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8-1

8-3 Hypothesis Testing 8.1Null and Alternative Hypotheses and Errors in Testing 8.2Large Sample Tests about a Mean: Testing a One-Sided Alternative Hypothesis 8.3Large Sample Tests about a Mean: Testing a Two-Sided Alternative Hypothesis 8.4Small Sample Tests about a Population Mean 8.5Hypothesis Tests about a Population Proportion *8.6Type II Error Probabilities and Sample Size Determination

8-4 8.1 Null and Alternative Hypotheses The null hypothesis, denoted H 0, is a statement of the basic proposition being tested. The statement generally represents the status quo and is not rejected unless there is convincing sample evidence that it is false. The alternative or research hypothesis, denoted H a, is an alternative (to the null hypothesis) statement that will be accepted only if there is convincing sample evidence that it is true.

8-5 Types of Hypothesis One-Sided, Less Than H 0 :   19.5H a :  < 19.5(Accounts Receivable) One-Sided, Greater Than H 0 :   50H a :  > 50(Trash Bag) Two-Sided, Not Equal To H 0 :  = 4.5H a :   4.5(Camshaft)

8-6 Type I and Type II Errors Type I Error Rejecting H 0 when it is true Type II Error Failing to reject H 0 when it is false

8-7 8.2 Large Sample Tests about a Mean: Testing a One-Sided Alternative Hypothesis Test Statistic If the sampled population is normal or if n is large, we can reject H 0 :  =  0 at the  level of significance (probability of Type I error equal to  ) if and only if the appropriate rejection point condition holds. AlternativeReject H 0 if: If  unknown and n is large, estimate  by s. : 0 0 :     a a H H   zz zz  

8-8 Example: One-Sided, Greater Than Testing H 0 :   50 versus Trash Bag H a :  > 50 for  = 0.05 and  = 0.01

8-9 Example: The p-Value for “Greater Than” Testing H 0 :   50 vs H a :  > 50 using rejection points and p- value. Trash Bag The p-value or the observed level of significance is the probability of observing a value of the test statistic greater than or equal to z when H 0 is true. It measures the weight of the evidence against the null hypothesis and is also the smallest value of  for which we can reject H 0.

8-10 Example: One-Sided, Less Than Testing H 0 :   19.5 versus Accts Rec H a :  < 19.5 for  = 0.01

8-11 Large Sample Tests about Mean: p-Values If the sampled population is normal or if n is large, we can reject H 0 :  =  0 at the  level of significance (probability of Type I error equal to  ) if and only if the appropriate rejection point condition holds or, equivalently, if the corresponding p-value is less than . Test Statistic If  unknown and n is large, estimate  by s. Alternative Reject H 0 if:p-Value

8-12 Five Steps of Hypothesis Testing 1)Determine null and alternative hypotheses 2)Specify level of significance (probability of Type I error)  3)Select the test statistic that will be used. Collect the sample data and compute the value of the test statistic. 4)Use the value of the test statistic to make a decision using a rejection point or a p-value. 5)Interpret statistical result in (real-world) managerial terms

8-13 Example: Two-Sided, Not Equal to Testing H 0 :  = 4.5 versus Camshaft H a :   4.5 for  = 0.05

8-14 8.5 Small Sample Tests about a Population Mean If the sampled population is normal, we can reject H 0 :  =  0 at the  level of significance (probability of Type I error equal to  ) if and only if the appropriate rejection point condition holds or, equivalently, if the corresponding p-value is less than . Test Statistic t , t  /2 and p-values are based on n – 1 degrees of freedom. Alternative Reject H 0 if:p-Value

8-15 Example: Small Sample Test about a Mean Testing H 0 :   18.8 vs H a :  < 18.8 using rejection points and p-value. Credit Card Interest Rates

8-16 8.5 Hypothesis Tests about a Population Proportion Test Statistic If the sample size n is large, we can reject H 0 : p = p 0 at the  level of significance (probability of Type I error equal to  ) if and only if the appropriate rejection point condition holds or, equivalently, if the corresponding p-value is less than . Alternative Reject H 0 if:p-Value

8-17 Example: Hypothesis Tests about a Proportion Testing H 0 : p  0.70 versus H a : p > 0.70 using rejection points and p-value. Using Phantol, proportion of patients with reduced severity and duration of viral infections.

8-18 *8.6 Type II Error Probabilities Testing H 0 :   vs H a :  < 3 (Amount of Coffee in 3-Pound Can) , Probability of Type II Error, Given  = 2.995,  = 0.05.

8-19 How Type II Error Varies Against Alternatives Testing H 0 :   vs H a :  < 3 (Amount of Coffee in 3-Pound Can) , Probability of Type II Error (  = 0.05) Given  = 2.995, Given  = 2.990, Given  = 2.985,

8-20 Summary: Selecting an Appropriate Test Statistic for a Test about a Population Mean

8-21 Hypothesis Testing 8.1Null and Alternative Hypotheses and Errors in Testing 8.2Large Sample Tests about a Mean: Testing a One-Sided Alternative Hypothesis 8.3Large Sample Tests about a Mean: Testing a Two-Sided Alternative Hypothesis 8.4Small Sample Tests about a Population Mean 8.5Hypothesis Tests about a Population Proportion *8.6Type II Error Probabilities and Sample Size Determination Summary: