Transcription
Chapter 6Hypothesis Testing
Key words : Null hypothesis H0, Alternative hypothesis HA , testinghypothesis , test statistic , P-valueText Book : Basic Concepts andMethodology for the Health Sciences2
Hypothesis Testing One type of statistical inference,estimation, was discussed in Chapter 5. The other type ,hypothesis testing ,isdiscussed in this chapter.Text Book : Basic Concepts andMethodology for the Health Sciences3
Definition of a hypothesisIt is a statement about one or morepopulations.It is usually concerned with the parameters ofthe population. e.g. the hospital administratormay want to test the hypothesis that theaverage length of stay of patients admitted tothe hospital is 5 daysText Book : Basic Concepts andMethodology for the Health Sciences4
Definition of Statistical hypothesisThey are hypothesis that are stated in such a waythat they may be evaluated by appropriate statisticaltechniques. There are two hypotheses involved in hypothesistesting Null hypothesis H0: It is the hypothesis to betested . Alternative hypothesis HA : It is a statement ofwhat we believe is true if our sample data cause usto reject the null hypothesis Text Book : Basic Concepts andMethodology for the Health Sciences5
Text Book : Basic Concepts andMethodology for the Health Sciences6
7.2 Testing a hypothesis about the mean of apopulation:We have the following steps:1.Data: determine variable, sample size (n), samplemean( x ) , population standard deviation orsample standard deviation (s) if is unknown2. Assumptions : We have two cases: Case1: Population is normally or approximatelynormally distributed with known or unknownvariance (sample size n may be small or large), Case 2: Population is not normal with known orunknown variance (n is large i.e. n 30). Text Book : Basic Concepts andMethodology for the Health Sciences7
3.Hypothesis:we have three casesCase I : H0: μ μ0HA: μ μ0 e.g. we want to test that the population mean isdifferent than 50 Case II : H0: μ μ0HA: μ μ0 e.g. we want to test that the population mean is greaterthan 50Case III : H0: μ μ0HA: μ μ0 e.g. we want to test that the population mean is lessthan 50Text Book : Basic Concepts andMethodology for the Health Sciences8
Testing hypothesis for the mean μ : When the value of sample size (n):population is normal or not normal( n 30 )σ is knownZ X 0 n population is normal(n 30)σ is not knownZ X 0Snσ is knownZ X 0 n σ is not knownT X 0SnText Book : Basic Concepts andMethodology for the Health Sciences9
Text Book : Basic Concepts andMethodology for the Health Sciences10
Text Book : Basic Concepts andMethodology for the Health Sciences11
The Use of P – Values in Decision Definition Making::
6.Decision : If we reject H0, we can conclude that HA istrue. If ,however ,we do not reject H0, we mayconclude that H0 is true. Text Book : Basic Concepts andMethodology for the Health Sciences14
An Alternative Decision Rule using thep - value Definition The p-value is defined as the smallest value of αfor which the null hypothesis can be rejected.If the p-value is less than or equal to α ,wereject the null hypothesis (p α) If the p-value is greater than α ,we do not rejectthe null hypothesis (p α) Text Book : Basic Concepts andMethodology for the Health Sciences15
Example 7.2.1 Page 223Researchers are interested in the mean age of acertain population. A random sample of 10 individuals drawn fromthe population of interest has a mean of 27. Assuming that the population is approximatelynormally distributed with variance 20,can weconclude that the mean is different from 30years ? (α 0.05) . If the p - value is 0.0340 how can we use it inmaking a decision? Text Book : Basic Concepts andMethodology for the Health Sciences16
Solution1-Data: variable is age, n 10, x 27 ,σ2 20,α 0.052-Assumptions: the population is approximatelynormally distributed with variance 203-Hypotheses: H0 : μ 30 HA: μ 30Text Book : Basic Concepts andMethodology for the Health Sciences17
