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Hypothesis Testing Developing Null and Alternative Hypotheses Developing Null and Alternative Hypotheses Type I and Type II Errors Type I and Type II Errors.

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Presentation on theme: "Hypothesis Testing Developing Null and Alternative Hypotheses Developing Null and Alternative Hypotheses Type I and Type II Errors Type I and Type II Errors."— Presentation transcript:

1 Hypothesis Testing Developing Null and Alternative Hypotheses Developing Null and Alternative Hypotheses Type I and Type II Errors Type I and Type II Errors One-Tailed Tests About a Population Mean: One-Tailed Tests About a Population Mean: Large-Sample Case Large-Sample Case Two-Tailed Tests About a Population Mean: Two-Tailed Tests About a Population Mean: Large-Sample Case Tests About a Population Mean: Tests About a Population Mean: Small-Sample Case

2 Developing Null and Alternative Hypotheses Hypothesis testing can be used to determine whether Hypothesis testing can be used to determine whether a statement about the value of a population parameter a statement about the value of a population parameter should or should not be rejected. should or should not be rejected. The null hypothesis, denoted by H 0, is a tentative The null hypothesis, denoted by H 0, is a tentative assumption about a population parameter. assumption about a population parameter. The alternative hypothesis, denoted by H a, is the The alternative hypothesis, denoted by H a, is the opposite of what is stated in the null hypothesis. opposite of what is stated in the null hypothesis. The alternative hypothesis is what the test is The alternative hypothesis is what the test is attempting to establish. attempting to establish.

3 Testing Research Hypotheses Developing Null and Alternative Hypotheses Hypothesis testing is proof by contradiction. Hypothesis testing is proof by contradiction. The research hypothesis should be expressed as The research hypothesis should be expressed as the alternative hypothesis. the alternative hypothesis. The conclusion that the research hypothesis is true The conclusion that the research hypothesis is true comes from sample data that contradict the null comes from sample data that contradict the null hypothesis. hypothesis.

4 Developing Null and Alternative Hypotheses Testing the Validity of a Claim Manufacturers’ claims are usually given the benefit Manufacturers’ claims are usually given the benefit of the doubt and stated as the null hypothesis. of the doubt and stated as the null hypothesis. The conclusion that the claim is false comes from The conclusion that the claim is false comes from sample data that contradict the null hypothesis. sample data that contradict the null hypothesis.

5 Testing in Decision-Making Situations Developing Null and Alternative Hypotheses A decision maker might have to choose between A decision maker might have to choose between two courses of action, one associated with the null two courses of action, one associated with the null hypothesis and another associated with the hypothesis and another associated with the alternative hypothesis. alternative hypothesis. Example: Accepting a shipment of goods from a Example: Accepting a shipment of goods from a supplier or returning the shipment of goods to the supplier or returning the shipment of goods to the supplier supplier

6 One-tailed(lower-tail)One-tailed(upper-tail)Two-tailed Summary of Forms for Null and Alternative Hypotheses about a Population Mean n The equality part of the hypotheses always appears in the null hypothesis. in the null hypothesis. In general, a hypothesis test about the value of a In general, a hypothesis test about the value of a population mean  must take one of the following population mean  must take one of the following three forms (where  0 is the hypothesized value of three forms (where  0 is the hypothesized value of the population mean). the population mean).

7 Null and Alternative Hypotheses 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. Example: Metro EMS

8 n Null and Alternative Hypotheses The director of medical services 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. Example: Metro EMS

9 Null and Alternative Hypotheses The emergency service is meeting the response goal; no follow-up action is necessary. The emergency service is not meeting the response goal; appropriate follow-up action is necessary. H 0 :  H a :  where:  = mean response time for the population of medical emergency requests of medical emergency requests

10 Type I and Type II Errors Because hypothesis tests are based on sample data, Because hypothesis tests are based on sample data, we must allow for the possibility of errors. we must allow for the possibility of errors. n A Type I error is rejecting H 0 when it is true. n The person conducting the hypothesis test specifies the maximum allowable probability of making a the maximum allowable probability of making a Type I error, denoted by  and called the level of Type I error, denoted by  and called the level of significance. significance.

