Bài giảng Statistical Techniques in Business and Economics - Chapter 10 Hypothesis Testing

Chapter 10Hypothesis Testing1Chapter GoalsDefine null and alternative hypothesis and hypothesis testingDefine Type I and Type II errorsDescribe the five-step hypothesis testing procedureDistinguish between a one-tailed and a two-tailed test of hypothesisWhen you have completed this chapter, you will be able to:and...2Conduct a test of hypothesis about a population meanChapter GoalsConduct a test of hypothesis about a population proportionExplain the relationship between hypothesis testing and confidence interval estimationCompute the probability of a Type II error, and power of a test103TerminologyHypothesisis a statement about a population distribution such that:Examplesthe mean monthly income for all systems analysts is $3569.35% of all customers buying coffee at Tim Horton’s return within a week.(i) it is either true or false, but never both, and(ii) with full knowledge of the population data, it is possible to identify, with certainty, whether it is true or false.4Terminologyis the complement of the alternative hypothesis. We accept the null hypothesis as the default hypothesis. It is not rejected unless there is convincing sample evidence against it.Null Hypothesis HoAlternative Hypothesis H1is the statement that we are interested in proving . It is usually a research hypothesis. Hypothesis Testing Steps5State the decision ruleIdentify the test statisticDo NOT reject H0Reject H0 and accept H1Compute the value of the test statistic and make a decisionStep 1Select the level of significanceStep 2Step 3Step 4Step 5 Hypothesis Testing State the null and alternate hypotheses6Keep in MindWhen a decision is based on analysis of sample data and not the entire population data, it is not possible to make a correct decision all the time.Our objective is to try to keep the probability of making a wrong decision as small as possible!7Two kinds of errorsLet’s look at the Canadian legal system for an analogy...1. the accused person is innocent2. the accused person is guiltyTwo hypotheses:After hearing from both the prosecution and the defence, a decision is made, declaring the accused either:Innocent!orBut do the courts always make the “right” decision?Guilty!8Person is “innocent”Person is “guilty”Person is declared ’not guilty’Person is declared “guilty”Correct DecisionCorrect DecisionErrorErrorH0: person is innocent H1: person is guiltyH0 is trueH1 is trueType II ErrorType I ErrorCourt DecisionRealityTwo kinds of errors9TerminologyLevel of Significanceis the probability of rejecting the null hypothesis when it is actually true, i.e. Type I Erroraccepting the null hypothesis when it is actually false.Type II Error10TerminologyTest Statisticis a value, determined from sample information, used to determine whether or not to reject the null hypothesis.Critical Valueis the dividing point between the region where the null hypothesis is rejected and the region where it is not rejected.11TestsOne-TailVs.Two-Tail120Critical z  = rejection region1-  = acceptance region One-Tail130 = rejection region1-  = acceptance regionTwo-Tailz/2-z/2/2 /2 14A test is one-tailed when the alternate hypothesis, H1, states a direction.H1: The mean yearly commissions earned by full-time realtors is more than $65,000. (µ>$65,000)H1: The mean speed of trucks traveling on the 407 in Ontario is less than 120 kilometres per hour. (µ1.650= 5% rejection region1-  = 95% acceptance region 1.6516A test is two-tailed when no direction is specified in the alternate hypothesis, H1 H1: The mean time Canadian families live in a particular home is not equal to 10 years. (µ10)H1: The average speed of trucks travelling on the 407 in Ontario is different than 120 kph. (µ120)H1: The percentage of repeat customers within a week at Tim Horton’s is not 50%. (p .50)ExamplesTests of SignificanceTwo-Tailed17Two-Tailed5% Level of SignificanceSampling DistributionReject Ho when z>1.96 or z 1.96 or z 400 = 0.05Because the sample is large, we use the test statistic ZReject H0 if z > 1.64542.2=-=nXzsm Reject the hypothesis. H0 . Lisa can