Tests of Hypotheses Using Statistics

1 Tests of Hypotheses Using StatisticsAdam Massey and Steven J. Miller Mathematics DepartmentBrown UniversityProvidence, RI 02912 AbstractWe present the various methods of hypothesis testing that one typically encounters in amathematical Statistics course. The focus will be on conditions for Using each test, the hypothesistested by each test, and the appropriate (and inappropriate) ways of Using each test. Weconclude by summarizing the different Tests (what conditions must be met to use them, whatthe test statistic is, and what the critical region is).Contents1 Types of Hypotheses and Test Introduction .. Types of Hypotheses .. Types of Statistics .. 32z- Tests Testing Means I: Large Sample Size or Known Variance .. Testing Means II: Small Sample Size and Unknown Variance .. 93 Testing the Variance124 Testing Testing Proportions I: One Proportion .. Testing Proportions II: K Proportions .. Testingr cContingency Tables .. Incompleter cContingency Tables Tables.

2 185 Normal Regression Analysis196 Non-parametric Tests of Signs .. Tests of Ranked Signs .. Tests Based on Runs .. 23 .. Tests comparing means .. Variance Test .. Proportions .. Contingency Tables .. Regression Analysis .. Signs and Ranked Signs .. Tests on Runs .. 311 Types of Hypotheses and Test IntroductionThe method of hypothesis testing uses Tests of significance to determine the likelihood that a state-ment (often related to the mean or variance of a given distribution) is true, and at what likelihoodwe would, as statisticians, accept the statement as true. While understanding the mathematicalconcepts that go into the formulation of these Tests is important, knowledge of how to appropri-ately use each test (and when to use which test) is equally important. The purpose here is on thelatter skill. To this end, we will examine each statistical test commonly taught in an introductorymathematical Statistics course, stressing the conditions under which one could use each test, thetypes of Hypotheses that can be tested by each test, and the appropriate way to use each test.

3 Inorder to do so, we must first understand how to conduct a statistical significance test (following thesteps indicated in [MM]), and we will then show how to adapt each test to this general begin by formulating the hypothesis that we want to test, called the alternative this hypothesis is derived from an attempt to prove an underlying theory (for example,attempting to show that women score, on average, higher on the SAT verbal section than men). Wedo this by testing against the null hypothesis, the negation of the alternative hypothesis ( Using oursame example, our null hypothesis would be that women do not, on average, score higher than menon the SAT verbal section). Finally, we set a probability level ; this value will be our significancelevel and corresponds to the probability that we reject the null hypothesis when it s in fact logic is to assume the null hypothesis is true, and then perform a study on the parameter inquestion. If the study yields results that would be unlikely if the null hypothesis were true (likeresults that would only occur with ), then we can confidently say the null hypothesisis not true and accept the alternative hypothesis.

4 Now that we have determined the hypothesesand the significance level, the data is collected (or in this case provided for you in the exercises).Once the data is collected, Tests of Hypotheses follow the following the sampling distribution of an appropriate test statistic, determine a critical region ofsize . the value of the test statistic from the sample whether the value of the test statistic falls within the critical region; if yes, we rejectthe null in favor of the alternative hypothesis, and if no, we fail to reject the null three steps are what we will focus on for every test; namely, what the appropriatesampling distribution for each test is and what test statistic we use (the third step is done bysimply comparing values). Types of HypothesesThere are two main types of Hypotheses we can test: one-tailed Hypotheses and two-tailed hypothe-ses. Our critical region will be constructed differently in each we wanted to test whether or not girls, on average, score higher than 600on the SAT verbal section.

5 Our underlying theory is that girls do score higher than 600, whichwould give us the following null (denotedH0) and alternative (denotedH1) Hypotheses :H0: 600H1: >600,( )where is the average score for girls on the SAT verbal section. This is an example of what is calleda one-tailed hypothesis. The name comes from the fact that evidence against the null hypothesiscomes from only one tail of the distribution (namely, scores above600). When constructing thecritical region of size , one finds a critical value in the sampling distribution so that the area underthe distribution in the interval(critical value, )is . We will explain how to find a critical valuein later instead that we wanted to see if girls scored significantly different than thenational average score on the verbal section of the SAT, and suppose that national average underlying theory is that girls do score significantly different than the national average, whichwould give us the following null and alternative Hypotheses :H0: = 500H1: 6= 500,( )where again is the average score for girls on the SAT verbal section.

