Statistial Inferene II: he Priniples of Interval Estimation and Hypothesis esting 5. Hypothesis ests About the Mean of a Normal Population When σ is Not Known In Setion 3 we used the t-distribution as a basis for onfidene interval estimation for the mean of a normal population when the population variane σ is not known. Similarly, when testing hypothesis, if σ is not known, we use a t-statisti. From (3.5) we know that b β t = ~ t( 1) (5.1) When testing the null hypothesis H : 0 β = against the alternative hypothesis H : 1 β the test statisti b t = ~ t( 1) (5.) if the null hypothesis is true. Following the same logi as in Setion 4, we rejet H 0 : β = if t t, or if the p-value is less than the level of signifiane α. he rejetion rules and ritial values from the t- distribution are shown in Figure 4. Figure 4 Rejetion region for testing H : 0 β = against H : β 1 (5.1) An Empirial Example of a wo-ailed est Let us illustrate by testing the null hypothesis that the population hip size is 16 inhes, against the alternative that it is not, using the hip data. We will follow the steps outlined in the testing format suggested in Setion 4. 1. he null hypothesis is H 0 : β = 16. he alternative hypothesis is H 1 : β 16. 17
Statistial Inferene II: he Priniples of Interval Estimation and Hypothesis esting b 16. he test statisti t = ~ t( 1) if the null hypothesis is true. 3. Let us selet α=.05. he ritial value t is.01 for a t-distribution with ( 1) = 49 degrees of freedom. hus we will rejet the null hypothesis in favor of the alternative if t.01 or t.01, or equivalently, if t.01 4. Using the hip data, the least squares estimate of β is b = 17.158, with estimated variane so σ= ˆ 1.807. he value of the test statisti is 17.158 16 t = = 4.531. 1.807 50 σ ˆ = 3.67, 5. Conlusion: Sine t=4.531 > t =.01 we rejet the null hypothesis. he sample information we have is inompatible with the hypothesis that β = 16, or, that the population mean hip size is 16 inhes. Equivalently, for this test the p-value is p=.000038 < α=.05 and on this basis we rejet the null hypothesis. (5.) A Relationship Between Hypothesis esting and Interval Estimation here is an algebrai relationship between two-tailed hypothesis tests and onfidene interval estimates that is sometimes useful. Suppose that we are testing the null hypothesis H : 0 β = against the alternative β. If we fail to rejet the null hypothesis at the α level of signifiane, then the value will fall within a (1 α) 100% onfidene interval estimate of β. Conversely, if we rejet the null hypothesis, then will fall outside the (1 α) 100% onfidene interval estimate of β. his algebrai relationship is true beause we fail to rejet the null hypothesis when t t t, or when b t t whih, when rearranged beomes b t b+ t he endpoints of this interval are the same as the endpoints of a (1 α) 100% onfidene interval estimate of β. hus for any value of within the interval we do not rejet H : 0 β = against the alternative 18
Statistial Inferene II: he Priniples of Interval Estimation and Hypothesis esting β. For any value of outside the interval we rejet H : 0 β = and aept the alternative β. his relationship an be handy if you are given only a onfidene interval and want to determine what the outome of a two-tailed test would be. You an verify that the test statistis z in able 4 lead us to rejet the null hypothesis for the same samples in able whih do not ontain the value β = 10. We have two omments about this relationship between interval estimation and hypothesis testing: 1. he relationship is between onfidene intervals and two-tailed tests. It does not apply to one-tailed tests.. A onfidene interval is an estimation tool; that is, it is an interval estimator. A hypothesis test about one or more parameters is a ompletely separate form of inferene, with the only onnetion being that the test statisti inorporates the least squares estimator. o test hypotheses you should arry out the steps outlined in Setion 4 and should not ompute and report an interval estimate. (5.3) One-ailed ests We have foused so far on testing hypotheses of the form H 0 : β = against the alternative H 1 : β. his kind of test is alled a two-tailed test sine portions of the rejetion region are found in both tails of the test statisti s distribution. One-tailed tests are used to test H 0 : β = against the alternative H 1 : β >, or H 1 : β <. he logi of one-tailed tests is idential to that for the two-tailed tests that we have studied. he test statisti is the same, and given by (5.). What is different is the seletion of the rejetion region and the omputation of the p-values. For example, to test H 0 : β = against the alternative H 1 : β > we selet the rejetion region to be values of the test statisti t that support the alternative hypothesis and whih are unlikely if the null hypothesis is true. Large values of the t-statisti are unlikely if the null hypothesis is true. We define the rejetion region to be values of t greater than a ritial value t, from a t-distribution with 1 degrees of freedom, suh that P( t ) =α, where α is the level of signifiane of the test and the probability of a ype I error. See Figure 5 below. t 19
