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Unformatted text preview: Chapter 9 Hypothesis Testing Overview
s s s s s Developing Null and Alternative Hypotheses Type I and Type II Errors σ Known: Tests About a Population Mean • LargeSample Case (n ≥ 30) • SmallSample Case (n < 30) σ Unknown: Tests About a Population Mean • LargeSample Case (n ≥ 30) • SmallSample Case (n < 30) Tests About a Population Proportion Slide 1 What is Hypothesis Testing?
To draw a conclusion on whether the sample information supports a hypothesis formed about a characteristic (parameter) of the population. Slide 2 Example: Metro EMS
s 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 city has maintained the service goal to respond to medical emergencies with a mean time of 12 minutes or less. In a recent survey, a group of residents believed that the average response time had been longer than 12 minutes because of high demand in the last several months. In order to find out whether the belief was true, a sample of 40 response times was collected, and the average response time was 13.25 minutes with a standard deviation of 3.2 minutes for this sample. Slide 3 s s Null and Alternative Hypotheses
s Null Hypothesis, H0 • • • An assertion about a population parameter True until we have sufficient evidence to conclude otherwise What is tested Opposite of what is stated in the null hypothesis The burden of proof lies with Ha Ha: µ > 12 H0: µ ≤ 12 Slide 4 s Alternative Hypothesis, Ha • • Define µ = the average response time Developing Null and Alternative Hypotheses
s Testing Research Hypotheses • The research hypothesis should be expressed as the alternative hypothesis. • The conclusion that the research hypothesis is true comes from sample data that contradict the null hypothesis. Slide 5 Developing Null and Alternative Hypotheses
s Testing the Validity of a Claim • Manufacturers’ claims are usually given the benefit 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. Slide 6 Examples
s Example 1: Suppose the mean sales volume of an automobile dealership is 15 automobile per month. If the manager of the dealership wants to conduct a study to see whether a new bonus plan increases the sales volume, then what should be the null and alternative hypotheses? Example 2: A production company claims that the proportion of defective items is no more than 2%. A consumer group wants to test this claim. Let p denote the proportion of defective items. What are the null and alternative hypotheses? s Slide 7 Procedure of Testing Hypotheses 1. 2. 3. 4. First set up the null and alternative hypotheses. Assume H0 is true. Find the teststatistic and its sampling distribution. Draw a conclusion using sample results. • • If the observed sample information is extreme (unusual) reject H0 and conclude Ha. If the observed sample information is not extreme, do not reject H0 and conclude that there is not sufficient evidence to reject H0 (or to support Ha). Slide 8 Three Forms for Null and Alternative Hypotheses about a Population Mean
s The equality part of the hypotheses always appears in the null hypothesis. Hypothesis test about a population mean µ (where µ 0 is the (where hypothesized value of the population mean). s H 0 : µ ≤ µ0 H a : µ > µ0 H 0 : µ ≥ µ0 H a : µ < µ0 H 0 : µ = µ0 H a : µ ≠ µ0 Onetailed Onetailed Twotailed (uppertail) (lowertail) Slide 9 Example: Metro EMS
Hypotheses Conclusion and Action H0: µ < 1 2 The emergency service is meeting the response goal; no followup action is necessary. Ha: µ > 1 2 The emergency service is not meeting the response goal; appropriate followup action is necessary. where: µ = mean response time for the population of medical emergency requests. Slide 10 Type I and Type II Errors
s s s s Since hypothesis tests are based on sample data, we must allow for the possibility of errors. Type I error: Rejecting H0 when it is true. Type II error: Accepting H0 when it is false. s s α = The maximum allowable probability of making a Type I error. Also called the level of significance. β = The probability of making a Type II error. We cannot control β . Statistician avoids the risk of making a Type II error by using “do not reject H0” and not “accept H0”. Slide 11 Example: Metro EMS
s Type I and Type II Errors Population Condition Conclusion Accept H0 (Conclude µ < 1 2 ) Reject H0 (Conclude µ > 1 2 ) Slide 12 H0 True Ha True (µ < 1 2 ) (µ > 1 2 ) Steps of Hypothesis Testing
Step 1. Develop the null and alternative hypotheses. Step 2. Specify the level of significance α. Step 3. Collect sample data and compute the test statistic. pValue Approach Step 4. Use the value of test statistic to compute the pvalue. Step 5. Reject H0 if pvalue < α . Critical Value Approach Step 4. Use the level of significance to determine the 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.
