chap10 - Chapter 10 Hypothesis Tests Developing Null and...

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Unformatted text preview: Chapter 10 Hypothesis Tests Developing Null and Alternative Hypotheses Type I and Type II Errors Hypothesis Tests for Population Mean: σ Known Hypothesis Tests for Population Mean: σ Unknown Slide 1 Summary of Forms for Null and Alternative Hypotheses about a Population Mean s 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 must take one of the following three forms (where µ0 is the hypothesized value of the population mean). 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 (lower­tail) One­tailed (upper­tail) Two­tailed Slide 2 Type I and Type II Errors Population Condition Conclusion H0 True (µ < µ0) Accept H0 (Conclude µ < µ0) Correct Decision Type II Error Type I Error Correct Decision Reject H0 (Conclude µ > µ0) H0 False (µ > µ0) Slide 3 Two Basic Approaches to Hypothesis Testing There are two basic approaches to conducting a hypothesis test: 1­ p­Value Approach, and 2­ Critical Value Approach Slide 4 1­ p­Value Approach to One­Tailed Hypothesis Testing In order to accept or reject the null hypothesis the p­value is In the computed using the test statistic ­­Actual Z value. Reject H0 if the p­value < α Do not reject (accept) H0 if the p­value > α Slide 5 2­ Critical Value Approach One­Tailed Hypothesis Testing Use the Z table to find the critical Z value, and Use the equation to find the actual Z­­Z statistics. s The rejection rule is: • Lower tail: Reject H0 if Actual z < Critical ­zα • Upper tail: Reject H0 if Actual z > Critical zα In other words, if the actual Z (Z statistics) is in the In actual rejection region, then reject the null hypothesis. rejection Equation for finding the Equation actual Z value: actual x −µ z= σ/ n Slide 6 Steps of Hypothesis Testing Step 1. Develop the null and alternative hypotheses. Step 2. Specify α and n. and Step 3. Compute critical Z and actual Z values. Step 3 Step 4. Use either of the following approaches to make conclusion: 1­ p­Value Approach, or 2­ Critical Approach Slide 7 One­Tailed Tests About a Population Mean: σ Known s Example: Metro EMS s 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. s The EMS director wants to perform a hypothesis test, with a 0.05 level of significance, to determine whether the service goal of the response time to be at most 12 minutes or less is being achieved. Slide 8 One­Tailed Tests About a Population Mean: σ Known: Solution p ­Value and Critical Value Approaches 1. Develop the hypotheses. H0: µ < 1 2 Ha: µ > 1 2 2. Level of significance and sample size are: α= .05 n = 40 3. Compute the value of the test statistic. x − µ 13.25 − 12 z= = = 2.47 σ / n 3.2 / 40 Actual z Actual Slide 9 One­Tailed Tests About a Population Mean: σ Known: Solution Continued p –Value Approach 4. Compute the p –value. From the Ztable the actual z = 2.47 using Z table, p–value = 0.5 ­ .4932 = .0068 5. Make conclusion about H0 Because p–value = .0068 < α = .05, we reject H0. s We are at least 95% confident that Metro EMS is not meeting the response goal of 12 minutes. Slide 10 Solution Continued Because p–value = .0068 < α = .05, we reject H0. α = .05 p­value = .0 0 6 8 z 0 Zc = 1.645 Za = 2.47 Slide 11 One­Tailed Tests About a Population Mean: σ Known: Solution Continued Critical Value Approach 4. Determine the critical value and rejection rule. For α = .05, z.05 = 1.645 Reject H0 if actual z > 1.645 5. Make conclusion about H0 Finding critical z value 0.5 – 0.05 = 0.45 Then, from table 1.64 + 1.65 3.29 / 2 = 1.645 Because actual z = 2.47 > Critical z = 1.645 we reject H0. s We are at least 95% confident that Metro EMS is not meeting the response goal of 12 minutes. Slide 12 One­Tailed Tests About a Population Mean: σ Known s Excel: SWStat Slide 13 One­Tailed Tests About a Population Mean: σ Known s