15+Ch09a - BIT 2405 Quantitative Methods I BIT 2405...

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Unformatted text preview: BIT 2405 Quantitative Methods I BIT 2405 Quantitative Methods I Chapter 9a Hypothesis Tests This Week This Week Lectures Help Session 11am Help Only. Only. GO HOKIES! Hawkes Today HW911 HW911 and REQUIRE REQUIRE D Quiz 3 will be will posted this weekend. weekend. I’ll Answer Questions on TEST 2 in the Help Sessions I’ll Today and Thursday Morning. Today Next Week and Beyond. Next Week and Beyond. Lectures Hawkes I suggest you suggest work on Hawkes EARLY! EARLY! 9.1 is not required 9.1 but you may want to do the instruct as an Intro to Hypothesis Testing Testing If You Struggled on Test 2, Get It Figured If You Struggled on Test 2, Out NOW, Because We Build on These Concepts from Here On! • Come to the Help Sessions – ASK QUESTIONS! • Hypothesis Testing BUILDS on the Material in Test 2. – – – Clearly knowing values versus areas/probabilities. Looking up values or areas. Keeping straight what you can look up versus what answers the question. • Figure out a better TEST TAKING Strategy if you had trouble finishing. Questions? Questions? Chapter 9 & 11.1 Chapter 9 & 11.1 Hypothesis Tests This Week KEY Chapter TEXT Quantitative Methods I, Anderson, Sweeney & Williams Today Section Topic 15 Ch09a.p pt 9.1 Hawkes Learning Systems: Statistics Certifications Section Topic 9.2 9.3 9.4 T X Type I and Type II Errors Population Mean: σ Known Population Mean: σ Y ou need to UNDERSTAND these so you can make Hypothesis Testing sense of the problems. sense Proportions: P Value X Developing Null and Alternative Hypotheses 9.2 16 Ch09b.p pt T Certifications due Certifications M onday, November 9th @ 11:59pm 11:59pm 9.3 Hypothesis Testing Proportions: z Value 9.4 Hypothesis Testing Means: P Value Hypothesis Testing 9.5 Chapter 9 Chapter 9 Hypothesis Testing Developing Null and Alternative Hypotheses Type I and Type II Errors Population Mean: σ Known Population Mean: σ Unknown Population Proportion How Would You Approach This Question Now? How Would You Approach This Question Now? During the Test, It was really HOT in the During the Test, It was really HOT in the room. How hot was it? I’d bet it was AT LEAST 75 degrees! I’d bet it was AT LEAST 75 degrees! I’d bet it was AT LEAST 75 degrees! A guess about µ µ ≥75 H0: µ ≥75 I’d bet it was AT LEAST 75 degrees! I’d bet it was AT LEAST 75 degrees! A guess about µ µ ≥75 H0: µ ≥75 Ha: µ < 75 I’d bet it was AT LEAST 75 degrees! I’d bet it was AT LEAST 75 degrees! H0: µ ≥75 We will conduct a TEST of the Null Hypothesis. Collect a sample and look at the sample mean x̅obs Ha: µ < 75 We call the We hypothesized mean µ0 so mean µ0 = 75 With the TEST STATISTIC (from the sample), we can either REJECT or NOT REJECT the we Null Hypothesis. z = xobs − µ obs σ/ n 0 1 σx̅ z=0 z=0 μ0=75 x̅ I’d bet it was AT LEAST 75 degrees! I’d bet it was AT LEAST 75 degrees! H0: µ ≥75 We will conduct a TEST of the Null Hypothesis. Collect a sample and look at the sample mean x̅obs Ha: µ < 75 With the TEST STATISTIC (from the sample), we can either REJECT or NOT REJECT the we Null Hypothesis. z = xobs − µ obs Level of Significance: α Z critical z=0 μ0=75 σ/ n 0 I’d bet it was AT LEAST 75 degrees! I’d bet it was AT LEAST 75 degrees! H0: µ is ≥75 We will conduct a TEST of the Null Hypothesis. Collect a sample and look at the sample mean x̅obs Ha: µ is < 75 With the TEST STATISTIC (from the sample), we can either REJECT or NOT REJECT the we Null Hypothesis. z = xobs − µ obs If zobs i s outside the critical value, we reject the value, null hypothesis null reject Z zobs critical z=0 μ0=75 σ/ n 0 I’d bet it was AT LEAST 75 degrees! I’d bet it was AT LEAST 75 degrees! H0: µ is ≥75 We will conduct a TEST of the Null Hypothesis. Collect a sample and look at the sample mean x Ha: µ is < 75 With the TEST STATISTIC (from the sample), we can either REJECT or NOT REJECT the we Null Hypothesis. z = x−µ obs If zobs i s NOT outside the critical value, we DO NOT reject the value, null hypothesis null do not reject Z critical zobs z=0 μ0=75 0 σ/ n Developing Hypotheses Developing Hypotheses ? Start with a QUESTION. “ How do we know she’ s a How witch?” witch?” Population L ook for Characteristics of Look Elements we can measure and a plausible explanation that can be tested. be “ I f she weighs the same as If a duck, she’ s a witch.” duck, H0: µ = µ0 weight-of-duck weight-of-duck Ha: µ ≠µ0 weight-of-duck Testing Hypotheses Testing Hypotheses H0 Null Hypothesis ? Ha1 , Ha2 ... Alternative Hypotheses Populationst Te • • •• • • • • le p m Sa • • Developing Null and Alternative Hypotheses Developing Null and Alternative Hypotheses 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 alternative hypothesis is what the test is attempting to establish. The Text Discusses 3 Types of Hypotheses The Text Discusses 3 Types of Hypotheses s s s Research Hypotheses Claim Hypotheses Decision­Making Hypotheses Hawkes Tends to Establish the Null Hypothesis According to the Text’s Research Approach to Hypotheses Developing Null and Alternative Hypotheses 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. Developing Null and Alternative Hypotheses 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. Developing Null and Alternative Hypotheses Developing Null and Alternative Hypotheses s Testing in Decision­Making Situations • A decision maker might have to choose between two courses of action, one associated with the null hypothesis and another associated with the alternative hypothesis. • Example: Accepting a shipment of goods from a supplier or returning the shipment of goods to the supplier 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, an 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 µ 0 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 One­tailed (lower­tail) One­tailed (upper­tail) Two­tailed Hypothesis Testing Hypothesis Testing Three Possible Forms The objective of the test is to REJECT the The claim of the Null Hypothesis. claim H0: μ ≥µ0 H0: μ ≤µ0 H0: μ = µ0 Ha: μ < µ0 Ha: μ > µ0 Ha: μ ≠µ0 Hypothesis Testing Hypothesis Testing Three Possible Forms The objective of the test is to REJECT the The claim of the Null Hypothesis. claim H0: μ ≥µ0 H0: μ ≤µ0 H0: μ = µ0 Ha: μ < µ0 Ha: μ > µ0 Ha: μ ≠µ0 Hypothesis Testing Three Possible Forms reject H0: μ ≥µ0 Ha: μ < µ0 The objective of the test is to REJECT the The claim of the Null Hypothesis. claim reject reject H0: μ ≤µ0 H0: μ = µ0 Ha: μ > µ0 Ha: μ ≠µ0 So in this case, our objective is see if the test sample mean (x̅1) i ndicates the sample is from a SIGNIFICANTLY DIFFERENT population with a mean μ that is LESS THAN the claimed value μ0 rather than from the claimed population with μ ≥μ0 . Null and Alternative Hypotheses Null and Alternative Hypotheses s 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. Null and Alternative Hypotheses Null and Alternative Hypotheses H0: µ < 1 2 Ha: µ > 1 2 The emergency service is meeting the response goal; no follow­up action is necessary. 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 Type I Error Type I Error Because hypothesis tests are based on sample data, we must allow for the possibility of errors. s A Type I error is rejecting H0 when it is true. s The probability of making a Type I error when the null hypothesis is true as an equality is called the level of significance. s Applications of hypothesis testing that only control the Type I error are often called significance tests. Type II Error Type II Error s A Type II error is accepting H0 when it is false. s It is difficult to control for the probability of making a Type II error. s Statisticians avoid the risk of making a Type II