conclude that the mean unpaid balance is greater than $400!Conclusion:=17238$400$407$-24Test Statistic to be used:Testing for the Population Mean: Small Sample, Population Standard Deviation UnknownnsXt/m-=25The current production rate for producing 5 amp fuses at Ned’s Electric Co. is 250 per hour. Testing for the Population Mean: Small Sample, Population Standard Deviation UnknownA new machine has been purchased and installed that, according to the supplier, will increase the production rate!A sample of 10 randomly selected hours from last month revealed the mean hourly production on the new machine was 256 units, with a sample standard deviation of 6 per hour.At the .05 significance level, can Ned conclude that the new machine is faster?26 Hypothesis Test State the null and alternate hypothesesStep 1Select the level of significanceStep 2Identify the test statisticStep 3State the decision ruleStep 4Compute the test statistic and make a decisionStep 5H0: µ = 250 H1: µ > 250 = 0.05Because the sample is small and  is unknown, we use the t-testReject H0 if t > 1.833162.3=-=nXtsm Reject the hypothesis. H0 . Ned can conclude that the new machine will increase the production rate!Conclusion:=106250256 - 10 -1 = 9 degrees of freedom27p-value in hypothesis testingA P -Value is the probability, (assuming that the null hypothesis is true) of finding a value of the test statistic at least as extreme as the computed value for the test!If the P-Value is smaller than the significance level, H0 is rejected.If the P-Value is larger than the significance level, H0 is not rejected.28Since P-value is smaller than  of 0.05, reject H0. The population mean is greater than $400.Rock’s Discount Store chain issues its own credit card. Lisa, the credit manager, wants to find out if the mean monthly unpaid balance is more than $400. The level of significance is set at .05. A random check of 172 unpaid balances revealed the sample mean to be $407 and the sample standard deviation to be $38. Should Lisa conclude that the population mean is greater than $400? = 0.05RecallP(z  2.42) =Previously determined .5 - .4922 = .007842.2=-=nsXzm29P-Value = p(z  |computed value|)P-Value = 2p(z  |computed value|)One-TailTwo-Tailp-value in hypothesis testing|....| means absolute value of30The processors of eye drop medication indicate on the label that the bottle contains 16 ml of medication. The standard deviation of the process is 0.5 ml. A sample of 36 bottles from last hour’s production revealed a mean weight of 16.12 ml per bottle. At the .05 significance level is the process out of control? That is, can we conclude that the mean amount per bottle is different from 16 ml? = 0.05Previously determinedP-Value = 2p(z  |computed value|)= 2p(z  |1.44|)= 2(.5 - .4251)= 2(.0749)= .1498Since .1498 > .05, do not reject H0.44.1=-=nXzsmRecall31Interpreting the Weight of Evidence against HoIf the P-value is less than .10 we have some evidence that Ho is not true.05 we have strong evidence that Ho is not true.01 we have very strong evidence that Ho is not true.001 we have extremely strong evidence that Ho is not true32If the P-value is less than.10 we have some evidence.05 we have strong evidence.01 we have very strong evidence.001 we have extremely strong evidence that Ho is not trueSince P-value is .0078 we have very strong evidenceto conclude that the population mean is greater than $400!33Tests concerning Proportions is the fraction or percentage that indicates the part of the population or sample having a particular trait of interestA Proportion is denoted by p is found by: Sample ProportionsampledNumber sample in the successes ofNumber =p34Testing a Single Population Proportion: Test Statistic to be used:where is the symbol for sample proportion is the symbol for population proportionpp^p0 represents a population proportion of interestnppppz)1(ˆ000--=35In the past, 15% of the mail order solicitations for a certain charity resulted in a financial contribution. At the .05 significance level can it be concluded that the new letter is more effective?A new solicitation letter that has been drafted is sent to a sample of 200 people and 45 responded with a contribution.36 Hypothesis Test State the null and alternate hypothesesStep 1Select the level of significanceStep 2Identify the test statisticStep 3State the decision ruleStep 4Compute the test statistic and make a decisionStep 5 = 0.05We will use the z-test Reject the hypothesis. More than 15% are responding with a pledge, therefore, the new letter is more effective!Conclusion:H1: p > .15H0: p = .15 Reject H0 if z > 1.645ppzˆnpp)1(ˆ--=200)15.1(.15-2004515.