deviation of 2 years. Is there evidence to support the claim at α = 0.05? Find the P-value. © The McGraw-Hill Companies, Inc., 2000
48.
9-48
9-3 P-Values - Example Step 1: State the hypotheses and identify the claim. H0: µ ≤ 24 H1: µ > 24 (claim) Step 2: Compute the test value. 24.7 − 24 z= = 2.10 2 36 © The McGraw-Hill Companies, Inc., 2000
49.
9-49
9-3 P-Values - Example Step 3: Using Table E in Appendix C, find the corresponding area under the normal distribution for z = 2.10. It is 0.4821 Step 4: Subtract this value for the area from 0.5000 to find the area in the right tail. 0.5000 – 0.4821 = 0.0179 Hence the P-value is 0.0179. © The McGraw-Hill Companies, Inc., 2000
50.
9-50
9-3 P-Values - Example Step 5: Make the decision. Since the P-value is less than 0.05, the decision is to reject the null hypothesis. Step 6: Summarize the results. There is enough evidence to support the claim that the average age of lifeguards in Ocean City is greater than 24 years. © The McGraw-Hill Companies, Inc., 2000
51.
9-51
9-3 P-Values - Example A researcher claims that the average wind speed in a certain city is 8 miles per hour. A sample of 32 days has an average wind speed of 8.2 miles per hour. The standard deviation of the sample is 0.6 mile per hour. At α = 0.05, is there enough evidence to reject the claim? Use the P-value method. © The McGraw-Hill Companies, Inc., 2000
52.
9-52
9-3 P-Values - Example Step 1: State the hypotheses and identify the claim. H0: µ = 8 (claim) H1: µ ≠ 8 Step 2: Compute the test value. 8.2 − 8 z= = 1.89 0.6 32 © The McGraw-Hill Companies, Inc., 2000
53.
9-53
9-3 P-Values - Example Step 3: Using table E, find the corresponding area for z = 1.89. It is 0.4706. Step 4: Subtract the value from 0.5000. 0.5000 – 0.4706 = 0.0294 © The McGraw-Hill Companies, Inc., 2000
54.
9-54
9-3 P-Values - Example Step 5: Make the decision: Since this test is two-tailed, the value 0.0294 must be doubled; 2(0.0294) = 0.0588. Hence, the decision is not to reject the null hypothesis, since the P-value is greater than 0.05. Step 6: Summarize the results. There is not enough evidence to reject the claim that the average wind speed is 8 miles per hour. © The McGraw-Hill Companies, Inc., 2000
55.
9-55
9-4 Small Sample Mean Test When the population standard deviation is unknown and n < 30, the z test is inappropriate for testing hypotheses involving means. The t test is used in this case. Properties for the t distribution are given in Chapter 8. © The McGraw-Hill Companies, Inc., 2000
56.
9-56
9-4 Small Sample Mean Test - Formula for t test X −µ t= s n where X = sample mean µ = hypothesized population mean s = sample standard deviation n = sample size degrees of freedom = n − 1 © The McGraw-Hill Companies, Inc., 2000
57.
9-57
9-4 Small Sample Mean Test - Example A job placement director claims that the average starting salary for nurses is $24,000. A sample of 10 nurses has a mean of $23,450 and a standard deviation of $400. Is there enough evidence to reject the director’s claim at α = 0.05? © The McGraw-Hill Companies, Inc., 2000
58.
9-58
9-4 Small Sample Mean Test - Example Step 1: State the hypotheses and identify the claim. H0: µ = $24,000 (claim) H1: µ ≠ $24,000 Step 2: Find the critical value. Since α = 0.05 and the test is a two-tailed test, the critical values are t = –2.262 and +2.262 with d.f. = 9. © The McGraw-Hill Companies, Inc., 2000
59.
9-59
9-4 Small Sample Mean Test - Example Step 3: Compute the test value. t = [23,450 – 24,000]/[400/√10] = – 4.35. Step 4: Reject the null hypothesis, since – 4.35 < – 2.262. Step 5: There is enough evidence to reject the claim that the starting salary of nurses is $24,000. © The McGraw-Hill Companies, Inc., 2000
60.
9-60
9-3 Small Sample Mean Test - Example −4.35 –2.262 2.262 © The McGraw-Hill Companies, Inc., 2000
61.
9-61
9-5 Proportion Test Since the normal distribution can be used to approximate the binomial distribution when np ≥ 5 and nq ≥ 5, the standard normal distribution can be used to test hypotheses for proportions. The formula for the z test for proportions is given on the next slide. © The McGraw-Hill Companies, Inc., 2000
62.
9-62
9-5 Formula for the z Test for Proportions X −µ X − np z= or z = σ npq where µ = np σ = npq © The McGraw-Hill Companies, Inc., 2000
63.
9-63
9-5 Proportion Test - Example An educator estimates that the dropout rate for seniors at high schools in Ohio is 15%. Last year, 38 seniors from a random sample of 200 Ohio seniors withdrew. At α = 0.05, is there enough evidence to reject the educator’s claim? © The McGraw-Hill Companies, Inc., 2000
64.
