Presentation on theme: "Chapter 9 Hypothesis Testing"— Presentation transcript:
1 Chapter 9 Hypothesis Testing Developing Null and Alternative HypothesesType I and Type II ErrorsPopulation Mean: s Known
2 Developing Null and Alternative Hypotheses Hypothesis testing can be used to determine whethera statement about the value of a population parametershould or should not be rejected.The null hypothesis, denoted by H0 , is a tentativeassumption about a population parameter.The alternative hypothesis, denoted by Ha, is theopposite of what is stated in the null hypothesis.The alternative hypothesis is what the test isattempting to establish.
3 Developing Null and Alternative Hypotheses Testing Research HypothesesThe research hypothesis should be expressed asthe alternative hypothesis.The conclusion that the research hypothesis is truecomes from sample data that contradict the nullhypothesis.
4 Developing Null and Alternative Hypotheses Testing the Validity of a ClaimManufacturers’ claims are usually given the benefitof the doubt and stated as the null hypothesis.The conclusion that the claim is false comes fromsample data that contradict the null hypothesis.
5 Developing Null and Alternative Hypotheses Testing in Decision-Making SituationsA decision maker might have to choose betweentwo courses of action, one associated with the nullhypothesis and another associated with thealternative hypothesis.Example: Accepting a shipment of goods from asupplier or returning the shipment of goods to thesupplier
6 Summary of Forms for Null and Alternative Hypotheses about a Population Mean The equality part of the hypotheses always appearsin the null hypothesis.In general, a hypothesis test about the value of apopulation mean must take one of the followingthree forms (where 0 is the hypothesized value ofthe population mean).One-tailed(lower-tail)One-tailed(upper-tail)Two-tailed
7 Null and Alternative Hypotheses Example: Metro EMSA major west coast city providesone of the most comprehensiveemergency medical services inthe world.Operating in a multiplehospital system withapproximately 20 mobile medicalunits, the service goal is to respond to medicalemergencies with a mean time of 12 minutes or less.
8 Null and Alternative Hypotheses Example: Metro EMSThe director of medical serviceswants to formulate a hypothesistest that could use a sample ofemergency response times todetermine whether or not theservice goal of 12 minutes or lessis being achieved.
9 Null and Alternative Hypotheses The emergency service is meetingthe response goal; no follow-upaction is necessary.H0: The emergency service is notmeeting the response goal;appropriate follow-up action isnecessary.Ha:where: = mean response time for the populationof medical emergency requests
10 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 thenull hypothesis is true as an equality is called thelevel of significance.Applications of hypothesis testing that only controlthe Type I error are often called significance tests.
11 Type II Error A Type II error is accepting H0 when it is false. It is difficult to control for the probability of makinga Type II error.Statisticians avoid the risk of making a Type IIerror by using “do not reject H0” and not “accept H0”.
12 Type I and Type II Errors Population ConditionH0 True(m < 12)H0 False(m > 12)ConclusionAccept H0(Conclude m < 12)CorrectDecisionType II ErrorReject H0(Conclude m > 12)Type I ErrorCorrectDecision
13 One-Tailed Hypothesis Testing p-Value Approach toOne-Tailed Hypothesis TestingThe p-value is the probability, computed using thetest statistic, that measures the support (or lack ofsupport) provided by the sample for the nullhypothesis.If the p-value is less than or equal to the level ofsignificance , the value of the test statistic is in therejection region.Reject H0 if the p-value < .
14 Lower-Tailed Test About a Population Mean: s Knownp-Value Approachp-Value < a ,so reject H0.a = .10Samplingdistributionofp-value72zz =-1.46-za =-1.28
15 Upper-Tailed Test About a Population Mean: s Knownp-Value Approachp-Value < a ,so reject H0.Samplingdistributionofa = .04p-Value11zza =1.75z =2.29
16 Critical Value Approach to One-Tailed Hypothesis Testing The test statistic z has a standard normal probabilitydistribution.We can use the standard normal probabilitydistribution table to find the z-value with an areaof a in the lower (or upper) tail of the distribution.The value of the test statistic that established theboundary of the rejection region is called thecritical value for the test.The rejection rule is:Lower tail: Reject H0 if z < -zUpper tail: Reject H0 if z > z
17 Lower-Tailed Test About a Population Mean: s KnownCritical Value ApproachSamplingdistributionofReject H0a 1Do Not Reject H0z-za = -1.28
18 Upper-Tailed Test About a Population Mean: s KnownCritical Value ApproachSamplingdistributionofReject H0Do Not Reject H0zza = 1.645
19 Steps of Hypothesis 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 ApproachStep 4. Use the value of the test statistic to compute thep-value.Step 5. Reject H0 if p-value < a.
20 Steps of Hypothesis Testing Critical Value ApproachStep 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 rejectionrule to determine whether to reject H0.