-=97.2=37Relationship Between Hypothesis Testing Procedure and Confidence Interval EstimationCase 1:Our decision rule can be restated as:Do not reject H0 if 0 lies in the (1-) confidence interval estimate of the population mean, computed from the sample dataTwo-TailTEST380 = rejection region1-  = Confidence Interval regionTwo-TailDo not reject Ho when z falls in the confidence interval estimate39Relationship Between Hypothesis Testing Procedure and Confidence Interval EstimationCase 2:Lower-tailed testOur decision rule can be restated as: Do not reject H0 if 0 is less than or equal to the (1-) upper confidence bound for , computed from the sample data.40Lower-Tailed0 = rejection region1-  = confidence level regionDo not rejectRelationship Between Hypothesis Testing Procedure and Confidence Interval Estimation41Relationship Between Hypothesis Testing Procedure and Confidence Interval EstimationCase 3:Upper-tailed testOur decision rule can be restated as: Do not reject H0 if 0 is greater than or equal to the (1-) lower confidence bound for , computed from the sample data.42Upper-Tailed0 = rejection region1- = acceptance region4310 - 44Level of Significanceis the probability of rejecting the null hypothesis when it is actually true, i.e. Type I Erroraccepting the null hypothesis when it is actually false.Type II ErrorType II Error44Calculating the Probability of a Type II Error10 - 45 A batch of 5000 light bulbs either belong to a superior type, with a mean life of 2400 hours, or to an inferior type, with a mean life of 2000 hours. (By default, the bulbs will be sold as the inferior type.) SolveSuppose we select a sample of 4 bulbs. Find the probability of a Type II error.Both bulb distributions are normal, with a standard deviation of 300 hours.  = 0.025. 45State the null and alternate hypothesesStep 1Select the level of significanceStep 2Identify the test statisticStep 3State the decision ruleStep 4H0: µ = 2000 H1: µ = 2400 = 0.025As populations are normal,  is known, we use the z-test Reject H0 if the computed z > 1.96, or stated another way,If the computed value x bar is greater than xu = 2000 +1.96(300/n), REJECT H0 in favour of H1Superior: =2400 Inferior: =2000=300 =0.025 46Suppose H0 is false and H1 is true. i.e. the true value of µ is 2400, then x bar is approximately normally distributed with a mean of 2400 and a standard deviation of /n = 300/nis the probability of not rejecting Hois the probability that the value of x bar obtained will be less than or equal to xuThe probability of a Type II ErrorXuX£47Suppose we select a sample of 4 bulbs. Then x bar has a mean of 2400 and a sd of 300/4 = 150Xu = 2000+1.96(300/4) = 2294 A1 = 0.2611, giving us a left tail area of 0.2470666.0-430024002294=-=-=nXzsm48The probability of a Type II error is 0.24 i.e.=0.2449 If we decrease the value of (alpha), the value z increases and the critical value xu moves to the right, and therefore the value of (beta) increases. Conversely, if we increase the value of (alpha), xu moves to the left, thereby decreasing the value of (beta) For a given value of (alpha), the value of (beta) can be decreased by increasing the sample size.50Power of a Test is defined as the probability of rejecting H0 when H0 is false, or the probability of correctly identifying a true alternative hypothesisit is equal to (1-)In previous example,  = 0.24Therefore, the test’s power is 1-0.24 = 0.7651Test your learning www.mcgrawhill.ca/college/lindClick onOnline Learning Centrefor quizzesextra contentdata setssearchable glossaryaccess to Statistics Canada’s E-Stat dataand much more!52This completes Chapter 1053