4-Test Statistic: Z -2.125.Decision Rule The alternative hypothesis isHA: μ 30 Hence we reject H0 if Z Z1-0.025 Z0.975or Z - Z1-0.025 - Z0.975 Z0.975 1.96(from table D)Text Book : Basic Concepts andMethodology for the Health Sciences18
6.Decision: We reject H0 ,since -2.12 is in therejection region . We can conclude that μ is not equal to 30 Using the p value ,we note that p-value 0.0340 0.05,therefore we reject H0Text Book : Basic Concepts andMethodology for the Health Sciences19
Example7.2.2 page227Referring to example 7.2.1.Suppose that theresearchers have asked: Can we concludethat μ 30.1.Data.see previous example2. Assumptions .see previous example3.Hypotheses: H0 μ 30 HِA: μ 30 Text Book : Basic Concepts andMethodology for the Health Sciences20
4.Test Statistic : X - oZ n 27 302010 -2.125. Decision Rule: Reject H0 if Z - Z 1-α, whereZ 1-α 1.645. (from table D)6. Decision: Reject H0 ,thus we can conclude that thepopulation mean is smaller than 30. Text Book : Basic Concepts andMethodology for the Health Sciences21
Example7.2.4 page232 Among 157 African-American men ,themean systolic blood pressure was 146mm Hg with a standard deviation of 27.We wish to know if on the basis of thesedata, we may conclude that the meansystolic blood pressure for a populationof African-American is greater than 140.Use α 0.01.Text Book : Basic Concepts andMethodology for the Health Sciences22
Solution1. Data: Variable is systolic blood pressure,n 157 , 146, s 27, α 0.01.2. Assumption: population is not normal, σ2 isunknown3. Hypotheses: H0 :μ 140HA: μ 1404.Test Statistic:146 1406 Z X - 2.78s27on2.1548157Text Book : Basic Concepts andMethodology for the Health Sciences23
5. Decision Rule:we reject H0 if Z Z1-α Z0.99 2.33(from table D)6. Decision: We reject H0.Hence we may conclude that the meansystolic blood pressure for a population ofAfrican-American is greater than 140.Text Book : Basic Concepts andMethodology for the Health Sciences24
ExercisesQ7.2.1:Escobar performed a study to validate a translatedversion of the Western Ontario and McMasterUniversity index (WOMAC) questionnaire used withspanish-speaking patient s with hip or kneeosteoarthritis . For the 76 women classified with severhip pain. The WOMAC mean function score was 70.7with standard deviation of 14.6 , we wish to know if wemay conclude that the mean function score for apopulation of similar women subjects with sever hippain is less than 75 . Let α 0.01Text Book : Basic Concepts andMethodology for the Health Sciences25
Solution :1.Data :2. Assumption :3. Hypothesis :4.Test statistic :Text Book : Basic Concepts andMethodology for the Health Sciences26
5.Decision Rule6. Decision :Text Book : Basic Concepts andMethodology for the Health Sciences27
ExercisesQ7.2.3:The purpose of a study by Luglie was to investigate the oral statusof a group of patients diagnosed with thalassemia major (TM) .One of the outcome measure s was the decayed , missing, filledteeth index (DMFT) . In a sample of 18 patients ,the mean DMFTindex value was 10.3 with standard deviation of 7.3 . Is thissufficient evidence to allow us to conclude that the mean DMFTindex is greater than 9 in a population of similar subjects?Let α 0.1Text Book : Basic Concepts andMethodology for the Health Sciences28
Solution :1.Data :2. Assumption :3. Hypothesis :4.Test statistic :Text Book : Basic Concepts andMethodology for the Health Sciences29
5.Decision Rule6. Decision :Text Book : Basic Concepts andMethodology for the Health Sciences30
For Q7.2.3:Take the p- value 0.22 , Use the P-value tomake your decision ?Text Book : Basic Concepts andMethodology for the Health Sciences31
Text Book : Basic Concepts and Methodology for the Health Sciences 16. Solution 1-Data: 2variable is age, n 10, 27 ,σ 20,α 0.05 2-Assumptions: the population is approximately nor