Excel: SWStat P Approach Critical Approach Because actual z = 2.47 > Because actual Critical z = 1.645 we Critical reject H0, or or Because p–value = .0068 Because < α = .05, we reject we H0 Slide 14 Example: Glow Toothpaste s Two­Tailed Test for Population Mean: σ Known The production line for Glow toothpaste is designed to fill tubes with a mean weight of 6 oz. Periodically, a sample of 30 tubes o will be selected in order to check the z.Glow 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. Slide 15 Example Continued: Glow Toothpaste s Two­Tailed Test for Population Mean: σ Known 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. o z.Glow Slide 16 Two­Tailed Tests About a Population Mean: σ Known: Solution Glow p –Value and Critical Value Approaches 1. Determine the hypotheses. H0: µ = 6 Ha: µ ≠ 6 2. Alpha and sample size are given α = .03 and n=30 3. Compute the value of the test statistic. x − µ0 6.1 − 6 z= = = 2.74 σ / n .2 / 30 Actual z Slide 17 Two­Tailed Tests About a Population Mean: σ Known: Solution Continued p –Value Approach Glow 4. Compute the p –value. For actual z = 2.74, the probability = 0.4969, thus p–value = 2(0.5 – 0.4969) = 2 (0.0031) = 0.0062 5. Determine whether to reject or to accept H0. Because p–value = .0062 < α = .03, we reject H0. s We are at least 97% confident that the mean filling weight of the toothpaste tubes is not 6 oz. Slide 18 Solution Continued Glow Because p–value = .0062 < α = .03, we reject H0. 1/2 p ­value = .0031 1/2 p ­value = .0031 α/2 = .015 α/2 = .015 z z = ­2.74 ­zα/2 = ­2.17 0 zα/2 = 2.17 za = 2.74 Slide 19 Two­Tailed Tests About a Population Mean: σ Known: Solution Continued Critical Value Approach To Find the Critical Z Value: G lo w 0.5 Given that α = 0.03, thus α/2 = .015 and 0.5 – 0.015 = 0.485 Then from the table we need to find the z value of 0.485. Critical zα/2 Locate 0.485 in the Z Table. Thus, the critical z value for 0.485 is 2.17 0.485 Critical zα/2 = 2.17 α/2 = .015 Slide 20 Two­Tailed Tests About a Population Mean: σ Known: Solution Continued Critical Value Approach G lo w Conclusion: Because actual z of 2.74 > critical z of 2.17, we reject H 0 s We are at least 97% confident that the mean filling weight of the toothpaste tubes is not 6 oz. Actual Z z= x − µ0 6.1 − 6 = = 2.74 σ / n .2 / 30 Slide 21 Critical Approach: Solution Continued Glow Because actual z of 2.74 > critical z of 2.17, we reject H0. Actual Z Value z= x − µ0 σ / n = 2.74 Reject H0 Reject H0 α/2 = .015 ­2.17 α/2 = .015 0 Critical Z values 2.17 z Slide 22 Two­Tailed Tests About a Population Mean: σ Known s Glow Excel: SWStat Slide 23 Two­Tailed Tests About a Population Mean: σ Known s Excel: SWStat P Approach Critical Approach Glow Slide 24 THOUGHT THOUGHT Confidence Intervals Versus Hypothesis Tests A standard confidence interval is equivalent to a two-tail hypothesis test. All two tails tests can be handled either as hypothesis tests or as confidence intervals. The confidence interval has the appeal of providing a graphic feeling for how close the hypothesized value lies to the ends of confidence interval. Rejection Rule: If the confidence interval does not contain H0 , we reject H0. Thinking Challenge Example 32 males between the ages of 40 and 69 years with 32 40 69 moderate carotid disease were tested at the Henry Hospital over 39-,months period. Their mean systolic pressure was 146.6 mmHg with a standard deviation of 146.6 17.3 mmHg. At a = 0.05, is this sample consistent with a 17.3 0.05 population mean of 140 mmHg, which is considered a 140 borderline for dangerously high blood pressure (note: borderline recent medical evidence suggests 130 as a borderline, but we will use the older benchmark)? we Apply confidence interval approach to test the hypothesis Thinking Challenge Example Solution Confidence Interval Approach: For this problem, the two­sided hypothesis