Slide 13 p Value
s The p value is the probability of obtaining a value of test statistic more extreme (< or/and > ) than the observed sample value given H0 is true. Reject H0 if the pvalue < α . Example: Metro EMS s s x − µ0 13.25 − 12 z= = = 2.47 s / n 3.2 / 40 p − value = P ( Z > 2.47) = 0.5 − .4932 = .0068 If α = 0.05, p − value < α . So reject H 0 : µ ≤ 12. Slide 14 Level of Significance
s s s Denoted by α Typical values are 1%, 5%, 10% Commonly used z values
α 1% 5% 10% zα 2.326 1.645 1.282 zα/2 2.576 1.960 1.645 Slide 15 Summary of Test Statistics to be Used in a Hypothesis Test about a Population Mean Slide 16 σ Known: OneTailed Tests about a Population Mean (Large & Small Samples)
s Hypotheses I II H0: µ < µ 0 or or H0: µ > µ 0 Ha: µ > µ 0 Ha: µ < µ 0 Test Statistic n ≥ 30 n < 30 Invoke CLT Assume Normal Popln. x − µ0 x − µ0 z= z= s/ n σ/ n Rejection Rule s s I) Reject H0 if pvalue < α OR Reject H0 if z > zα
II) Reject H0 if pvalue < α OR Reject H0 if z < zα II) Reject H Slide 17 σ Known: UpperTailed Test About a Population Mean (Large & Small Samples)
s pValue Approach Sampling distribution x − µ0 z of = σ/ n pValue < α , so reject H0. α = .05 pValue = .0 1 1 z 0 zα = 1.645 z = 2.29 Slide 18 σ Known: UpperTailed Test About a Population Mean (Large & Small Samples)
s Critical Value Approach Sampling distribution x − µ0 z of = σ/ n Do Not Reject H0 0 Reject H0 α = .0 5 zα = 1.645 z Slide 19 σ Known: LowerTailed Test About a Population Mean (Large & Small Samples)
s pValue Approach pValue < α , so reject H0. α = .10 pvalue = .0 7 2 Sampling distribution x − µ0 z of = σ/ n z z = zα = 1.46 1.28 0 Slide 20 σ Known: LowerTailed Test About a Population Mean (Large & Small Samples)
s Critical Value Approach Reject H0 Sampling distribution x − µ0 z= of σ/ n Do Not Reject H0 0 α = .1 0 − α = − z 1.28 z Slide 21 σ Known: TwoTailed Tests about a Population Mean (Large & Small Samples)
s Hypotheses Ha: µ µ0 H0: µ = µ 0 ≠ s Test Statistic n ≥ 30 n < 30 Invoke CLT Assume Normal Popln. x − µ0 x − µ0 z= z= σ/ n s/ n Rejection Rule (I) Reject Ho if pvalue < α (II) Reject Ho if z < zα/2 or z > zα/2 s Slide 22 σ Unknown: Tests about a Population Mean: LargeSample Case (n ≥30)
Test Statistic (You can use either Z or t distribution) x − µ0 z = x −µ σ / n OR t = 1 degrees of The t test statistic has a t distribution with n s / n freedom. (Assume population is Normal) s Rejection Rule (when σ is Unknown) is Unknown) OneTailed TwoTailed
s
0 H0: µ < µ 0 Reject H0 if t > tα H0: µ > µ 0 Reject H0 if t < tα H0: µ = µ 0 Reject H0 if t > tα/2 Slide 23 σ Unknown: Tests about a Population Mean: SmallSample Case (n < 30)
