1. 1 Dr. Jerrell T. Stracener EMIS 7370 STAT 5340 Probability and Statistics for Scientists and Engineers Department of Engineering Management, Information and Systems Tests of Hypothesis Basic Concepts & Tests of Proportions

2. 2 Tests of Hypothesis - Basic Concepts

3. 3 In many situations the reason for gathering and analyzing data is to provide a basis for deciding on a course of action. Let us assume that either of two courses of action is possible: A 1 or A 2, and that we would be clear whether one or the other is the better action, if only we knew the nature of a certain population - that is, if we knew the probability distribution of a certain random variable. Testing Hypotheses

4. 4 The whole population or the distribution of probability is usually unattainable, therefore, we are forced to settle for such information as can be gleaned from a sample and to make our choice between the two actions on the basis of the sample. 1. Obtain random sample of size n 2. Apply decision rule to data Testing Hypotheses

5. 5 Statistical Hypothesis - is a statement about a probability distribution and is usually a statement about the values of one or more parameters of the distribution. For example, a company may want to test the hypothesis that the true average lifetime of a certain type of TV is at least 500 hours, i.e., that   500. Testing Hypotheses

6. 6 The hypothesis to be tested is called the null hypothesis and is denoted by H 0. To construct a criterion for testing a given null hypothesis, an Alternative hypotheses, must be formed denoted by H 1 or H A. Remark: To test the validity of a statistical hypothesis the test is conducted, and according to the test plan the hypothesis is rejected if the results are improbable under the hypothesis. If not, the hypothesis is accepted. The test leads to one of two possible actions: accept H 0 or reject H 0 (accept H 1 ) Testing Hypotheses

7. 7 Test Statistic - The statistic upon which a test of a statistical hypothesis is based. Critical Region - The range of values of a test statistic which, for a given test, requires the rejection of H 0. Remark: Acceptance or rejection of a statistical hypothesis does not prove nor disprove the hypothesis! Whenever a statistical hypothesis is accepted or rejected on the basis of a sample, there is always the risk of making a wrong decision. The uncertainty with which a decision is made is measured in terms of probability. Testing Hypotheses

8. 8 There are two possible decision errors associated with testing a statistical hypothesis: A Type I error is made when a true hypothesis is rejected. A Type II error is made when a false hypothesis is accepted. True Situation DecisionH 0 trueH 0 false Accept H 0 correctType II error Reject H 0 Type I error correct (Accept H 1 ) Testing Hypotheses

9. 9 The decision risks are measured in terms of probability.  = P(Type I error) = P(reject H 0 |H 0 is true) = Producers risk  = P(Type II error) = P(accept H 0 |H 1 is true) = Consumers risk Remark:   100% is commonly referred to as the significance level of a test. Testing Hypotheses

10. 10 Note: For fixed n,  increases as  decreases, and vice versa, as n increases, both  and  decrease. Testing Hypotheses

11. 11 The Type I error and Type II error are related. A decrease in the probability of one generally results in an increase in the probability of the other. The size of the critical region, and therefore the probability of committing a Type I error, can always be reduced by adjusting the critical value(s). Important Properties

12. 12 An increase in the sample size n will reduce  and  simultaneously. If the null hypothesis is false,  is a maximum when the true value of a parameter approaches the hypothesized value. The greater the distance between the true value and the hypothesized value, the smaller  will be. Important Properties

13. 13 A p-value is the lowest level of significance at which the observed value of the test statistic is significant. Reject H 0 if the computed p-value is less than or equal to the desired level of significance . P-Value

14. 14 Before applying a test procedure, i.e., a decision rule, we need to analyze its discriminating power, i.e., how good the test is. A function called the power function enables us to make this analysis. Power Function = PF() = P R () = P(rejecting H 0 |true parameter value) Power Function

15. 15 A plot of the power function vs the test parameter value is called the power curve and 1 - power curve is the OC curve. 1 0 PR()PR() ideal power curve H0H0 H1H1  Power Function

16. 16 The power of a test can be computed as 1 -  Often different types of tests are compared by contrasting power properties Power of a Test