would be: H0: µ = 1 4 0 Ha: µ ≠ 1 4 0 The 95% confidence interval (α=0.05) for µ is: for s x ± tα / 2 Margin of Error Margin n 146 – + (2.040) 17.3 /5.657 Since interval 140.36 < µ < 152.84 does not contain µ =140 we =140 would reject the hypothesis H0: µ = 1 4 0 in favor of Ha: µ ≠ 1 4 0 . Slide 28 Hypothesis Tests About a Population Mean: σ Unknown s Test Statistic Actual t Value x − µ0 t= s/ n This test statistic has a t distribution with n ­ 1 degrees of freedom. Slide 29 Example: Highway Patrol s One­Tailed 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. Slide 30 Example Continued: Highway Patrol At Location F on I­75, a sample of 64 vehicles shows a mean speed of 66.2 mph with a sample standard deviation of 4.2 mph. Use α = .05 to test the hypothesis. Use Excel Slide 31 Using SWStat Slide 32 Solution Using SWStat P Approach Critical Approach H0: µ < 65 Since p=0.0128 < α=0.05 Since =0.05 we reject H0 we The locations where H0 is rejected are deemed rejected the best locations for radar the best traps. traps. s Slide 33 One­Tailed Test About a Population Mean: σ Unknown: Solution Continued Reject H0 Do Not Reject H0 0 α = .0 5 critical tα = 1.669 t t Statistic = Actual t = 2.286 Slide 34 Thinking Challenge The current rate for producing 5 amp fuses at Ariana Electric Co. is 250 per hour. A new machine has been purchased and installed that, according to the supplier, will increase the production rate. The production hours are normally distributed. A sample of 10 randomly selected hours from last month revealed that the mean hourly production on the new machine was 256 units, with a sample standard deviation of 6 per hour. and Solution At the .05 significance level can Ariana Electric Co. conclude that the new machine is faster? Slide 35 The null hypothesis is rejected if t > 1.833 or, using the p-value, the null hypothesis is rejected if p ≤ 0.05 Step 4 State the decision rule. There are 10 – 1 = 9 degrees of freedom. Step 1 State the null and alternate hypotheses. H0: µ < 250 H1: µ > 250 Step 3 Find a test statistic. Use the t distribution since σ is not known and n < 30. Step 2 Select the level of significance. It is .05. Slide 36 Step 5 Make a decision and interpret the results. t= X −µ s Actual t Computed t (or actual t) of 3.162 > critical t of 1.833 and From Excel, p of .0058 < α = .0 5 So we reject Ho n = 256 − 250 6 10 = 3.162 The p(t >3.162) is .0058 for a one-tailed test. Conclusion The mean number of amps produced by the new machine is more than 250 per hour. Slide 37 Solution Using SWStat Slide 38 Solution Continued H0: µ < 250 Since computed t (or actual t) of 3.162 > critical t of 1.833 and since p of .0058 < α = .0 5 thus, we reject Ho Hence, we conclude that the mean number of amps produced by the new machine is more than 250 per hour. Slide 39 and Solution Thinking Challenge s A group of young businesswomen wish to open a high fashion boutique in a vacant store, but only if the average income of households in the area is more than $45,000. A random sample of 9 households showed the following results. $48,000 $44,000 $46,000 $43,000 $47,000 $46,000 $44,000 $42,000 $45,000 Slide 40 Thinking Challenge (Continued) s Use the statistical techniques in Excel (SWStat) to advise the group on whether or not they should locate the boutique in this store. Use a 0.05 level of significance. (Assume the population is normally distributed). Slide 41 Thinking Challenge 4 (Solution) Slide 42 Summary of Selecting an Appropriate Test Statistic for a Test about a Population Mean Slide 43 Cartoon Classics Cartoon End of Chapter 10 The End You can wait for music to end, or…. Press Esc to end the presentation. Hope you enjoyed the show and had fun ! ...
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This document was uploaded on 11/25/2011.

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