MGTS 103 Ch 9 Slides (1) PDF

Statistics for Business and Economics (14e) Hypothesis Testing Hypothesis testing can be used to determine whether a statement about the value o...

Statistics for Business and Economics (14e) Hypothesis Testing Hypothesis testing can be used to determine whether a statement about the value of a population parameter 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 hypothesis testing procedure uses data from a sample to test the two competing statements indicated by H0 and Ha. © 2020 Cengage. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 1  Statistics for Business and Economics (14e) Developing Null and Alternative Hypotheses (1 of 4) It is not always obvious how the null and alternative hypotheses should be formulated. Care must be taken to structure the hypotheses appropriately so that the test conclusion provides the information the researcher wants. The context of the situation is very important in determining how the hypotheses should be stated. In some cases it is easier to identify the alternative hypothesis first. In other cases the null is easier. Correct hypothesis formulation will take practice. © 2020 Cengage. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 2  Statistics for Business and Economics (14e) Developing Null and Alternative Hypotheses (2 of 4) Alternative Hypothesis as a Research Hypothesis Many applications of hypothesis testing involve an attempt to gather evidence in support of a research hypothesis. In such cases, it is often best to begin with the alternative hypothesis and make it the conclusion that the researcher hopes to support. The conclusion that the research hypothesis is true is made if the sample data provides sufficient evidence to show that the null hypothesis can be rejected. Example: A new teaching method is developed that is believed to be better than the current method. Null Hypothesis: The new method is no better than the old method. Alternative Hypothesis: The new teaching method is better. © 2020 Cengage. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 3  Statistics for Business and Economics (14e) Developing Null and Alternative Hypotheses (3 of 4) Example: A new sales force bonus plan is developed in an attempt to increase sales. Null Hypothesis: The new bonus plan will not increase sales. Alternative Hypothesis: The new bonus plan will increase sales. Example: A new drug is developed with the goal of lowering blood pressure more than the existing drug. Null Hypothesis: The new drug does not lower blood pressure more than the existing drug. Alternative Hypothesis: The new drug lowers blood pressure more than the existing drug. © 2020 Cengage. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 4  Statistics for Business and Economics (14e) Developing Null and Alternative Hypotheses (4 of 4) Null Hypothesis as an Assumption to be Challenged We might begin with a belief or assumption that a statement about the value of a population parameter is true. We then use a hypothesis test to challenge the assumption and determine if there is statistical evidence to conclude that the assumption is incorrect. In these situations, it is helpful to develop the null hypothesis first. Example: The label on a soft drink bottle states that it contains 67.6 fluid ounces. © 2020 Cengage. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 5  Statistics for Business and Economics (14e) Summary of Forms for Null and Alternative Hypotheses The equality part of the hypotheses always appears in the null hypothesis. In general, a hypothesis test about the value of a population mean μ must take one of the following three forms (where μ0 is the hypothesized value of the population mean). © 2020 Cengage. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 6  Statistics for Business and Economics (14e) Null and Alternative Hypotheses (1 of 2) 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. 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. © 2020 Cengage. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 7  Statistics for Business and Economics (14e) Null and Alternative Hypotheses (2 of 2) © 2020 Cengage. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 8  Statistics for Business and Economics (14e) 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 for the Type I error are often called significance tests. © 2020 Cengage. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 9  Statistics for Business and Economics (14e) 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” rather than “accept H0”. © 2020 Cengage. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 10  Statistics for Business and Economics (14e) Type I and Type II Errors © 2020 Cengage. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 11  Statistics for Business and Economics (14e) 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 α, the value of the test statistic is in the rejection region. Reject H0 if the p-value ≤ α. © 2020 Cengage. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 12  Statistics for Business and Economics (14e) Suggested Guidelines for Interpreting p-Values Less than 0.01: Overwhelming evidence to conclude Ha is true. Between 0.01 and 0.05: Strong evidence to conclude Ha is true. Between.05 and.10: Weak evidence to conclude Ha is true. Greater than.10: Insufficient evidence to conclude Ha is true. © 2020 Cengage. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 13  Statistics for Business and Economics (14e) Lower-Tailed Test About a Population Mean: σ Known (1 of 2) p-Value Approach © 2020 Cengage. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 14  Statistics for Business and Economics (14e) Upper-Tailed Test About a Population Mean: σ Known (1 of 2) p-Value Approach © 2020 Cengage. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 15  Statistics for Business and Economics (14e) Steps of Hypothesis Testing Step 1. Develop the null and alternative hypotheses. Step 2. Specify the level of significance α. Step 3. Collect the sample data and compute the value of the test statistic. p-Value Approach Step 4. Use the value of the test statistic to compute the p-value. Step 5. Reject H0 if p-value ≤ α. Step 6. Conclusion statement Critical Value Approach (won’t be tested on) Step 4. Use the level of significance α 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. © 2020 Cengage. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed 16 with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (14e) One-Tailed Tests About a Population Mean: σ Known (1 of 5) 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 being achieved. © 2020 Cengage. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 17  Statistics for Business and Economics (14e) One-Tailed Tests About a Population Mean: σ Known (2 of 5) 1. Develop the hypotheses. 2. Specify the level of significance. α =.05 3. Compute the value of the test statistic. © 2020 Cengage. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 18  Statistics for Business and Economics (14e) One-Tailed Tests About a Population Mean: σ Known (3 of 5) p –Value Approach 4. Compute the p –value. For z = 2.47, the cumulative probability is 0.9932. p-value = 1 – 0.9932 = 0.0068 5. Determine whether to reject H0. Because p-value = 0.0068 ≤ α = 0.05, we reject H0. There is sufficient statistical evidence to infer that Metro EMS is not meeting the response goal of 12 minutes. © 2020 Cengage. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 19  Statistics for Business and Economics (14e) One-Tailed Tests About a Population Mean: σ Known (4 of 5) p –Value Approach © 2020 Cengage. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 20  Statistics for Business and Economics (14e) p-Value Approach to Two-Tailed Hypothesis Testing Compute the p-value using the following three steps: 1. Compute the value of the test statistic. 2. If is in the upper tail ( > 0), compute the probability that is greater than or equal to the value of the test statistic. If is in the lower tail ( < 0), compute the probability that is less than or equal to the value of the test statistic. 3. Double the tail area obtained in step 2 to obtain the p-value. The rejection rule: Reject H0 if the p-value ≤ α. © 2020 Cengage. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 21  Statistics for Business and Economics (14e) Two-Tailed Tests About a Population Mean: σ Known (1 of 6) Example: Glow Toothpaste 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. Assume that a sample of 30 toothpaste tubes provides a sample mean of 6.1 oz. The population standard deviation is believed to be 0.2 oz. Perform a hypothesis test, at the 0.03 level of significance, to help determine whether the filling process should continue operating or be stopped and corrected. © 2020 Cengage. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed 22 with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.  Statistics for Business and Economics (14e) Two-Tailed Tests About a Population Mean: σ Known (2 of 6) 1. Develop the hypotheses. 2. Specify the level of significance. α = 0.03 3. Compute the value of the test statistic. © 2020 Cengage. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 23  Statistics for Business and Economics (14e) Two-Tailed Tests About a Population Mean: σ Known (3 of 6) p –Value Approach 4. Compute the p –value. For z = 2.74, the cumulative probability is 0.9969. p-value = 2(1 – 0.9969) = 0.0062 5. Determine whether to reject H0. Because p-value = 0.0062 ≤ α = 0.03, we reject H0. There is sufficient statistical evidence to infer that the alternative hypothesis is true (i.e. the mean filling weight is not 6 ounces). © 2020 Cengage. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 24  Statistics for Business and Economics (14e) Two-Tailed Tests About a Population Mean: σ Known (4 of 6) p-Value Approach © 2020 Cengage. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 25  Statistics for Business and Economics (14e) Confidence Interval Approach to Two-Tailed Tests About a Population Mean (1 of 2) Select a simple random sample from the population and use the value of the sample meanto develop the confidence interval for the population mean μ. (Confidence intervals are covered in Chapter 8.) If the confidence interval contains the hypothesized value μ0, do not reject H0. Otherwise, reject H0. (Actually, H0 should be rejected if μ0 happens to be equal to one of the end points of the confidence interval.) © 2020 Cengage. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 26  Statistics for Business and Economics (14e) Confidence Interval Approach to Two-Tailed Tests About a Population Mean (2 of 2) The 97% confidence interval for μ is Because the hypothesized value for the population mean, μ0 = 6, is not in this interval, the hypothesis-testing conclusion is that the null hypothesis, H0: μ = 6, can be rejected. © 2020 Cengage. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 27  Statistics for Business and Economics (14e) Tests About a Population Mean: σ Unknown (1 of 2) Test Statistic: This test statistic has a t distribution with n – 1 degrees of freedom. © 2020 Cengage. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 28  Statistics for Business and Economics (14e) Tests about a Population Mean: σ Unknown (2 of 2) Rejection Rule: p-value approach Rejection Rule: Critical value approach © 2020 Cengage. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 29  Statistics for Business and Economics (14e) p -Values and the t Distribution The format of the t distribution table provided in most statistics textbooks does not have sufficient detail to determine the exact p-value for a hypothesis test. However, we can still use the t distribution table to identify a range for the p-value. An advantage of computer software packages is that the computer output will provide the p-value for the t distribution. © 2020 Cengage. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 30  Statistics for Business and Economics (14e) Example: Highway Patrol One-Tailed Test About a Population Mean: σ Unknown 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 H0: μ ≤ 65. The locations where H0 is rejected are deemed the best locations for radar traps. At Location F, a sample of 64 vehicles shows a mean speed of 66.2 mph with a standard deviation of 4.2 mph. Use α = 0.05 to test the hypothesis. © 2020 Cengage. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 31  Statistics for Business and Economics (14e) One-Tailed Test About a Population Mean: σ Unknown (1 of 4) 1. Develop the hypotheses. 2. Specify the level of significance. α =.05 3. Compute the value of the test statistic. © 2020 Cengage. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 32  Statistics for Business and Economics (14e) One-Tailed Test About a Population Mean: σ Unknown (2 of 4) p –Value Approach 4. Compute the p –value. 5. Determine whether to reject H0. Because p-value < α = 0.05, we reject H0. We are at least 95% confident that the mean speed of vehicles at Location F is greater than 65 mph. © 2020 Cengage. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 33  Statistics for Business and Economics (14e) End of Chapter 9 © 2020 Cengage. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 34