英文统计学1.pptx

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1、,Chapter 9,Part A Hypothesis Testing,Developing Null and Alternative Hypotheses,Type I and Type II Errors,Population Mean:s Known,Population Mean:s Unknown,Population Proportion,Hypothesis Testing,Hypothesis testing can be used to determine whether a statement about the value of a population paramet

2、er should or should not be rejected.,The null hypothesis,denoted by H0,is a tentative assumption about a population parameter.,The alternative hypothesis,denoted by Ha,is the opposite of what is stated in the null hypothesis.,The hypothesis testing procedure uses data from a sample to test the two c

3、ompeting statements indicated by H0 and Ha.,Developing Null and Alternative Hypotheses,It is not always obvious how the null and alternative hypotheses should be formulated.,Care must be taken to structure the hypotheses appropriately so that the test conclusion provides the information the research

4、er wants.,The context of the situation is very important in determining how the hypotheses should be stated.,In some cases it is easier to identify the alternative hypothesis first.In other cases the null is easier.,Correct hypothesis formulation will take practice.,Alternative Hypothesis as a Resea

5、rch Hypothesis,Developing Null and Alternative Hypotheses,Many applications of hypothesis testing involve an attempt to gather evidence in support of a research hypothesis.,In such cases,it is often best to begin with the alternative hypothesis and make it the conclusion that the researcher hopes to

6、 support.,The conclusion that the research hypothesis is true is made if the sample data provide sufficient evidence to show that the null hypothesis can be rejected.,Alternative Hypothesis as a Research Hypothesis,Developing Null and Alternative Hypotheses,Example:A new teaching method is developed

7、 that is believed to be better than the current method.,Alternative Hypothesis:The new teaching method is better.,Null Hypothesis:The new method is no better than the old method.,Alternative Hypothesis as a Research Hypothesis,Developing Null and Alternative Hypotheses,Example:A new sales force bonu

8、s plan is developed in an attempt to increase sales.,Alternative Hypothesis:The new bonus plan increase sales.,Null Hypothesis:The new bonus plan does not increase sales.,Alternative Hypothesis as a Research Hypothesis,Developing Null and Alternative Hypotheses,Example:A new drug is developed with t

9、he goal of lowering blood pressure more than the existing drug.,Alternative Hypothesis:The new drug lowers blood pressure more than the existing drug.,Null Hypothesis:The new drug does not lower blood pressure more than the existing drug.,Developing Null and Alternative Hypotheses,Null Hypothesis as

10、 an Assumption to be Challenged,We might begin with a belief or assumption that a statement about the value of a population parameter is true.,We then using a hypothesis test to challenge the assumption and determine if there is statistical evidence to conclude that the assumption is incorrect.,In t

11、hese situations,it is helpful to develop the null hypothesis first.,Developing Null and Alternative Hypotheses,Example:The label on a soft drink bottle states that it contains 67.6 fluid ounces.,Null Hypothesis:The label is correct.m 67.6 ounces.,Alternative Hypothesis:The label is incorrect.m 67.6

12、ounces.,Null Hypothesis as an Assumption to be Challenged,One-tailed(lower-tail),One-tailed(upper-tail),Two-tailed,Summary of Forms for Null and Alternative Hypotheses about a Population Mean,The equality part of the hypotheses always appears in the null hypothesis.,In general,a hypothesis test abou

13、t the value of a population mean must take one of the following three forms(where 0 is the hypothesized value of the population mean).,Example:Metro EMS,Null and Alternative Hypotheses,A major west coast city provides one of the mostcomprehensive emergency medical services in theworld.Operating in a

14、 multiple hospital systemwith approximately 20 mobile medical units,theservice goal is to respond to medical emergencieswith a mean time of 12 minutes or less.,The director of medical services wants toformulate a hypothesis test that could use a sampleof emergency response times to determine whether

15、or not the service goal of 12 minutes or less is beingachieved.,Null and Alternative Hypotheses,The emergency service is meetingthe response goal;no follow-upaction is necessary.,The emergency service is notmeeting the response goal;appropriate follow-up action isnecessary.,H0:,Ha:,where:=mean respo

16、nse time for the population of medical emergency requests,Type I Error,Because hypothesis tests are based on sample data,we must allow for the possibility of errors.,A Type I error is rejecting H0 when it is true.,The probability of making a Type I error when the null hypothesis is true as an equali

17、ty is called the level of significance.,Applications of hypothesis testing that only control the Type I error are often called significance tests.,Type II Error,A Type II error is accepting H0 when it is false.,It is difficult to control for the probability of making a Type II error.,Statisticians a

18、void the risk of making a Type II error by using“do not reject H0”and not“accept H0”.,Type I and Type II Errors,CorrectDecision,Type II Error,CorrectDecision,Type I Error,Reject H0(Conclude m 12),Accept H0(Conclude m 12),H0 True(m 12),H0 False(m 12),Conclusion,Population Condition,Example:Metro EMS,

