STAT 400 Lecture AL1 1 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 Fall 2014 Dalpiaz Examples for 8 3 8 1 cure rate p 0 60 vs p 0 60 CDF x p n x 20 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 a 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 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 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 0 60 0 000 0 000 0 000 0 000 0 000 0 002 0 006 0 021 0 057 0 128 0 245 0 404 0 584 0 750 0 874 0 949 0 984 0 996 0 999 1 000 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 A null hypothesis denoted by H 0 is an assertion about one or more population parameters This is the assertion we hold as true until we have sufficient statistical evidence to conclude otherwise The alternative hypothesis denoted by H 1 is the assertion of all situations not covered by the null hypothesis The test is designed to assess the strength of the evidence against the null hypothesis H 0 true Accept H 0 Do NOT Reject H 0 Reject H 0 H 0 false Type II Error Type I Error significance level P Type I Error P Reject H 0 H 0 is true P Type II Error P Do Not Reject H 0 H 0 is NOT true Power 1 P Type II Error P Reject H 0 H 0 is NOT true Testing Hypotheses about a Population Proportion Null p Alternative H 0 p p0 vs H1 p p0 Left tailed H 0 p p0 vs H1 p p0 Right tailed H 0 p p0 vs H1 p p0 Two tailed Test Statistic z p p0 p0 1 p0 n z Y n p0 n p0 1 p0 where Y is the number of S s in n independent trials Rejection Region H 0 p p0 H 0 p p0 H 0 p p0 H 1 p p0 H 1 p p0 H 1 p p0 Left tailed Right tailed Two tailed Reject H 0 if Reject H 0 if Reject H 0 if Z z Z z Z z 2 or Z z 2 If the value of the Test Statistic falls into the Rejection Region then Reject H 0 otherwise Accept H 0 Do NOT Reject H 0 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 H0 vs H1 Test Statistic Rejection Region Decision b Perform an appropriate test at a 5 level of significance 0 05 Rejection Region Decision The P value observed level of significance is the probability computed assuming that H 0 is true of obtaining a value of the test statistic as extreme as or more extreme than the observed value The smaller the p value is the stronger is evidence against H 0 provided by the data P value P value Do NOT Reject H 0 Accept H 0 Reject H 0 Computing P value H 0 p p0 H 0 p p0 H 0 p p0 H 1 p p0 H 1 p p0 H 1 p p0 Left tailed Right tailed Two tailed Area to the left of the observed test statistic Area to the right of the observed test statistic 2 area of the tail c Find the p value of the appropriate test d Using the p value from part c state your decision Accept H0 or Reject H0 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 H0 vs H1 Test Statistic Rejection Region Decision b Find the p value of the test in part a c Using the p value from part b state your decision Accept H 0 or Reject H 0 for 0 05
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