6 This is an example of a two-tailed hypothesis. The name comes from the fact that evidence against the null hypothesis can comefrom either tail of the sampling distribution (namely, scores significantly above AND significantlybelow500can offer evidence against the null hypothesis). When constructing the critical regionof size , one finds two critical values (when assuming the null is true, we take one above themean and one below the mean) so that the region under the sampling distribution over the interval( , critical value1) (critical value2, )is . Often we choose symmetric regions so that thearea in the left tail is /2and the area in the right tail is /2; however, this is not required. Thereare advantages in choosing critical regions where each tail has equal will be several types of Hypotheses we will encounter throughout our work, but almostall of them may be reduced to one of these two cases, so understanding each of these types willprove to be critical to understanding hypothesis Types of StatisticsThere are many different Statistics that we can investigate.

7 We describe a common situation. LetX1, .. , XNbe independent identically distributed random variables drawn from a population withdensityp. This means that for eachi {1, .. , N}we have that the probability of observing avalue ofXilying in the interval [a, b] is justProb(Xi [a, b]) = bap(x)dx.( )3We often useXto denote a random variable drawn from this population andxa value of therandom variableX. We denote the mean of the population by and its variance by 2: = xp(x)dx=E[X] 2= (x )2p(x)dx=E[X2] E[X]2.( )IfXis in meters then the variance is in meters squared; the square root of the variance, called thestandard deviation, is in meters. Thus it makes sense that the correct scale to study fluctuationsis not the variance, but the square root of the variance. If there are many random variables withdifferent underlying distributions, we often add a subscript to emphasize which mean or standarddeviation we are some quantity we are interested in studying, we shall often study the related quantityY Mean(Y)StDev(Y)=Y Y Y.

8 ( )For example, ifY= (X1+ +XN)/N, thenYis an approximation to the mean. If we observevaluesx1, .. , xNforX1, .. , XN, then the observed value of the sample mean isy= (x1+ +xN)/N. We have (assuming the random variables are independently and identically distributedfrom a population with mean Xand standard deviation X), that Y=E[Y]=E(1NN i=1Xi)=1NN i=1E[Xi]=1N N X= X,( )and 2Y= Var(Y)= Var(1NN i=1Xi)=1N2N i=1 Var(Xi)=1N2 NVar(X) = 2XN;( )thus Y= StDev(Y) = X/ N.( )Thus, asN , we see thatYbecomes more and more concentrated about X; this is becausethe mean ofYis Xand its standard deviation is X/ N, which tends to zero withN. If webelieve X= 5, say, then forNlarge the observed value ofYshould be close to 5. If it is, this4provides evidence supporting our hypothesis that the population has mean 5; if it does not, thenwe obtain evidence against this it is imperative that we know what the the distribution ofYis. While the exact distrib-ution ofYis a function of the underlying distribution of theXi s, in many cases the Central LimitTheorem asserts thatYis approximately normally distributed with mean 0 and variance 1.

9 Thisis trivially true if theXiare drawn from a normal distribution; for more general distributions thisapproximation is often fairly good forN example is typical of the Statistics we shall study below. We have some random variableYwhich depends on random variablesX1, .. , XN. If we observe values ofx1, .. , xNfor theX1, .. , XN, we say these are the sample values. Given these observations we calculate the valueofY; in our case above whereY= (X1+ +XN)/Nwe would observey= (x1+ +xN) then normalizeYand look atZ=Y Mean(Y)StDev(Y)=Y Y Y.( )The advantage is thatZhas mean 0 and variance 1. This facilitates Using a table to analyze theresulting example, consider a normal distribution with mean 0 and standard deviation . Are wesurprised if someone says they randomly chose a number according to this distribution and observedit to be 100? We are if = 1, as this is over 100 standard deviations away from the mean; however,if = 1000 then we are not surprised at all. If we do not have any information about the scaleof the fluctuations, it is impossible to tell if something is large or small we have no basis forcomparison.

10 This is one reason why it is useful to study Statistics such asZ= (Y Y)/ Y,namely wemustdivide by the standard reason why it is useful to study quantities such asZ= (Y Y)/ Yis thatZhasmean 0 and variance 1. This allows us to create justonelookup table. If we just studiedY Y,we would need a lookup table for each possible standard deviation. This is similar to logarithmtables. It is enough to have logarithm tables in one base because of the change of base formula:logbx=logcxlogcb.( )In particular, if we can calculate logarithms baseewe can calculate logarithms inanybase. Theimportance of this formula cannot be overstated. It reduced the problem of tabulating all logarithms(with any base!) to just finding logarithms in one the probability of observing a value of 100 or larger if it is drawn froma normal distribution with mean 0 and variance 1. One may approximate the integrals directly, oruse Chebyshev s Testing Means I: Large Sample Size or Known VarianceThe first type of test we explore is the most basic: testing the mean of a distribution in which wealready know the population variance 2.