19、The EMS director wants to perform a hypothesistest,with a.05 level of significance,to determinewhether the service goal of 12 minutes or less is being achieved.,The response times for a random sample of 40medical emergencies were tabulated.The samplemean is 13.25 minutes.The population standarddevia

20、tion is believed to be 3.2 minutes.,One-Tailed Tests About a Population Mean:s Known,Critical Value Approach to One-Tailed Hypothesis Testing,The test statistic z has a standard normal probability distribution.,We can use the standard normal probability distribution table to find the z-value with an

21、 area of a in the lower(or upper)tail of the distribution.,The value of the test statistic that established the boundary of the rejection region is called the critical value for the test.,The rejection rule is:Lower tail:Reject H0 if z z,a 1,0,-za=-1.28,Reject H0,Do Not Reject H0,z,Samplingdistribut

22、ion of,Lower-Tailed Test About a Population Mean:s Known,Critical Value Approach,0,za=1.645,Reject H0,Do Not Reject H0,z,Samplingdistribution of,Upper-Tailed Test About a Population Mean:s Known,Critical Value Approach,p-Value Approach toOne-Tailed Hypothesis Testing,Reject H0 if the p-value.,The p-

23、value is the probability,computed using the test statistic,that measures the support(or lack of support)provided by the sample for the null hypothesis.,If the p-value is less than or equal to the level of significance,the value of the test statistic is in the rejection region.,p-Value Approach,p-val

24、ue 72,0,-za=-1.28,a=.10,z,z=-1.46,Lower-Tailed Test About a Population Mean:s Known,Samplingdistribution of,p-Value a,so reject H0.,p-Value Approach,p-Value 11,0,za=1.75,a=.04,z,z=2.29,Upper-Tailed Test About a Population Mean:s Known,Samplingdistribution of,p-Value a,so reject H0.,Steps of Hypothes

25、is Testing,Step 1.Develop the null and alternative hypotheses.,Step 2.Specify the level of significance.,Step 3.Collect the sample data and compute the test statistic.,p-Value Approach,Step 4.Use the value of the test statistic to compute the p-value.,Step 5.Reject H0 if p-value a.,Critical Value Ap

26、proach,Step 4.Use the level of significanceto determine the critical value and the rejection rule.,Step 5.Use the value of the test statistic and the rejection rule to determine whether to reject H0.,Steps of Hypothesis Testing,12,x,13.25,One-Tailed Tests About a Population Mean:s Known,1.Develop th

27、e hypotheses.,2.Specify the level of significance.,a=.05,H0:Ha:,p-Value and Critical Value Approaches,One-Tailed Tests About a Population Mean:s Known,3.Compute the value of the test statistic.,5.Determine whether to reject H0.,p Value Approach,One-Tailed Tests About a Population Mean:s Known,4.Comp

28、ute the p value.,For z=2.47,cumulative probability=.9932.pvalue=1-.9932=.0068,Because pvalue=.0068 a=.05,we reject H0.,There is sufficient statistical evidenceto infer that Metro EMS is not meetingthe response goal of 12 minutes.,p Value Approach,p-value,0,za=1.645,a=.05,z,z=2.47,One-Tailed Tests Ab

29、out a Population Mean:s Known,Samplingdistribution of,5.Determine whether to reject H0.,There is sufficient statistical evidenceto infer that Metro EMS is not meetingthe response goal of 12 minutes.,Because 2.47 1.645,we reject H0.,Critical Value Approach,One-Tailed Tests About a Population Mean:s K

30、nown,For a=.05,z.05=1.645,4.Determine the critical value and rejection rule.,Reject H0 if z 1.645,p-Value Approach toTwo-Tailed Hypothesis Testing,The rejection rule:Reject H0 if the p-value.,Compute the p-value using the following three steps:,3.Double the tail area obtained in step 2 to obtain the

31、 p value.,2.If z is in the upper tail(z 0),find the area under the standard normal curve to the right of z.If z is in the lower tail(z 0),find the area under the standard normal curve to the left of z.,1.Compute the value of the test statistic z.,Critical Value Approach to Two-Tailed Hypothesis Test

32、ing,The critical values will occur in both the lower and upper tails of the standard normal curve.,The rejection rule is:Reject H0 if z z/2.,Use the standard normal probability distribution table to find z/2(the z-value with an area of a/2 in the upper tail of the distribution).,Example:Glow Toothpa

33、ste,Quality assurance procedures call for thecontinuation of the filling process if the sampleresults are consistent with the assumption that themean filling weight for the population of toothpastetubes is 6 oz.;otherwise the process will be adjusted.,The production line for Glow toothpaste isdesign

34、ed to fill tubes with a mean weight of 6 oz.Periodically,a sample of 30 tubes will be selected inorder to check the filling process.,Two-Tailed Tests About a Population Mean:s Known,Perform a hypothesis test,at the.05 level ofsignificance,to help determine whether the fillingprocess should continue