11 Type I and Type II Errors n A Type II error is accepting H 0 when it is false. n It is difficult to control for the probability of making a Type II error, denoted by . a Type II error, denoted by . n Statisticians avoid the risk of making a Type II error by using “do not reject H 0 ” and not “accept H 0 ”. error by using “do not reject H 0 ” and not “accept H 0 ”.

12 Type I and Type II Errors CorrectDecision Type II Error CorrectDecision Type I Error Reject H 0 (Conclude  > 12) Accept H 0 (Conclude  < 12) H 0 True (  < 12) H 0 False (  > 12) Conclusion Population Condition

13 Using the Test Statistic The test statistic z has a standard normal probability The test statistic z has a standard normal probability distribution. distribution. We can use the standard normal probability We can use the standard normal probability distribution table to find the z -value with an area distribution table to find the z -value with an area of  in the lower (or upper) tail of the distribution. of  in the lower (or upper) tail of the distribution. The value of the test statistic that established the The value of the test statistic that established the boundary of the rejection region is called the boundary of the rejection region is called the critical value for the test. critical value for the test. n The rejection rule is: Lower tail: Reject H 0 if z < z . Lower tail: Reject H 0 if z < z . Upper tail: Reject H 0 if z > z . Upper tail: Reject H 0 if z > z .

14 Using the p -Value Reject H 0 if the p -value < . Reject H 0 if the p -value < . The p -value is the probability of obtaining a sample The p -value is the probability of obtaining a sample result that is at least as unlikely as what is observed. result that is at least as unlikely as what is observed. If the p -value is less than the level of significance , If the p -value is less than the level of significance , the value of the test statistic is in the rejection region. the value of the test statistic is in the rejection region.

15 Steps of Hypothesis Testing 1. Determine the null and alternative hypotheses. 2. Specify the level of significance . 3. Select the test statistic that will be used to test the hypothesis. hypothesis. Using the Test Statistic 4. Use  to determine the critical value for the test statistic and state the rejection rule for H 0. statistic and state the rejection rule for H Collect the sample data and compute the value of the test statistic. of the test statistic. 6. Use the value of the test statistic and the rejection rule to determine whether to reject H 0. rule to determine whether to reject H 0.

16 Steps of Hypothesis Testing Using the p -Value 4. Collect the sample data and compute the value of the test statistic. the test statistic. 5. Use the value of the test statistic to compute the p -value. p -value. 6. Reject H 0 if p -value < .

17 One-Tailed Tests about a Population Mean: Large-Sample Case ( n > 30) n Hypotheses n Test Statistic n Rejection Rule Reject H 0 if | z| > z  H 0 :   H a :   H 0 :   H a :    Known  Unknown or

18  0 0 z  = Reject H 0 Do Not Reject H 0 One-Tailed Test about a Population Mean: Large-Sample Case ( n > 30) z Sampling distribution of Sampling distribution of Let  =.05

19  0 0  z  =  1.28 Reject H 0 Do Not Reject H 0 One-Tailed Test about a Population Mean: Large-Sample Case ( n > 30) z Sampling distribution of Sampling distribution of Let  =.10

20 Example: Metro EMS n Null and Alternative Hypotheses The response times for a random The response times for a random sample of 40 medical emergencies were tabulated. The sample mean is minutes and the sample standard deviation is 3.2 minutes. The director of medical services The director of medical services wants to perform a hypothesis test, with a.05 level of significance, to determine whether or not the service goal of 12 minutes or less is being achieved.

