STAT 400 Lecture AL1 1 Spring 2015 Dalpiaz A scientist wishes to test if a new treatment has a better cure rate than the traditional treatment which cures only 60 of the patients In order to test whether the new treatment is more effective or not a test group of 20 patients were given the new treatment Assume that each personal result is independent of the others Trying to decide a Answers for 8 3 8 1 cure rate p 0 60 vs p 0 60 If the new treatment has the same success rate as the traditional what is the probability that at least 14 out of 20 patients 14 or more will be cured P X 14 p 0 60 1 CDF 13 p 0 60 1 0 750 0 250 b Suppose that 14 out of 20 patients in the test group were cured Based on the answer for part a is there a reason to believe that the new treatment has a better cure rate than the traditional treatment If p 0 60 then 25 of all possible samples would have 14 or more patients cured out of 20 Thus it is not unusual to see 14 out of 20 patients cured for a treatment that cures 60 of the patients We have no reason to believe that the new treatment has a better cure rate than the traditional treatment if X 14 c If the new treatment has the same success rate as the traditional what is the probability that at least 17 out of 20 patients 17 or more will be cured P X 17 p 0 60 1 CDF 16 p 0 60 1 0 984 0 016 d Suppose that 17 out of 20 patients in the test group were cured Based on the answer for part c is there a reason to believe that the new treatment has a better cure rate than the traditional treatment If p 0 60 then only 1 6 of all possible samples would have 17 or more patients cured out of 20 Thus it is fairly unusual to see 17 out of 20 patients cured for a treatment that cures 60 of the patients We have a good reason to believe that the new treatment has a better cure rate than the traditional treatment if X 17 2 A certain automobile manufacturer claims that at least 80 of its cars meet the tough new standards of the Environmental Protection Agency EPA Let p denote the proportion of the cars that meet the new EPA standards The EPA tests a random sample of 400 its cars suppose that 308 of the 400 cars in our sample meet the new EPA standards a Perform an appropriate test at a 10 level of significance 0 10 Claim p 0 80 H 0 p 0 80 Y 308 n 400 Test Statistic Z Decision H 1 p 0 80 Y 308 0 77 p n 400 p p0 0 77 0 80 1 50 p0 1 p0 0 80 0 20 400 n Left tailed Rejection Region 0 10 vs Reject H 0 if Z z Reject H 0 if Z 1 282 z 0 10 1 282 The value of the test statistic DOES fall into the Rejection Region Reject H 0 at 0 10 b Perform an appropriate test at a 5 level of significance 0 05 Left tailed Rejection Region 0 05 Decision Reject H 0 if Z z Reject H 0 if Z 1 645 z 0 05 1 645 The value of the test statistic does NOT fall into the Rejection Region Do NOT Reject H 0 at 0 05 c Find the p value of the appropriate test Left tailed d P value P Z 1 50 0 0668 Using the p value from part c state your decision Accept H0 or Reject H0 at 0 08 0 0668 p value 0 08 Reject H 0 at 0 08 3 Alex wants to test whether a coin is fair or not Suppose he observes 477 heads in 900 tosses Let p denote the probability of obtaining heads a Perform the appropriate test using a 10 level of significance Claim p 0 50 H 0 p 0 50 Y 477 n 900 Test Statistic Z Y 477 0 53 p n 900 p p0 0 53 0 50 1 80 p0 1 p0 0 50 0 50 n 900 Two tailed Rejection Region Reject H 0 if Z z 0 10 H 1 p 0 50 vs 2 or Z z 2 z 0 05 1 645 2 0 05 Reject H 0 if Z 1 645 or Z 1 645 Decision The value of the test statistic does fall into the Rejection Region Reject H 0 at 0 10 OR P value Two tailed P value P Z 1 80 2 0 0359 0 0718 Decision 0 0718 p value 0 10 b Find the p value of the test in part a Two tailed c Reject H 0 at 0 10 P value P Z 1 80 2 0 0359 0 0718 Using the p value from part b state your decision Accept H 0 or Reject H 0 for 0 05 0 0718 p value 0 05 Do NOT Reject H 0 at 0 05 4 H 0 p 0 20 vs H 1 p 0 20 Y 72 n 300 Compute the p value State your decision at 0 05 Y 72 0 24 p n 300 p p0 Test Statistic Z P value Rightt tailed 0 0418 p value 0 05 p0 1 p0 n 0 24 0 20 0 20 0 80 300 1 73 P value P Z 1 73 0 0418 Reject H 0 at 0 05
View Full Document
Unlocking...