Statistial Inferene II: he Priniples of Interval Estimation and Hypothesis esting Figure 5 Critial Value for One-ailed est H 0 : β = vs. H 1 : β > he deision rule for this one-tailed test is, Rejet H 0 : β = and aept the alternative H 1 : β > if t t. If t < t then we do not rejet the null hypothesis. Computation of the p-value is similarly onfined to one tail of the distribution of the test statisti, though its interpretation is exatly as before. For testing H 0 : β = against the alternative H 1 : β > the p- value is omputed by finding the probability that the test statisti is greater than or equal to the omputed sample value of the test statisti. (5.4) An Empirial Example of a One-ailed est Let us illustrate by testing the null hypothesis that the population hip size is 16 inhes, against the alternative that it is greater than 16 inhes, using the hip data. We will follow the steps outlined in the testing format suggested in Setion 4. 6. he null hypothesis is H 0 : β = 16. he alternative hypothesis is H 1 : β > 16 b 16 7. he test statisti t = ~ t( 1) if the null hypothesis is true. 8. Let us selet α=.05. he ritial value t is 1.68 for a t-distribution with ( 1) = 49 degrees of freedom. hus we will rejet the null hypothesis in favor of the alternative if t 1.68. 9. Using the hip data, the least squares estimate of β is b = 17.158, with estimated variane so σ= ˆ 1.807. he value of the test statisti is 17.158 16 t = = 4.531. 1.807 50 σ ˆ = 3.67, 10. Conlusion: Sine t=4.531 > t =1.68 we rejet the null hypothesis. he sample information we have is inompatible with the hypothesis that β = 16, or, that the population mean hip size is 16 inhes. hus we aept the alternative that the population mean hip size is greater than 16 inhes, based on 0
Statistial Inferene II: he Priniples of Interval Estimation and Hypothesis esting our test whih has α=.05 probability of a ype I error. Equivalently, for this test the p-value is p=.000019 < α=.05 and on this basis we rejet the null hypothesis. (5.5) A Comment on Stating Null and Alternative Hypotheses We have noted in the previous setions that a statistial test proedure annot prove the truth of a null hypothesis. When we fail to rejet a null hypothesis, all the hypothesis test an establish is that the information in a sample of data is ompatible with the null hypothesis. On the other hand, a statistial test an lead us to rejet the null hypothesis, with only a small probability, α, of rejeting the null hypothesis when it is atually true. hus rejeting a null hypothesis is a stronger onlusion than failing to rejet it. he null hypothesis is usually stated in suh a way that if our theory is orret, then we will rejet the null hypothesis. For example, our airplane seat designer has been operating under the assumption (the maintained or null hypothesis) that the population mean hip width is 16 inhes. Casual observation suggests that people are getting larger all the time. If we are larger, and if the airline wants to ontinue aommodate the same perentage of the population, then the seat widths must be inreased. his ostly hange should be undertaken only if there is statistial evidene that the population size is indeed larger. When using a hypothesis test we would like to find out that there is statistial evidene against our urrent theory, or if the data are ompatible with it. With this goal, we set up the null hypothesis that the population mean is 16 inhes, H 0 : β = 16against the alternative that it is greater than 16 inhes, H 1 : β > 16. You may view the null hypothesis to be too limited in this ase, sine it is feasible that the population mean hip width is now smaller than 16 inhes, or that β < 16. he hypothesis testing proedure for the testing the null hypothesis that H 0 : β 16 against the alternative hypothesis H 1 : β > 16 is exatly the same as testing H 0 : β = 16 against the alternative hypothesis H 1 : β > 16. he test statisti, rejetion region and p-value are exatly the same. For a one-tailed test you an form the null hypothesis in either of these ways. What ounts is that the alternative hypothesis is properly speified. Finally, it is important to set up the null and alternative hypotheses before you arry out the regression analysis. Failing to do so an lead to errors in formulating the alternative hypothesis. Suppose that we wish to test whether β > 16 and the least squares estimate of β is b= 15.5. Does that mean we should set up the alternative β < 16, to be onsistent with the estimate? he answer is no. he alternative is formed to state the onjeture that we wish to establish, β > 16. 1