Statistics 511: Statistical MethodsDr. LevinePurdue UniversitySpring 2011Lecture 14: Introduction to Hypothesis TestingDevore: Section 8.1March, 2011Page 1Statistics 511: Statistical MethodsDr. LevinePurdue UniversitySpring 2011What is statistical hypothesis?• A statistical hypothesis is a claim about the value of aparameter(s) or about the form of a distribution as a whole.• As an example, consider a normal distribution with the mean µ.Then, the statement µ = .75 is a hypothesis.March, 2011Page 2Statistics 511: Statistical MethodsDr. LevinePurdue UniversitySpring 2011Null and Alternative Hypotheses• Usually, two contradictory hypotheses are under consideration.For example, we may have µ = .75 and µ 6= .75. Alternatively,for a probability of success of some binomial distribution, wemay have p ≥ .10 and p ≤ .10.1. The null hypothesis H0is the one that is initially assumed tobe true.2. The alternative hypothesis Hais the assertion contrary toH0.March, 2011Page 3Statistics 511: Statistical MethodsDr. LevinePurdue UniversitySpring 2011• We reject the null hypothesis in favor of the alternativehypothesis if the sample evidence suggests so. If the sampledoes not contradict H0, we continue to believe it is true.• Thus, the two possible conclusions from a hypothesis-testinganalysis are reject H0or fail to reject H0.March, 2011Page 4Statistics 511: Statistical MethodsDr. LevinePurdue UniversitySpring 2011An Example of the Test• A test of hypothesis is a method for using sample data to decidewhether the null hypothesis should be rejected.• How exactly do we formulate a test? It depends on what ourgoals are...• Consider a company that wants to introduce an expensive newproduct to its line-up of existing ones. Clearly, there has to bean extensive evidence in favor of this new product. If it is, forexample, a new type of the lightbulb, we need to ensure that itsaverage lifetime is much longer than the one for existing typesbefore adopting it.March, 2011Page 5Statistics 511: Statistical MethodsDr. LevinePurdue UniversitySpring 2011• A reasonable test would be to test H0: µ = a vs. Ha: µ > awhere a is some predetermined threshold.• Clearly, the alternatives Ha: µ < a or H0: µ 6= a are of nointerest in this case.• Ha: µ < a and Ha: µ > a are called one-sided alternatives;H0: µ 6= a is called a two-sided alternative.• The value a that separates null hypothesis from an alternative iscalled a null value.March, 2011Page 6Statistics 511: Statistical MethodsDr. LevinePurdue UniversitySpring 2011Testing Procedure• A test procedure is specified by1. A test statistic, a function of the sample data on which thedecision will be based2. A rejection region, a set of all test statistic values for whichH0will be rejected (null hypothesis rejected iff(=if and onlyif) the test statistic value falls in this region.)March, 2011Page 7Statistics 511: Statistical MethodsDr. LevinePurdue UniversitySpring 2011Example• Example. Consider the claim about the average nicotine contentof a cigarette brand being at most 1.5 mg. In this case, the bestsetup would be to test H0: µ = 1.5 vs. Ha: µ > 1.5. Why?We only care if this content is exceeded!• Let¯X be a sample average nicotine content. Then, evidenceagainst H0would be provided by ¯x > 1.5.• Note that the choice of the rejection region is somewhatarbitrary...We could have selected ¯x > 1.55 as a rejectionregion.March, 2011Page 8Statistics 511: Statistical MethodsDr. LevinePurdue UniversitySpring 2011Error Types• Example It is possible that for some sample ¯x = 1.8 evenwhen H0is true.• A Type I error consists of rejecting the null hypothesis H0whenit is true• Example It is also possible that ¯x = 1.5 for a particular sampleeven if H0is false• A Type II error involves not rejecting H0when it is falseMarch, 2011Page 9Statistics 511: Statistical MethodsDr. LevinePurdue UniversitySpring 2011• The only way to get rid of both errors is to use the entirepopulation! In reality, a different procedure has to be followed.• Assume that 25% of the time automobiles have no visibledamage in 10mph crash tests. Denote p the proportion of all 10mph crashes that results in no visible damage to the newbumper. Then, H0: p = .25 vs. Ha: p > .25. Theexperiment is based on n = 20 independent crashes withprototype of the new design.March, 2011Page 10Statistics 511: Statistical MethodsDr. LevinePurdue UniversitySpring 2011Type I Error analysis• Consider the following procedure:1. Test Statistic is X - the number of crashes with no visibledamage2. Rejection region R8= {8, 9, . . . , 20}; in other words,reject H0if x ≥ 8.• Thus, the probability of Type I error isα = P ( Type I Error ) = P (X ≥ 8 when X ∼ Bin(20, .25))= 1 − B(7; 20, .25) = .102March, 2011Page 11Statistics 511: Statistical MethodsDr. LevinePurdue UniversitySpring 2011Type II Error Analysis• In contrast to Type I Error, there is no single β; instead, thereare different β’s for different values of p• Suppose the true value of p is p = 0.3. Then,β(.3) = P ( Type II Error when p = 0.3)= P (X ≤ 7 when X ∼ Bin(20, .3))= B(7; 20, .3) = .772• It is easy to understand that β decreases as p grows moredifferent from the null value .25March, 2011Page 12Statistics 511: Statistical MethodsDr. LevinePurdue UniversitySpring 2011• Now consider a different rejection regionR9= {9, 10, . . . , 20}.• Since X ∼ Bin(20, p), we haveα = P (H0rejected when p = .25 )= P (X ≥ 9 when X ∼ Bin(20, .25)) = 1 − B(8; 20, .25) = .041• Note that the Type I error probability has gone down.March, 2011Page 13Statistics 511: Statistical MethodsDr. LevinePurdue UniversitySpring 2011• At the same time,β(.3) = P (H0is not rejected when X ∼ Bin(20, .3))= P (X ≤ 8 when X ∼ Bin(20, .3))= B(8; 20, .3) = .887which is larger than before. Think of it as an equilibrium...March, 2011Page 14Statistics 511: Statistical MethodsDr. LevinePurdue UniversitySpring 2011Proposition• If an experiment and a sample size are fixed, decreasing thesize of the rejection region to obtain a smaller value of α alwaysresults in a larger value of β for any parameter value consistentwith with the alternative hypothesis Ha.• The usual approach is to specify the largest value of α that canbe tolerated and find a rejection region for it. This makes β assmall as possible subject to the bound on α. Such a value of αis called the significance level of the test.• Traditional choices