Ch10-Hypothesis-Testing--One-Population-Mean-or-Pr.ppt

Click to edit Master title style,Click to edit Master text styles,Second Level,Third Level,Fourth Level,Fifth Level,CHAPTER 10:Hypothesis Testing,One Population Mean or Proportion,to accompany,Introduction to Business Statistics,fourth edition,by Ronald M.Weiers,Presentation by Priscilla Chaffe-Stengel,Donald N.Stengel,2002 The Wadsworth Group,Chapter 10-Learning Objectives,Describe the logic of and transform verbal statements into null and alternative hypotheses.,Describe what is meant by Type I and Type II errors.,Conduct a hypothesis test for a single population mean or proportion.,Determine and explain the,p,-value of a test statistic.,Explain the relationship between confidence intervals and hypothesis tests.,2002 The Wadsworth Group,Null and Alternative Hypotheses,Null Hypotheses,H,0,:Put here what is,typical,of the population,a term that characterizes“business as usual”where nothing out of the ordinary occurs.,Alternative Hypotheses,H,1,:Put here what is the,challenge,the view of some characteristic of the population that,if it were true,would trigger some new action,some change in procedures that had previously defined“business as usual.”,2002 The Wadsworth Group,Beginning an Example,When a robot welder is in adjustment,its mean time to perform its task is 1.3250 minutes.Past experience has found the standard deviation of the cycle time to be 0.0396 minutes.An incorrect mean operating time can disrupt the efficiency of other activities along the production line.For a recent random sample of 80 jobs,the mean cycle time for the welder was 1.3229 minutes.Does the machine appear to be in need of adjustment?,2002 The Wadsworth Group,Types of Error,No error,Type II error:,b,Type Ierror:,a,No error,State of Reality,H,0,True,H,0,False,H,0,True,H,0,False,Test,Says,2002 The Wadsworth Group,Types of Error,Type I Error:,Saying you reject H,0,when it really is true.,Rejecting a true H,0,.,Type II Error:,Saying you do not reject H,0,when it really is false.,Failing to reject a false H,0,.,2002 The Wadsworth Group,Acceptable Error for the Example,Decision makers frequently use a 5%significance level.,Use,a,=0.05.,An,a,-error means that we will decide to adjust the machine when it does not need adjustment.,This means,in the case of the robot welder,if the machine is running properly,there is only a 0.05 probability of our making the mistake of concluding that the robot requires adjustment when it really does not.,2002 The Wadsworth Group,Nondirectional,Two-Tail Tests,H,0,:pop parameter=value,H,1,:pop parameter,value,2002 The Wadsworth Group,Directional,Right-Tail Tests,H,0,:pop parameter,value,H,1,:pop parameter value,2002 The Wadsworth Group,The Logic of Hypothesis Testing,Step 1.,A claim is made.,A new claim is asserted,that challenges existing thoughts about a population characteristic.,Suggestion:,Form the alternative hypothesis first,since it embodies the challenge.,2002 The Wadsworth Group,The Logic of Hypothesis Testing,Step 3.,If the null hypothesis were true,what would you expect to see?,Assume the null hypothesis is true.,This is a very powerful statement.The test is always referenced to the null hypothesis.,Form the rejection region,the areas in which the decision maker is willing to reject the presumption of the null hypothesis.,2002 The Wadsworth Group,The Logic of Hypothesis Testing,Step 4.,What did you actually see?,Compute the sample statistic.,The sample provides a set of data that serves as a window to the population.The decision maker computes the sample statistic and calculates how far the sample statistic differs from the presumed distribution that is established by the null hypothesis.,2002 The Wadsworth Group,The Logic of Hypothesis Testing,Step 6.,What are the implications of the decision for future actions?,State what the decision means in terms of the business situation.,The decision maker must draw out the implications of the decision.Is there some action triggered,some change implied?What recommendations might be extended for future attempts to test similar hypotheses?,2002 The Wadsworth Group,Hypotheses for the Example,The hypotheses are:,H,0,:,=1.3250 minutes,The robot welder is in adjustment.,H,1,:,1.3250 minutes,The robot welder is not in adjustment.,This is a,nondirectional,two-tail test.,2002 The Wadsworth Group,Identifying the Appropriate Test Statistic,Ask the following questions:,Are the data the result of a measurement(a continuous variable)or a count(a discrete variable)?,Is,s,known?,What shape is the distribution of the population parameter?,What is the sample size?,2002 The Wadsworth Group,Continuous Variables,Continuous data are the result of a,measurement process,.Each element of the data set is a measurement representing one sampled individual element.,Test of a mean,Example:,When a robot welder is in adjustment,its mean time to perform its task is 1.3250 minutes.For a recent sample of 80 jobs,the mean cycle time for the welder was 1.3229 minutes.,Note that time to complete each of the 80 jobs was measured.The sample average was computed.,2002 The Wadsworth Group,Test of,s,Known,Population Normally Distributed,Test Statistic:,where,is the sample statistic.,0,is the value identified in the null hypothesis.,s,is known.,n,is the sample size.,n,x,z,s,0,m,=,2002 The Wadsworth Group,Test of,s,Unknown,Population Normally Distributed,Test Statistic:,where,is the sample statistic.,0,is the value identified in the null hypothesis.,s,is unknown.,n,is the sample size,degrees of freedom on,t,are,n,1.,x,x,m,n,s,t,0,=,2002 The Wadsworth Group,Test of,s,Unknown,Population Shape Not Known/Not Normal,If,n,30,Test Statistic:,If,n,30,use a distribution-free test(see Chapter 14).,2002 The Wadsworth Group,The Formal Hypothesis Test for the Example,s,Known,I.Hypotheses,H,0,:,=1.3250 minutes,H,1,:,1.3250 minutes,II.Rejection Region,a,=0.05,Decision Rule:,If,z,1.96,reject H,0,.,2002 The Wadsworth Group,The Formal Hypothesis Test,cont.,V.Implications,This is not sufficient evidence to conclude that the robot welder is out of adjustment.,2002 The Wadsworth Group,Discrete Variables,Discrete data are the result of a,counting process,.The sampled elements are sorted,and the elements with the characteristic of interest are counted.,Test of a proportion,p,Example:,The career services director of Hobart University has said that 70%of the schools seniors enter the job market in a position directly related to their undergraduate field of study.In a sample of 200 of last years graduates,132 or 66%have entered jobs related to their field of study.,2002 The Wadsworth Group,Test of,p,Sample Not Sufficiently Large,If either,n,p,5 or,n,(1,p,)5,convert the proportion to the underlying binomial distribution.,Note there is no,t,-test on a population proportion.,2002 The Wadsworth Group,Observed Significance Level,A,p,-value is:,the exact level of significance of the test statistic.,the smallest value,a,can be and still allow us to reject the null hypothesis.,the amount of area left in the tail beyond the test statistic for a one-tailed hypothesis test or,twice the amount of area left in the tail beyond the test statistic for a two-tailed test.,the probability of getting a test statistic from another sample that is at least as far from the hypothesized mean as this sample statistic is.,2002 The Wadsworth Group,