STAT 400 Lecture AL1 Examples for 8.3, 8.1 Fall 2014 Dalpiaz 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: cure rate p 0.60 vs. p > 0.60. 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? CDF @ x p n x 0.60 20 0 0.000 1 0.000 2 0.000 3 0.000 4 0.000 5 0.002 6 0.006 7 0.021 8 0.057 9 0.128 10 0.245 11 0.404 12 0.584 13 0.750 14 0.874 15 0.949 16 0.984 17 0.996 18 0.999 19 1.000 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?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 H 0 false Accept H 0 ( Do NOT Reject H 0 ) Type II Error Reject H 0 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 p Null Alternative H 0 : p p 0 vs. H 1 : p < p 0 Left – tailed. H 0 : p p 0 vs. H 1 : p > p 0 Right – tailed. H 0 : p p 0 vs. H 1 : p p 0 Two – tailed. Test Statistic: nppppz )1(000ˆ. )1( 0 00ppnpn Yz, where Y is the number of S’s in n independent trials. Rejection Region: H 0 : p p 0 H 1 : p p 0 Left - tailed. H 0 : p p 0 H 1 : p > p 0 Right - tailed. H 0 : p = p 0 H 1 : p p 0 Two - tailed. Reject H 0 if Z < – z Reject H 0 if Z > z Reject H 0 if 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: H 0 : vs. H 1 : 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 > Do NOT Reject H 0 (Accept H 0). P-value < Reject H 0. Computing P-value: H 0 : p p 0 H 1 : p p 0 Left - tailed. H 0 : p p 0 H 1 : p > p 0 Right - tailed. H 0 : p = p 0 H 1 : p p 0 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: H 0 : vs. H 1 : 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 =
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