s Test Statistic (You have to use t distribution only) x − µ0 t= s/ n The t test statistic has a t distribution with n 1 degrees of freedom. (Assume population is Normal) s Rejection Rule (when σ is Unknown) is Unknown) OneTailed TwoTailed H0: µ < µ 0 Reject H0 if t > tα H0: µ > µ 0 Reject H0 if t < tα H0: µ = µ 0 Reject H0 if t > tα/2 Slide 24 p Values and the t Distribution s The t distribution table can not help to determine the exact p value for a hypothesis test. However, we will use the t distribution table to identify a range for the pvalue. We reject H0 if • Uppertailed test: pvalue = P(t > t obs ) < α . • Lowertailed test: pvalue = P(t < t obs ) < α . • Twotailed test: pvalue = 2P(t > t obs ) < α. Software packages will provide the pvalue for the t distribution. Slide 25 s s OneTailed Tests About a Population Mean
Example 1: Metro EMS
s 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 city has maintained the service goal to respond to medical emergencies with a mean time of 12 minutes or less. In a recent survey, a group of residents believed that the average response time had been longer than 12 minutes because of high demand in the last several months. In order to find out whether the residents’ belief was true, a random sample of 40 response times was collected, and the average response time was 13.25 minutes with a standard deviation of 3.2 minutes for this sample. Slide 26 s s OneTailed Tests About a Population Mean
Example 1: Metro EMS a. Develop the null and alternative hypotheses. b. Compute the value of the test statistic. Slide 27 OneTailed Tests About a Population Mean Example 1: Metro EMS p –Value Approach c. Compute the p–value. d. What is the conclusion of the test at 5% level of significance? e. Express the statistical decision in terms of the problem.
We are 95% confident that Metro EMS is not meeting the response goal of 12 minutes; appropriate action should be taken to improve service. Slide 28 OneTailed Tests about a Population Mean Example 1: Metro EMS Critical Value Approach c. Determine the critical value and rejection rule (use 5% level of significance). d. What is the conclusion of the test? Slide 29 OneTailed Tests about a Population Mean
Example 1: Metro EMS
s Using the pvalue and Critical Value to Test the Hypothesis Reject H0 Do Not Reject H0 pvalue= .0 0 6 8 z Slide 30 0 1.645 2.47 TwoTailed Tests About a Population Mean Example 2: Glow Toothpaste
s The production line for Glow toothpaste is designed to fill tubes of toothpaste with a target mean weight of 6 ounces. 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 ounces; otherwise the filling process will be stopped and adjusted. One day, a sample of 30 toothpaste tubes provides a mean of 6.1 oz. with a standard deviation of 0.2 oz. 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. Slide 31 s s TwoTailed Tests About a Population Mean
Example 2: Glow Toothpaste
Glow a. Write down the appropriate hypotheses. b. Compute the value of the test statistic. Slide 32 TwoTailed Tests About a Population Mean
Example 2: Glow Toothpaste p –Value Approach c. Compute the p–value. d. What is the conclusion of the test? e. Express the statistical decision in terms of the problem.