21 One-Tailed Tests About a Population Mean: s KnownExample: Metro EMSThe response times for a randomsample of 40 medical emergencieswere tabulated. The sample meanis minutes. The populationstandard deviation is believed tobe 3.2 minutes.The EMS director wants toperform a hypothesis test, with a.05 level of significance, to determinewhether the service goal of 12 minutes or less is beingachieved.
22 One-Tailed Tests About a Population Mean: s Knownp -Value and Critical Value Approaches1. Develop the hypotheses.H0: Ha:2. Specify the level of significance.a = .053. Compute the value of the test statistic.
23 One-Tailed Tests About a Population Mean: s Knownp –Value Approach4. Compute the p –value.For z = 2.47, probability =p–value = =5. Determine whether to reject H0.Because p–value = < a = .05, we reject H0.We are at least 95% confident that Metro EMS is not meeting the response goal of 12 minutes.
24 One-Tailed Tests About a Population Mean: s Knownp –Value ApproachSamplingdistributionofa = .05p-valuezza =1.645z =2.47
25 One-Tailed Tests About a Population Mean: s KnownCritical Value Approach4. Determine the critical value and rejection rule.For a = .05, z.05 = 1.645Reject H0 if z > 1.6455. Determine whether to reject H0.Because 2.47 > 1.645, we reject H0.We are at least 95% confident that Metro EMS is not meeting the response goal of 12 minutes.
26 Two-Tailed Hypothesis Testing p-Value Approach toTwo-Tailed Hypothesis TestingCompute the p-value using the following three steps:1. Compute the value of the test statistic z.2. If z is in the upper tail (z > 0), find the area underthe standard normal curve to the right of z.If z is in the lower tail (z < 0), find the area underthe standard normal curve to the left of z.3. Double the tail area obtained in step 2 to obtainthe p –value.The rejection rule:Reject H0 if the p-value < .
27 Critical Value Approach to Two-Tailed Hypothesis Testing The critical values will occur in both the lower andupper tails of the standard normal curve.Use the standard normal probability distributiontable to find z/2 (the z-value with an area of a/2 inthe upper tail of the distribution).The rejection rule is:Reject H0 if z < -z/2 or z > z/2.
28 Example: Glow Toothpaste Two-Tailed Test About a Population Mean: s Knownoz.GlowThe production line for Glow toothpasteis designed to fill tubes with a mean weightof 6 oz. Periodically, a sample of 30 tubeswill be selected in order to check thefilling process.Quality assurance procedures call forthe continuation of the filling process if thesample results are consistent with the assumption thatthe mean filling weight for the population of toothpastetubes is 6 oz.; otherwise the process will be adjusted.
29 Example: Glow Toothpaste Two-Tailed Test About a Population Mean: s Knownoz.GlowAssume that a sample of 30 toothpastetubes provides a sample mean of 6.1 oz.The population standard deviation isbelieved to be 0.2 oz.Perform a hypothesis test, at the .03level of significance, to help determinewhether the filling process should continueoperating or be stopped and corrected.
30 Two-Tailed Tests About a Population Mean: s KnownGlowp –Value and Critical Value Approaches1. Determine the hypotheses.H0: = 6Ha:2. Specify the level of significance.a = .033. Compute the value of the test statistic.
31 Two-Tailed Tests About a Population Mean: s Known Glowp –Value Approach4. Compute the p –value.For z = 2.74, cumulative probability = .4969p–value = 2( ) =5. Determine whether to reject H0.Because p–value = < a = .03, we reject H0.We are at least 97% confident that the mean filling weight of the toothpaste tubes is not 6 oz.
32 Two-Tailed Tests About a Population Mean: s KnownGlowp-Value Approach1/2p -value= .00311/2p -value= .0031a/2 =.015a/2 =.015zz = -2.74z = 2.74-za/2 = -2.17za/2 = 2.17
33 Two-Tailed Tests About a Population Mean: s Known GlowCritical Value Approach4. Determine the critical value and rejection rule.For a/2 = .03/2 = .015, z.015 = 2.17Reject H0 if z < or z > 2.175. Determine whether to reject H0.Because 2.47 > 2.17, we reject H0.We are at least 97% confident that the mean filling weight of the toothpaste tubes is not 6 oz.
34 Two-Tailed Tests About a Population Mean: s KnownGlowCritical Value ApproachSamplingdistributionofReject H0Do Not Reject H0Reject H0a/2 = .015a/2 = .015z-2.172.17
35 Confidence Interval Approach to Two-Tailed Tests About a Population Mean Select a simple random sample from the populationand use the value of the sample mean to developthe confidence interval for the population mean .(Confidence intervals are covered in Chapter 8.)If the confidence interval contains the hypothesizedvalue 0, do not reject H0. Otherwise, reject H0.
36 Confidence Interval Approach to Two-Tailed Tests About a Population Mean GlowThe 97% confidence interval for isor toBecause 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.