error by using “do not reject H0” and not “accept H0”. Type I and Type II Errors Type I and Type II Errors Population Condition Conclusion H0 True (µ < 12) H0 False (µ > 12) Accept H0 (Conclude µ < 12) Correct Decision Type II Error Type I Error Correct Decision Reject H0 (Conclude µ > 12) There Are Two Approaches to Testing the Null There Are Two Approaches to Testing the Null Hypothesis. s p­Value Approach • Determine the test statistic • Determine the p­value which is an area to the left or right of the test statistic and in some cases 2 x area. • Compare the p­value to α. • Determine to reject or not reject the null hypothesis. s Critical Value Approach • Use α to find the critical value (a value with α to the left or right or α/2 in some cases) • Compare the test statistic to the critical value. • Determine to reject or not reject the null hypothesis. p­Value Approach to One­Tailed Hypothesis Testing The p­value 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 p­value 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 p­value < α . p-value -value Think PROBABILITY Think p-value is a probability (area under the curve) Lower­Tailed Test About a Population Mean: σ Known s p­Value Approach p­Value < α , so reject H0. α = .10 Sampling distribution x−µ of zobs = σ / n 0 p­value = .0 7 2 z z = ­zα = ­1.28 obs ­1.46 0 Upper­Tailed Test About a Population Mean: σ Known s p­Value Approach p­Value < α , so reject H0. Sampling distribution x−µ z of obs = σ / n α = .04 0 p­Value = .0 1 1 z 0 zα = 1.75 z = obs 2.29 Critical Value Approach to Critical Value Approach to One­Tailed Hypothesis Testing The test statistic zobs has a standard normal probability distribution. We can use the standard normal probability distribution table to find the z­value with an area of α in the lower (or upper) tail of the distribution. The value of the test statistic that established the boundary of the rejection region is called the critical value for the test. s The rejection rule is: • Lower tail: Reject H0 if zobs < ­zα • Upper tail: Reject H0 if zobs > zα Lower­Tailed Test About a Population Mean: σ Known s Critical Value Approach Sampling distribution zobs = x − µ of σ / n 0 Reject H0 α = .1 0 − α = − z 1.28 Do Not Reject H0 0 z Upper­Tailed Test About a Population Mean: σ Known s Critical Value Approach Sampling distribution x−µ z of obs = σ / n 0 Reject H0 Do Not Reject H0 0 α = .0 5 zα = 1.645 z Steps of Hypothesis Testing 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 test statistic. p­Value Approach Step 4. Use the value of the test statistic to compute the p­value. Step 5. Reject H0 if p­value < α. Steps of Hypothesis Testing Critical Value Approach 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. One­Tailed Tests About a Population Mean: σ Known s 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. One­Tailed Tests About a Population Mean: σ Known p ­Value and Critical Value Approaches 1. Develop the hypotheses. 1. Develop the hypotheses. H0: µ < 1 2 Ha: µ > 1 2 2. Specify the level of significance. α = .05 3. Compute the value of the test statistic. z obs x − µ 0 13.25 − 12 = = = 2.47 σ / n 3.2 / 40 One­Tailed Tests About a Population Mean: σ Known p –Value Approach 4. Compute the p –value. For zobs = 2.47, cumulative probability = .9932. p–value = 1 − .9932 = .0068 5. Determine whether to reject H0. 5. Determine whether to reject Because p–value = .0068 < α = .05, we reject H0. There is sufficient statistical evidence to infer that Metro EMS is not meeting the response goal of 12 minutes. One­Tailed Tests About a Population Mean: σ Known s p –Value Approach Sampling distribution x−µ z of obs = σ / n α = .05 0 p­value = .0 0 6 8 z 0 zα = 1.645 z = obs 2.47 One­Tailed Tests About a Population Mean: σ Known Critical Value Approach 4. Determine the critical value and rejection rule. For α = .05, z.05 = 1.645 Reject H0 if zobs > 1.645 5. Determine whether to reject H0. 