17. 17 The power function of a statistical test of hypothesis is the probability of rejecting H 0 as a function of the true value of the parameter being tested, say , i.e., PF(  ) = PR(  ) = P(reject H 0 |  ) = P(test statistic falls in C R | ) Power Function

18. 18 The operating characteristic function of a statistical test of hypothesis is the probability of accepting H 0 as a function of the true value of the parameter being tested, say , i.e., OC(  )= P A (  ) = P(accept H 0 |  ) = P(test statistic falls in C A | ) Note that OC(  )=1-PF(  ) Operating Characteristic Function

19. 19 1. State the null hypothesis H 0 that  =  0 2. Choose an appropriate alternative hypothesis H 1 from one of the alternatives   0, or    0 3. Choose a significance level of size . 4. Select the appropriate test statistic and establish the critical region. (If the decision is to be based on a P-value, it is not necessary to state the critical region.) 5. Compute the value of the test statistic from the sample data 6. Carry out the test of hypothesis and make a Decision. Reject H 0 if the test statistic has a value in the critical region (or if the computed P-value is less than or equal to the desired significance level  ); otherwise, do not reject H 0 Summary Procedures for Hypothesis Testing

20. 20 Tests of Proportions

21. 21 Let X 1, X 2,..., X n be a random sample of size n from B(n, p). Case 1: small sample sizes To test the Null Hypothesis H 0 : p = p 0, a specified value, against the appropriate Alternative Hypothesis 1. H A : p < p 0, or 2. H A : p > p 0, or 3. H A : p  p 0, Tests of Proportions

22. 22 at the   100% Level of Significance, calculate the value of the test statistic using X ~ B(n, p = p 0 ). Find the number of successes and compute the appropriate P-Value, depending upon the alternative hypothesis and reject H 0 if P  , where 1. P = P(X  x|p = p 0 ), or 2. P = P(X  x|p = p 0 ), or 3. P = 2P(X  x|p = p 0 ) if x < np 0, or P = 2P(X  x|p = p 0 ) if x > np 0, Tests of Proportions

23. 23 Case 2: large sample sizes with p not extremely close to 0 or 1. To test the Null Hypothesis H 0 : p = p 0, a specified value, against the appropriate Alternative Hypothesis 1. H A : p < p 0, or 2. H A : p > p 0, or 3. H A : p  p 0, Tests of Proportions

24. 24 Calculate the value of the test statistic and reject H 0 if 1., or 2., or 3. or, depending on the alternative hypothesis. Tests of Proportions

25. 25 A missile manufacturer claims that the probability of success for missile MX is 0.80. To demonstrate its reliability 25 missiles are fired. If 15 are successful, do you agree with the manufacturers claim? Example

26. 26 We want to test H 0 : p=0.8 vs H 1 : p  0.8 At the 10% level of significance (Since a value was given, I selected 10% to illustrate the process) The test statistic is X, the observed number of successes, and X~B(25,p) Solution

27. 27 We have to determine values L and U where and Note that is we cannot find integer values of L and U, select those values of L and U which make the value of each of the summations as large as possible without exceeding  /2. Now and Solution

28. 28 Here, L=16for S L =0.0468 U=23 for S u =0.0274 We reject H 0 if X U. Since X=15, we reject H 0 and conclude that at the 7.42% level of significance that the manufacturer’s claim is incorrect. Note that L=17 and U=22, result in S L = 0.109 and Su=0.0982 Solution

29. 29 Using case 2, the value of the test statistic is The critical region is Z 1.64. Since -2.5 is less than -1.64, reject H 0 and conclude that the manufacturer’s claim is incorrect. Solution

30. 30 A manufacturing company has submitted a claim that 90% of items produced by a certain process are non-defective. An improvement in the process is being considered that they feel will lower the proportion of defectives below the current 10%. In an experiment 100 items are produced with the new process and 5 are defective. Is this evidence sufficient to conclude that the method has been improved? Use a 0.05 level of significance. Example

31. 31 Follow the six-step procedure: 1.H 0 : p=0.9 2.H 1 :p>0.9  =0.05 4.Critical region: Z>1.645 5.Computations: x=95, n=100, np 0 =(100)(0.90)=90, and 6.Conclusion: Reject H 0 and conclude that the improvement has reduced the proportion of defects. Solution