We are 95% confident that the mean filling weight of the toothpaste tubes is not 6 ounces. The filling process should be stopped and the filling mechanism adjusted. Slide 33 TwoTailed Tests About a Population Mean
Example 2: Glow Toothpaste Critical Value Approach c. Determine the critical value and rejection rule. d. What is the conclusion of the test? Slide 34 TwoTailed Tests About a Population Mean
Example 2: Glow Toothpaste pvalue and critical value of the test 1/2 p value = .0031 1/2 p value = .0031 α /2 = .025
z = 2.74 0 α /2 = .025 z
zα/2 = 1.96 zα/2 = 1.96 z = 2.74 Slide 35 Confidence Interval Approach to a TwoTailed Test about a Population Mean
100(1α)% confidence interval for the population mean µ .: s s x ± z x ±t OR
s α/ 2 n α/2 n s If the confidence interval contains the hypothesized value µ 0, do not reject H0. Otherwise, reject H0. Slide 36 Example 2: Glow Toothpaste
a. Develop the 95% confidence interval for µ . b. What is the conclusion of the test? Since the hypothesized value for the population mean, µ 0 = 6, is / is not in the interval, the conclusion is that the null hypothesis, H0: µ = 6, is / is not rejected. Slide 37 OneTailed Test About a Population Mean Example 3: Highway Patrol
s A State Highway Patrol periodically samples vehicle speeds at various locations on a particular highway to see whether the average speed of the vehicles is significantly higher than the posted speedlimit of 65 mph. Recently, at Location F, a random sample of 16 vehicles shows a mean speed of 68.2 mph with a standard deviation of 5.12 mph. Perform a hypothesis test, at the .05 level of significance, to determine whether the average speed of vehicles at location F exceed the posted speedlimit of 65 mph (that is, location F is a good spot for a radar trap). Slide 38 s s OneTailed Test About a Population Mean Example 3: Highway Patrol
a. Determine the appropriate hypotheses. b. Compute the value of the test statistic. Slide 39 OneTailed Test About a Population Mean Example 3: Highway Patrol Critical Value Approach c. Determine the critical value and rejection rule. d. What is the conclusion of the test? e. Express the statistical decision in terms of the problem.
We are 95% confident that the average speed of vehicles at Location F is higher than 65 mph; Location F is a good spot for a radar trap. Slide 40 OneTailed Test About a Population Mean Example 3: Highway Patrol p –Value Approach c. Find the p –value. d. What is the conclusion? Slide 41 OneTailed Test About a Population Mean Example 3: Highway Patrol Reject H0 Do Not Reject H0 α = .0 5 0 tα = 1.753 t Slide 42 Summary of Test Statistics to be Used in a Hypothesis Test about a Population Mean Slide 43 s The equality part of the hypotheses always appears in the null hypothesis. Hypothesis tests about a population proportion p (where p0 is the hypothesized value of the population proportion). H0: p < p0 Null and Alternative Hypotheses about a Population Proportion s H0: p > p0 H0: p = p0 Ha: p > p0 Ha: p < p0 Ha: p p0 ≠ Slide 44 Tests about a Population Proportion: LargeSample Case (np > 5 and n(1 p) > 5)
s Test Statistic p − p0 z= σp where:
s σp = p0 (1 − p0 ) n Rejection Rule OneTailed TwoTailed H0: p < p0 Reject H0 if z > zα H0: p > p0 Reject H0 if z < zα H0: p = p0 Reject H0 if z > zα/2 Slide 45 Test about a Population Proportion Example: National Security Council s For a Christmas and New Year’s week, the National Safety Council estimated that 500 people would be killed and 25,000 injured on the nation’s roads. The NSC claimed that at least 50% of the accidents would be caused by drunk driving. A sample of 120 accidents showed that 54 were caused by drunk driving. Use these data to test the NSC’s claim with α = 0.10. s Slide 46 Test about a Population Proportion Example: National Security Council a. Determine the appropriate hypotheses. b. Compute the value of the test statistic. a common error is using p in the formula Slide 47 Test about a Population Proportion Example: National Security Council p−Value Approach c. Compute the pvalue. d. What is the conclusion? e. Express the statistical decision in terms of the problem.
There is no sufficient evidence to conclude that the NSC’s claim of at least 50% of the accidents would be caused by drunk driving is not right at the .10 significance level. Slide 48 Test about a Population Proportion Example: National Security Council Critical Value Approach c. Determine the critical value and rejection rule. d. What is the conclusion? Slide 49 Summary Null and alternative hypotheses: onetailed and twotailed Test statistic Sampling distribution Type I and Type II errors Significance level pvalue Statistical decisions Slide 50 ...
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This note was uploaded on 12/04/2009 for the course MGMT 305 taught by Professor Priya during the Summer '08 term at Purdue UniversityWest Lafayette.
 Summer '08
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