5. Determine whether to reject Because 2.47 > 1.645, we reject H0. There is sufficient statistical evidence to infer that Metro EMS is not meeting the response goal of 12 minutes. One­Tailed Tests About a Population Mean: σ Known s p –Value Approach Sampling distribution x−µ z of obs = σ / n 0 z 0 zα = 1.645 z = obs 2.47 p­Value Approach to Two­Tailed Hypothesis Testing Compute the p­value using the following three steps: 1. Compute the value of the test statistic zobs. 2. If zobs is in the upper tail (zobs > 0), find the area under the standard normal curve to the right of zobs. If zobs is in the lower tail (zobs < 0), find the area under the standard normal curve to the left of zobs. 3. Double the tail area obtained in step 2 to obtain the p –value. The rejection rule: Reject H0 if the p­value < α . Critical Value Approach to Critical Value Approach to Two­Tailed Hypothesis Testing he critical values will occur in both the lower and pper tails of the standard normal curve. Use the standard normal probability distribution table to find zα/2 (the z­value with an area of α /2 in the upper tail of the distribution). s The rejection rule is: Reject H0 if zobs < ­zα/2 or zobs > zα/2. Two­Tailed Tests About a Population Mean: σ Known s 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. Two­Tailed Tests About a Population Mean: σ Known s Example: Glow Toothpaste 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 .03 level of significance, to help determine whether the filling process should continue operating or be stopped and corrected. Two­Tailed Tests About a Population Mean: σ Known p –Value and Critical Value Approaches 1. Determine the hypotheses. 1. Determine the hypotheses. H0: µ = 6 Ha: µ ≠ 6 2. Specify the level of significance. α = .03 3. Compute the value of the test statistic. z obs x − µ0 6.1 − 6 = = = 2.74 σ / n .2 / 30 Two­Tailed Tests About a Population Mean: Two­Tailed Tests About a Population Mean: σ Known p –Value Approach 4. Compute the p –value. For zobs = 2.74, cumulative probability = .9969 p–value = 2(1 − .9969) = .0062 5. Determine whether to reject H0. Because p–value = .0062 < α = .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). Two­Tailed Tests About a Population Mean: Two­Tailed Tests About a Population Mean: σ Known p­Value Approach 1/2 p ­value = .0031 1/2 p ­value = .0031 α /2 = .015 α /2 = .015 z = ­2.74 ­zα/2 = ­2.17 obs z 0 zα/2 = 2.17 z = 2.74 obs Two­Tailed Tests About a Population Mean: σ Known Critical Value Approach 4. Determine the critical value and rejection rule. For α /2 = .03/2 = .015, z.015 = 2.17 Reject H0 if zobs < ­2.17 or zobs > 2.17 5. Determine whether to reject H0. Because 2.74 > 2.17, 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). Two­Tailed Tests About a Population Mean: σ Known Critical Value Approach Sampling distribution x−µ of zobs = σ / n 0 Reject H0 Reject H0 Do Not Reject H0 α/2 = .015 ­2.17 α/2 = .015 0 2.17 z Confidence Interval Approach to Confidence Interval Approach to Two­Tailed Tests About a Population Mean Select a simple random sample from the population x and use the value of the sample mean to develop the confidence interval for the population mean µ . (Confidence intervals are covered in Chapter 8.) If the confidence interval contains the hypothesized value µ 0, do not reject H0. Otherwise, reject H0. Confidence Interval Approach to Confidence Interval Approach to Two­Tailed Tests About a Population Mean The 97% confidence interval for µ is x ± zα / 2 σ = 6.1 ± 2.17(.2 n 30) = 6.1 ± .07924 or 6.02076 to 6.17924 Because the hypothesized value for the population mean, µ 0 = 6, is not in this interval, the hypothesis­testing conclusion is that the null hypothesis, H0: µ = 6, can be rejected. 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