35、operating or be stopped andcorrected.,Assume that a sample of 30 toothpaste tubesprovides a sample mean of 6.1 oz.The populationstandard deviation is believed to be 0.2 oz.,Two-Tailed Tests About a Population Mean:s Known,Example:Glow Toothpaste,1.Determine the hypotheses.,2.Specify the level of sig

36、nificance.,3.Compute the value of the test statistic.,a=.05,p Value and Critical Value Approaches,H0:=6Ha:,Two-Tailed Tests About a Population Mean:s Known,Two-Tailed Tests About a Population Mean:s Known,5.Determine whether to reject H0.,p Value Approach,4.Compute the p value.,For z=2.74,cumulative

37、 probability=.9969pvalue=2(1-.9969)=.0062,Because pvalue=.0062 a=.05,we reject H0.,There is sufficient statistical evidence toinfer that the alternative hypothesis is true(i.e.the mean filling weight is not 6 ounces).,Two-Tailed Tests About a Population Mean:s Known,a/2=.025,0,za/2=1.96,z,a/2=.025,p

38、-Value Approach,-za/2=-1.96,z=2.74,z=-2.74,1/2p-value=.0031,1/2p-value=.0031,Critical Value Approach,Two-Tailed Tests About a Population Mean:s Known,5.Determine whether to reject H0.,There is sufficient statistical evidence toinfer that the alternative hypothesis is true(i.e.the mean filling weight

39、 is not 6 ounces).,Because 2.74 1.96,we reject H0.,For a/2=.05/2=.025,z.025=1.96,4.Determine the critical value and rejection rule.,Reject H0 if z 1.96,a/2=.025,0,1.96,Reject H0,Do Not Reject H0,z,Reject H0,-1.96,Critical Value Approach,Samplingdistribution of,Two-Tailed Tests About a Population Mea

40、n:s Known,a/2=.025,Confidence Interval Approach toTwo-Tailed Tests About a Population Mean,Select a simple random sample from the population and use the value of the sample mean to develop the confidence interval for the population mean.(Confidence intervals are covered in Chapter 8.),If the confide

41、nce interval contains the hypothesized value 0,do not reject H0.Otherwise,reject H0.(Actually,H0 should be rejected if 0 happens to be equal to one of the end points of the confidence interval.),The 95%confidence interval for is,Confidence Interval Approach toTwo-Tailed Tests About a Population Mean

42、,Because the hypothesized value for thepopulation mean,0=6,is not in this interval,the hypothesis-testing conclusion is that thenull hypothesis,H0:=6,can be rejected.,or 6.0284 to 6.1716,Test Statistic,Tests About a Population Mean:s Unknown,This test statistic has a t distribution with n-1 degrees

43、of freedom.,Rejection Rule:p-Value Approach,H0:,Reject H0 if t t,Reject H0 if t-t,Reject H0 if t t,H0:,H0:,Tests About a Population Mean:s Unknown,Rejection Rule:Critical Value Approach,Reject H0 if p value a,p-Values and the t Distribution,The format of the t distribution table provided in most sta

44、tistics textbooks does not have sufficient detail to determine the exact p-value for a hypothesis test.,However,we can still use the t distribution table to identify a range for the p-value.,An advantage of computer software packages is that the computer output will provide the p-value for the t dis

45、tribution.,A State Highway Patrol periodically samplesvehicle speeds at various locations on a particularroadway.The sample of vehicle speeds is used totest the hypothesis H0:m 65.,Example:Highway Patrol,One-Tailed Test About a Population Mean:s Unknown,The locations where H0 is rejected are deemed

46、thebest locations for radar traps.At Location F,asample of 64 vehicles shows a mean speed of 66.2mph with a standard deviation of 4.2 mph.Use a=.05 to test the hypothesis.,One-Tailed Test About a Population Mean:s Unknown,1.Determine the hypotheses.,2.Specify the level of significance.,3.Compute the

47、 value of the test statistic.,a=.05,p Value and Critical Value Approaches,H0:65,One-Tailed Test About a Population Mean:s Unknown,p Value Approach,5.Determine whether to reject H0.,4.Compute the p value.,For t=2.286,the pvalue must be less than.025(for t=1.998)and greater than.01(for t=2.387).01 pva

48、lue.025,Because pvalue a=.05,we reject H0.,We are at least 95%confident that the mean speed of vehicles at Location F is greater than 65 mph.,Critical Value Approach,5.Determine whether to reject H0.,We are at least 95%confident that the mean speed of vehicles at Location F is greater than 65 mph.Lo

49、cation F is a good candidate for a radar trap.,Because 2.286 1.669,we reject H0.,One-Tailed Test About a Population Mean:s Unknown,For a=.05 and d.f.=64 1=63,t.05=1.669,4.Determine the critical value and rejection rule.,Reject H0 if t 1.669,0,ta=1.669,Reject H0,Do Not Reject H0,t,One-Tailed Test About a Population Mean:s Unknown,End of Chapter 9,Part A,