9-64
9-5 Proportion Test - Example Step 1: State the hypotheses and identify the claim. H0: p = 0.15 (claim) H1: p ≠ 0.15 Step 2: Find the mean and standard deviation. µ = np = (200)(0.15) = 30 and σ = √(200)(0.15)(0.85) = 5.05. Step 3: Find the critical values. Since α = 0.05 and the test is two-tailed the critical values are z = ±1.96. © The McGraw-Hill Companies, Inc., 2000
65.
9-65
9-5 Proportion Test - Example Step 4: Compute the test value. z = [38 – 30]/[5.05] = 1.58. Step 5: Do not reject the null hypothesis, since the test value falls outside the critical region. © The McGraw-Hill Companies, Inc., 2000
66.
9-66
9-5 Proportion Test - Example Step 6: Summarize the results. There is not enough evidence to reject the claim that the dropout rate for seniors in high schools in Ohio is 15%. Note: For one-tailed test for proportions, follow procedures for the large sample mean test. © The McGraw-Hill Companies, Inc., 2000
67.
9-67
9-5 Proportion Test - Example 1.58 -1.96 1.96 © The McGraw-Hill Companies, Inc., 2000
68.
9-68
9-6 Variance or Standard Deviation Test - Formula (n − 1) s 2 χ = 2 with d . f .= n –1 σ 2 where n = sample size σ = population variance 2 s = sample variance 2 © The McGraw-Hill Companies, Inc., 2000
69.
9-69
9-6 Assumptions for the Chi-Square Test for a Single Variance The sample must be randomly selected from the population. The population must be normally distributed for the variable under study. The observations must be independent of each other. © The McGraw-Hill Companies, Inc., 2000
70.
9-70
9-6 Variance Test - Example An instructor wishes to see whether the variation in scores of the 23 students in her class is less than the variance of the population. The variance of the class is 198. Is there enough evidence to support the claim that the variation of the students is less than the population variance (σ 2 = 225) at α = 0.05? © The McGraw-Hill Companies, Inc., 2000
71.
9-71
9-6 Variance Test - Example Step 1: State the hypotheses and identify the claim. H0: σ 2 ≥ 225 H1: σ 2 < 225 (claim) Step 2: Find the critical value. Since this test is left-tailed and α = 0.05, use the value 1 – 0.05 = 0.95. The d.f. = 23 – 1 = 22. Hence, the critical value is 12.338. © The McGraw-Hill Companies, Inc., 2000
72.
9-72
9-6 Variance Test - Example Step 3: Compute the test value. χ 2 = (23 – 1)(198)/225 = 19.36. Step 4: Make a decision. Do not reject the null hypothesis, since the test value falls outside the the critical region. © The McGraw-Hill Companies, Inc., 2000
73.
9-73
9-6 Variance Test - Example 0.05 12.338 19.36 © The McGraw-Hill Companies, Inc., 2000
74.
9-74
9-6 Variance Test Note: When the test is two-tailed, you will need to find χ 2(left) and χ 2(right) and check whether the test value is less than χ 2(left) or whether it is greater than χ 2(right) in order to reject the null hypothesis. © The McGraw-Hill Companies, Inc., 2000
75.
9-7 Confidence Intervals
and 9-75 Hypothesis Testing - Example Sugar is packed in 5-pound bags. An inspector suspects the bags may not contain 5 pounds. A sample of 50 bags produces a mean of 4.6 pounds and a standard deviation of 0.7 pound. Is there enough evidence to conclude that the bags do not contain 5 pounds as stated, at α = 0.05? Also, find the 95% confidence interval of the true mean. © The McGraw-Hill Companies, Inc., 2000
76.
9-7 Confidence Intervals
and 9-76 Hypothesis Testing - Example H0: µ = 5 H1: µ ≠ 5 (claim) The critical values are +1.96 and – 1.96 = 4.6 − 5.00 = − 4.04 The test value is z . 0 7 50 Since – 4.04 < –1.96, the null hypothesis is rejected. There is enough evidence to support the claim that the bags do not weigh 5 pounds. © The McGraw-Hill Companies, Inc., 2000
77.
9-7 Confidence Intervals
and 9-77 Hypothesis Testing - Example The 95% confidence interval for the mean is given by s s X −z ⋅ <µ<X+z ⋅ α2 n α2 n 0.7 < µ < 4.6 + (1.96) 0.7 4.6 − (1.96) 50 50 4.406 < µ < 4.794 Notice that the 95% confidence interval of µ does not contain the hypothesized value µ = 5. Hence, there is agreement between the hypothesis test and the confidence interval. © The McGraw-Hill Companies, Inc., 2000
Hinweis der Redaktion
1 1
2 2 2
3 3 2
4 4
5 5
5 5
7 7
8 8
9 9
10 10
11 11
12 12
13 13
14 14
15 15
16 16
17 17
18 18
19 19
20 20
21 21
22 22
23 23
24 24
25 25
26 26
27 27
28 28
29 29
30 30
31 31
32 32
33 33
34 34
35 35
36 36
37 37
38 38
39 39
40 40
41 41
42 42
43 43
44 44
45 45
46 46
47 47
48 48
49 49
50 50
51 51
50 50
53 53
54 54
51 51
52 52
53 53
54 54
55 55
56 56
57 57
58 58
59 59
60 60
61 61
62 62
63 63
64 64
65 65
66 66
67 67
68 68
69 69
70 70
70 70
70 70
70 70
Jetzt herunterladen