advertisement

ST 361 Ch8.2 Testing Hypotheses about Means (Part I: Testing for a Population Mean ) Topics: Hypothesis testing with population means ► One-sample problem: Testing for a population mean 1. Assume population SD is known: use a z test statistic 2. Assume population SD is not known: use a t test statistic ► A Special Case: the Paired t test ► Two-sample problem: : Testing for 2 population means 1 and 2 -------------------------------------------------------------------------------------------------------------------► Steps for Testing for a Population mean ----- Need x ~Normal !!! Step 1. Specify H 0 and H a H 0 : 0 vs. H a : 0 (this is referred to as _______________________) H a : 0 (this is referred to as _______________________) H a : 0 (this is referred to as _______________________) Step 2. Determine the test level (also called significance level) Step 3. Compute the test statistic A test statistic is defined as_________________________________________________ When the population SD is known, a test statistic is ________________________ When the population SD is NOT known, a test statistic is _____________________ Step 4. Calculate the p-value Step 5. Draw conclusions If p-value < __________________ and draw conclusion based on _______ Otherwise we _________________________and draw conclusion based on ______ 1 ► One-sample problem: Testing for a population mean assume the population SD is known A Working Example: (adapted from 8.14 p.355 of the textbook) Light bulbs of a certain type are advertised as having an average lifetime of 750 hours. The price of these bulbs is very favorable, so a potential customer has decided to go ahead with a purchase arrangement unless the true average lifetime is smaller than what is advertised. A random sample of 50 bulbs was selected. The sample data and result are presented below: (Assume the population SD of the bulbs lifetime is 38.2.) What conclusion would be appropriate for a significance level of 0.05? Variable Lifetime n 50 Z p-value Mean X SE of Mean X 738.44 5.4 -2.14 0.016 Sample Step 1: parameter of interest = H0 : Ha : Step 2: significance level = Step 3: test statistic = Step 4: p-value = Step 5: Conclusion: ► The p-value Rationale: The p-value quantifies how the null hypothesis ( H 0 ) is supported by the observed data, i.e., The smaller the p-value is, the more contradictory is the data to H 0 . To get p-values, one begins with assuming H 0 is true (e.g., in the working example, it means ______________________________________________________). We then calculate that when H o is true, how likely is it to observe a sample with X = 738.44? i.e., want to calculate P( X 738.444 | H 0 is true (i.e., μ 750) ) = ___________ 2 To resolve this problem, one will instead calculate the probability of observing X = 738.44 or the more extreme cases under H 0 . Note that the “more extreme cases” are determined by the _____________hypothesis. Here H a : ___ 750 , so we calculate Such probability is called “p-value” Calculation of p-value (If the population SD is known….) p-value = P( X 738.444 | μ 750 ) ► In general, the calculation of p-value can be simplified in the following steps: Let X * = First calculate the test statistic z * X * 0 X . Then, the p-value = ______________ Interpretation of p-value: (1) If the p-value, P( X 738.444 | μ 750 ), is very small (i.e., _______________), it implies that ________________________________________________________ _______________________________________. In other words, the sample data ______________ support H0. Reject / Do not reject (pick one) H0. (2) If the p-value, P( X 738.444 | μ 750 ), is not very small (i.e., _____________), it implies that the chance of observing sample with Y =738.44 or smaller when = 750 is ___________________ In other words, the sample data and H0 ___________________________________________ Reject / Do not reject (pick one) H0. 3 In summary, 1. The p-value describes the probability of seeing your data or more extreme IF the null hypothesis is true. Recall that the more extreme cases are determined by_______. If H a : 0 , p-value = =________________ If H a : 0 , p-value = =________________ If H a : 0 , p-value = = ________________ 2. To get p-value, all we need is to calculate the test statistic z * X * 0 X X * 0 n (where X * is the observed sample mean) and then find the corresponding p-value by P( Z z* ), or P( Z z* ), or 2 P( Z z * ), depending on the alternative hypothesis. 3. To draw conclusion, compare p-value with the test level : Reject H 0 if p-value___________, and conclude based on H a . Otherwise we do not reject H 0 , and conclude the test based on H 0 . 4 Ex1. Let X = the scores on the Verbal SAT exam this year. The score X varies according to a normal distribution with mean and variance 80. A sample of 64 students was collected, and x 580. Records showed that the mean SAT score of two years ago is 570. Based on the data, has the average SAT score increased over the two years? Perform a 0.05-level of test. Step 1: parameter of interest = H0 : Ha Step 2: significance level = Step 3: test statistic = Step 4: p-value = Step 5: Conclusion: 5 Ex2. Consider the true mean stopping distances at 50 mph for cars equipped with the braking system of brand A. It is known that the average stopping distances for braking system B is 120 inches. Result based on 36 cars have mean 115. Assume the population SD of the stopping distance is 20. Do the stopping distances of the two systems differ? Perform a 0.01-level of test. Step 1: parameter of interest = H0 : Ha Step 2: significance level = Step 3: test statistic = Step 4: p-value = Step 5: Conclusion: 6 ► One-sample problem: Testing for a Population mean when is unknown If the population SD is unknown, the testing procedure is the same as what we do when the population SD known, except that 1. the __________________ is used 2. (as a result of 1,) the test statistic is a ___________________ , which is a _________instead of a z statistic Need to know how to use Table VI to find p-values EX. (From textbook Question 8.17) 1. Upper-tailed test, df=8, t=2.0 2. Lower-tailed test, df=11, t= -2.4 3. Two-tailed test, df=15, t= -1.6 4. Two tailed test, df=40, t=4.8 7 Ex1. Life of electric bulb: Industrial standard for the bulb life is 6000 hours. A company claims that their bulbs are better than the industrial standard. To test their claim, a sample of 16 light bulbs was collected and has mean 6.5 (unit=1000 hours) and SD 1 (unit = 1000 hours). (a) Perform a test at 5% level. (b) What assumption do we need to conduct a hypothesis? (a) Step 1: parameter of interest = H0 : Ha Step 2: significance level = Step 3: test statistic: Step 4: p-value: Step 5: Conclusion: (b) Assumption needed: 8 Ex2. A certain pen has been designed so that the true average writing lifetime is 10 hours. A random sample of 18 pens is selected and the writing lifetime of each is determined: the mean lifetime of the 18 pens is 10.5 hours with SD=1.2 hours. Perform a 0.01 level of test to examine if the design specification has been satisfied. (a) Step 1: parameter of interest = H0 : Ha Step 2: significance level = Step 3: test statistic: Step 4: p-value: Step 5: Conclusion: (b) Assumption needed: (select any that apply) _______ The sample mean lifetime follows a normal distribution _______ The lifetime follows a normal distribution 9