21 One-Tailed Tests about a Population Mean: Large-Sample Case ( n > 30) 1. Determine the hypotheses. 2. Specify the level of significance. 3. Select the test statistic.  =.05 H 0 :  H a :  4. State the rejection rule. Reject H 0 if z > (  is not known) Using the Test Statistic Using the Test Statistic

22 One-Tailed Tests about a Population Mean: Large-Sample Case ( n > 30) 5. Compute the value of the test statistic. 6. Determine whether to reject H 0. We are 95% confident that Metro EMS is We are 95% confident that Metro EMS is not meeting the response goal of 12 minutes. Because 2.47 > 1.645, we reject H 0. Using the Test Statistic Using the Test Statistic

23 One-Tailed Tests about a Population Mean: Large-Sample Case ( n > 30) Using the p  Value Using the p  Value 4. Compute the value of the test statistic. 5. Compute the p –value. For z = 2.47, cumulative probability = p –value = 1 .9932 = Determine whether to reject H 0. Because p –value =.0068 <  =.05, we reject H 0.

24 Using the p-value p -value  p -value  0 0 z  = z  =  =.05 z z z = 2.47 z = 2.47 One-Tailed Tests about a Population Mean: Large-Sample Case ( n > 30)

25 Using Excel to Conduct a One-Tailed Hypothesis Test n Formula Worksheet Note: Rows are not shown.

26 Using Excel to Conduct a One-Tailed Hypothesis Test n Value Worksheet Note: Rows are not shown.

27 Two-Tailed Tests about a Population Mean: Large-Sample Case ( n > 30) n Hypotheses n Test Statistic n Rejection Rule  Known  Unknown Reject H 0 if | z | > z 

28 Example: Glow Toothpaste Two-Tailed Tests about a Population Mean: Large n The production line for Glow toothpaste is designed to fill tubes with a mean weight of 6 oz. Periodically, a sample of 30 tubes will be selected in order to check the filling process. Quality assurance procedures call for the continuation of the filling process if the sample results are consistent with the assumption that the mean filling weight for the population of toothpaste tubes is 6 oz.; otherwise the process will be adjusted. oz. Glow

29 Example: Glow Toothpaste oz. Glow n Two-Tailed Tests about a Population Mean: Large n Assume that a sample of 30 toothpaste Assume that a sample of 30 toothpaste tubes provides a sample mean of 6.1 oz. and standard deviation of 0.2 oz. Perform a hypothesis test, at the.05 Perform a hypothesis test, at the.05 level of significance, to help determine whether the filling process should continue operating or be stopped and corrected.

30 1. Determine the hypotheses. 2. Specify the level of significance. 3. Select the test statistic.  = State the rejection rule. Reject H 0 if | z| > 1.96 (  is not known) Using the Test Statistic Using the Test Statistic Two-Tailed Tests about a Population Mean: Large-Sample Case ( n > 30) Glow H 0 :  H a : (two-tailed test)

31  Reject H 0 Do Not Reject H 0 z z Reject H  Glow Two-Tailed Tests about a Population Mean: Large-Sample Case (n > 30) Using the Test Statistic Using the Test Statistic Sampling distribution of Sampling distribution of

32 Glow Two-Tailed Tests about a Population Mean: Large-Sample Case ( n > 30) Using the Test Statistic Using the Test Statistic 5. Compute the value of the test statistic. 6. Determine whether to reject H 0. We are 95% confident that the mean filling weight of the toothpaste tubes is not 6 oz. We are 95% confident that the mean filling weight of the toothpaste tubes is not 6 oz. Because 2.74 > 1.96, we reject H 0.

33 Using the p-Value Two-Tailed Tests about a Population Mean: Large-Sample Case (n > 30) Glow Suppose we define the p -value for a two-tailed test Suppose we define the p -value for a two-tailed test as double the area found in the tail of the distribution. With z = 2.74, the cumulative standard normal With z = 2.74, the cumulative standard normal probability table shows there is a =.0031 probability of a z –score greater than 2.74 in the upper tail of the distribution. Considering the same probability of a z -score less Considering the same probability of a z -score less than –2.74 in the lower tail of the distribution, we have p -value = 2(.0031) = p -value = 2(.0031) = The p -value.0062 is less than  =.05, so H 0 is rejected.

