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Chapter 9 Hypothesis Testing Developing Null and Alternative Hypotheses Type I and Type II Errors Population Mean Known Population Mean Unknown Population Proportion 2006 Thomson South Western 1 Developing Null and Alternative Hypotheses Hypothesis testing can be used to determine wheth a statement about the value of a population param 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 alternative hypothesis is what the test is attempting to establish 2006 Thomson South Western 2 Developing Null and Alternative Hypotheses Testing Research Hypotheses The research hypothesis should be expressed as the alternative hypothesis The conclusion that the research hypothesis is tru comes from sample data that contradict the null hypothesis 2006 Thomson South Western 3 Developing Null and Alternative Hypotheses Testing the Validity of a Claim Manufacturers claims are usually given the bene of the doubt and stated as the null hypothesis The conclusion that the claim is false comes from sample data that contradict the null hypothesis 2006 Thomson South Western 4 Developing Null and Alternative Hypotheses Testing in Decision Making Situations A decision maker might have to choose between two courses of action one associated with the nu hypothesis and another associated with the alternative hypothesis Example Accepting a shipment of goods from a supplier or returning the shipment of goods to th supplier 2006 Thomson South Western 5 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 about the value of a population mean must take one of the following three forms where 0 is the hypothesized value of the population mean H 0 0 H a 0 H 0 0 H a 0 H 0 0 H a 0 One tailed One tailed lower tail upper tail Two tailed 2006 Thomson South Western 6 Null and Alternative Hypotheses Example Metro EMS A major west coast city provides one of the most comprehensive emergency medical services in the world Operating in a multiple hospital system with approximately 20 mobile medical units the service goal is to respond to medical emergencies with a mean time of 12 minutes or less 2006 Thomson South Western 7 Null and Alternative Hypotheses Example Metro EMS The director of medical services wants to formulate a hypothesis test that could use a sample of emergency response times to determine whether or not the service goal of 12 minutes or less is being achieved 2006 Thomson South Western 8 Null and Alternative Hypotheses H0 12 The emergency service is meeting the response goal no follow up action is necessary Ha The emergency service is not meeting the response goal appropriate follow up action is necessary where mean response time for the population of medical emergency requests 2006 Thomson South Western 9 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 equality is called the level of significance Applications of hypothesis testing that only control the Type I error are often called significance tests 2006 Thomson South Western 10 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 avoid the risk of making a Type II error by using do not reject H0 and not accept H 2006 Thomson South Western 11 Type I and Type II Errors Population Condition Conclusion H0 True 12 H0 False 12 Accept H0 Conclude 12 Correct Decision Type II Error Type I Error Correct Decision Reject H0 Conclude 12 2006 Thomson South Western 12 p Value Approach to One Tailed Hypothesis Testing The p 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 th rejection region Reject H0 if the p value 2006 Thomson South Western 13 Lower Tailed Test About a Population Mean Known p Value Approach p Value so reject H0 10 Sampling distribution x 0 of z n p value 072 z z z 1 46 1 28 2006 Thomson South Western 0 14 Upper Tailed Test About a Population Mean Known p Value so reject H0 p Value Approach Sampling distribution x 0 of z n 04 p Value 011 z 0 2006 Thomson South Western z 1 75 z 2 29 15 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 area inofthe or upper tail of the The of value thelower test statistic that established the distribution boundary of the rejection region is called the critical value for the test The rejection rule is Lower tail Reject H0 if z z Upper tail Reject H0 if z z 2006 Thomson South Western 16 Lower Tailed Test About a Population Mean Known Critical Value Approach Sampling distribution x 0 z of n Reject H0 Do Not Reject H0 z z 1 28 2006 Thomson South Western 0 17 Upper Tailed Test About a Population Mean Known Critical Value Approach Sampling distribution x 0 of z n Reject H0 Do Not Reject H0 05 z 0 2006 Thomson South Western z 1 645 18 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 Approach 4 Use the value of the test statistic to compute the p value Step 5 Reject H0 if p value 2006 Thomson South Western 19 Steps of Hypothesis Testing Critical Value Approach Step 4 Use the level of significance to 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 2006 Thomson South Western 20 One Tailed Tests About a Population Mean Known Example Metro EMS The response times for a random sample of 40 medical emergencies were tabulated The sample mean is 13 25 minutes The population standard deviation is believed to be 3 2 minutes The EMS director wants to perform a hypothesis test with a 05 level of significance to determine whether the service goal of 12 minutes or less is