Introduction to Hypothesis TestingIntroductionHypothesis testingStepsHypotheses H0, HaSignificance levelp-valueFormal decisionFinal conclusionType I & II errorsPowerSingle sample proportion testUseComputationSimple example using test statisticA complete exampleDiscussionSummaryAdditional ExamplesIntroductory Statistics LecturesIntroduction to Hypothesis TestingTesting a claim about a population proportionAnthony TanbakuchiDepartment of MathematicsPima Community CollegeRedistribution of this material is prohibitedwithout written permission of the author© 2009(Compile date: Tue May 19 14:50:31 2009)Contents1 Introduction to Hypoth-esis Testing 11.1 Introduction . . . . . . . 11.2 Hypothesis testing . . . 4Steps . . . . . . . . . . . 4Hypotheses H0, Ha. . . 5Significance level . . . . 6p-value . . . . . . . . . . 6Formal decision . . . . . 7Final conclusion . . . . 7Type I & II errors . . . 7Power . . . . . . . . . . 81.3 Single sample propor-tion test . . . . . . . . . 9Use . . . . . . . . . . . . 9Computation . . . . . . 9Simple example usingtest statistic . . . 10A complete example . . 111.4 Discussion . . . . . . . . 131.5 Summary . . . . . . . . 131.6 Additional Examples . . 141 Introduction to Hypothesis Testing1.1 IntroductionExample 1. A 2001 study estimated 56% of people in the US wear correctivelenses.1However, you believe the proportion of people in the US who wear correctivelenses is less than 56 percent.1Source: Walker, T.C. and Miller, R.K. 2001 Health Care Business Market ResearchHandbook, fifth edition, Norcross (GA): Richard K. Miller & Associates, Inc., 2001. Studyestimated about 160 million people in US wear glasses. 2001 population was estimated to be286 million.12 of 14 1.1 IntroductionQuestion 1. How could you support your claim?Question 2. You conduct a study of our class and find the proportion of stu-dents who wear corrective lenses is 55.6%. Does this support our hypothesisthat the proportion of people in the US who wear corrective lenses is less than56 percent? Why?Question 3. What would we need to know to support our hypothesis thatthe proportion of people in the US who wear corrective lenses is less than 56percent?Goal• Find probability of observing a sample proportion at least as extremeas ˆp = 0.556.• If we can determine that it is unlikely to observe ˆp = 0.556 assumingp0= 0.56 then the rare event rule would make us question our assumptionthat p0= 0.56 and allow us to support our claim that p < 0.56.Sampling distribution of ˆpIf np and nq ≥ 5 then p will have a normal distribution2and the CLT tells usthat ˆp is approximately normally distribution where:µˆp= p (1)σˆp=rpqn(2)Probability of observing our sample data.In our case p = 0.56, n = 18. We want to find the probability of observinga sample proportion at least as extreme as 0.556: P (ˆp < 0.556).2Normal approximation of binomial.Anthony Tanbakuchi MAT167Introduction to Hypothesis Testing 3 of 140.0 0.2 0.4 0.6 0.8 1.00.0 1.0 2.0 3.0Sampling distribution of pp^The above plot is the sampling distribution for ˆp assuming µˆp= p = 0.56and the shaded area 0.5.Since p-value = 0.5:Question 4. Using the rare event rule, would it be unusual to observe a sampleproportion at least as extreme as 0.556 if the true population value is 0.56?Question 5. Can we support our claim that the proportion of people in the USwho wear corrective lenses is less than 56 percent?Question 6. If we decided to support our claim that the proportion of people inthe US who wear corrective lenses is less than 56 percent, what is the probabilitythat we made the wrong decision? In other words, what is the probability thatwe would observe ˆp = 0.556 from a random sample drawn from a populationwith p = 0.56Question 7. Under what conditions can we support our claim via therare event rule?Question 8. Under what conditions can’t we support our claim via therare event rule?Anthony Tanbakuchi MAT1674 of 14 1.2 Hypothesis testing1.2 Hypothesis testingGoal of hypothesis testingThe basic conceptual steps for hypothesis testing are:1. Assume the status quo — the null hypothesis — is true.2. Calculate the probability of observing the sample data assuming the thenull hypothesis is true — the p-value.3. If the p-value is small it is unlikely that we would have observed oursample data if the null hypothesis is true. Thus, we can reject the nullhypothesis and we have evidence to support our claim — the alternativehypothesis.4. If the p-value is not small it is not surprising to observe our sample dataunder the assumption that the null hypothesis is true. We cannot supportour alternative hypothesis.Two key concepts in hypothesis testing.1. A hypothesis test is designed to disprove the null hypothesis.We don’t prove anything. We simply show that the null hypothesis isstatistically unlikely in light of sample data and the data supports ouralternative hypothesis.2. The null hypothesis always involves equality.We never support claims with equality!3STEPSEight simple steps0. Write down what is known.1. Determine which type of hypothesis test to use.2. Check the test’s requirements.3. Formulate the hypothesis: H0, Ha4. Determine the significance level α.5. Find the p-value.6. Make the decision.7. State the final conclusion.You must know by heart and write down all eight steps whenworking problems!4K-T-R-H-S-P-D-C: “Know The Right Hypothesis So People Don’t Com-plain”5Step 1: Determine which type of hypothesis test to use.3If we wish to do so, we must go beyond the content of this course and calculate theprobability of a Type II error β.4Note: I am showing you the p-value method for hypothesis testing. The book discussesit as well as the critical-value and confidence interval methods. The p-value method providesmore information, is more precise (using R), and is more meaningful as compared to thecritical-value method. Use the p-value method.5Thanks to Maria Starzk for the nemonic.Anthony Tanbakuchi MAT167Introduction to Hypothesis Testing 5 of 14Some common tests:Single sample tests : Test for1. Population proportion (H0: p = p0)2. Population mean (H0: µ = µ0)3. Population std. dev. (H0: σ = σ0)4. No correlation (H0: ρ = 0)5. Normality (H0: pop. is normally dist.)Where p0, µ0, σ0are all constants, (the status quo).Two sample tests : Test for1. Equality of two proportions (H0: ∆p = 0)2. Equality of two mean (H0: ∆µ = 0)3. Equality of two std. devs. (H0: ∆σ = 0)Step 2: Check the test’s