34 Two-Tailed Tests about a Population Mean: Large-Sample Case ( n > 30) Glow     0 0 z  /2 = 1.96 z z     Using the p -Value Using the p -Value -z  /2 = z = 2.74 z = /2 p -value = /2 p -value = /2 p -value = /2 p -value =.0031

35 Using Excel to Conduct a Two-Tailed Hypothesis Test n Formula Worksheet Note: Rows are not shown. Glow

36 n Value Worksheet Using Excel to Conduct a Two-Tailed Hypothesis Test Note: Rows are not shown. Glow

37 Confidence Interval Approach to a Two-Tailed Test about a Population Mean Select a simple random sample from the population Select a simple random sample from the population and use the value of the sample mean to develop and use the value of the sample mean to develop the confidence interval for the population mean . the confidence interval for the population mean . (Confidence intervals are covered in Chapter 8.) (Confidence intervals are covered in Chapter 8.) If the confidence interval contains the hypothesized If the confidence interval contains the hypothesized value  0, do not reject H 0. Otherwise, reject H 0. value  0, do not reject H 0. Otherwise, reject H 0.

38 The 95% confidence interval for  is Confidence Interval Approach to a Two-Tailed Test about a Population Mean Glow Because the hypothesized value for the Because the hypothesized value for the population mean,  0 = 6, is not in this interval, the hypothesis-testing conclusion is that the null hypothesis, H 0 :  = 6, can be rejected. or to

39 Tests about a Population Mean: Small-Sample Case (n < 30) Test Statistic This test statistic has a t distribution with n - 1 degrees of freedom.  Known  Unknown

40 n Rejection Rule Tests about a Population Mean: Small-Sample Case ( n < 30) H 0 :   Reject H 0 if t > t  Reject H 0 if t < - t  Reject H 0 if | t | > t  H 0 :   H 0 :  

41 p -Values and the t Distribution The format of the t distribution table provided in most The format of the t distribution table provided in most statistics textbooks does not have sufficient detail statistics textbooks does not have sufficient detail to determine the exact p -value for a hypothesis test. to determine the exact p -value for a hypothesis test. However, we can still use the t distribution table to However, we can still use the t distribution table to identify a range for the p -value. identify a range for the p -value. An advantage of computer software packages is that An advantage of computer software packages is that the computer output will provide the p -value for the the computer output will provide the p -value for the t distribution. t distribution.

42 Example: Highway Patrol One-Tailed Test about a Population Mean: Small n A State Highway Patrol periodically samples vehicle speeds at various locations on a particular roadway. The sample of vehicle speeds is used to test the hypothesis H 0 :  < 65 The locations where H 0 is rejected are deemed the best locations for radar traps.

43 Example: Highway Patrol n One-Tailed Test about a Population Mean: Small n At Location F, a sample of 16 vehicles shows a mean speed of 68.2 mph with a standard deviation of 3.8 mph. Use  =.05 to test the hypothesis.

44 One-Tailed Test about a Population Mean: Small-Sample Case ( n < 30) 1. Determine the hypotheses. 2. Specify the level of significance. 3. Select the test statistic.  = State the rejection rule. Reject H 0 if t > (  is not known) Using the Test Statistic Using the Test Statistic H 0 :  < 65 H a :  > 65 (d.f. = 16-1 = 15)

45  Reject H 0 Do Not Reject H 0 (Critical value) t One-Tailed Test about a Population Mean: Small-Sample Case ( n < 30)

46 Using the Test Statistic Using the Test Statistic 5. Compute the value of the test statistic. 6. Determine whether to reject H 0. We are at least 95% confident that the mean speed of vehicles at Location F is greater than 65 mph. Location F is a good candidate for a radar trap. Because 3.37 > 1.753, we reject H 0. One-Tailed Test about a Population Mean: Small-Sample Case ( n < 30)

47 n Formula Worksheet Note: Rows are not shown. Using Excel to Conduct a One-Tailed Using Excel to Conduct a One-Tailed Hypothesis Test: Small-Sample Case Hypothesis Test: Small-Sample Case

48 n Value Worksheet Using Excel to Conduct a One-Tailed Using Excel to Conduct a One-Tailed Hypothesis Test: Small-Sample Case Hypothesis Test: Small-Sample Case Note: Rows are not shown.


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