Download Statistical Inference II: Hypothesis Testing & Interval Estimation for Unknown Variance and more Study notes Introduction to Econometrics in PDF only on Docsity! Statistical Inference II: The Principles of Interval Estimation and Hypothesis Testing 17 5. Hypothesis Tests About the Mean of a Normal Population When σ2 is Not Known In Section 3 we used the t-distribution as a basis for confidence interval estimation for the mean of a normal population when the population variance 2σ is not known. Similarly, when testing hypothesis, if 2σ is not known, we use a t-statistic. From (3.5) we know that ( )1~ˆ T b t t T − − β= σ (5.1) When testing the null hypothesis 0 :H cβ = against the alternative hypothesis 1 :H cβ ≠ the test statistic ( )1~ˆ T b c t t T − −= σ (5.2) if the null hypothesis is true. Following the same logic as in Section 4, we reject H0: β = c if |t| ≥ tc, or if the p-value is less than the level of significance α. The rejection rules and critical values from the t- distribution are shown in Figure 4. Figure 4 Rejection region for testing 0 :H cβ = against 1 :H cβ ≠ (5.1) An Empirical Example of a Two-Tailed Test Let us illustrate by testing the null hypothesis that the population hip size is 16 inches, against the alternative that it is not, using the hip data. We will follow the steps outlined in the testing format suggested in Section 4. 1. The null hypothesis is H0: β = 16. The alternative hypothesis is H1: β ≠ 16. Statistical Inference II: The Principles of Interval Estimation and Hypothesis Testing 18 2. The test statistic ( 1) 16 ~ ˆ T b t t T − −= σ if the null hypothesis is true. 3. Let us select α=.05. The critical value tc is 2.01 for a t-distribution with (T−1) = 49 degrees of freedom. Thus we will reject the null hypothesis in favor of the alternative if 2.01 or 2.01t t≥ ≤ − , or equivalently , if | | 2.01t ≥ 4. Using the hip data, the least squares estimate of β is b = 17.158, with estimated variance 2ˆ 3.267σ = , so ˆ 1.807σ = . The value of the test statistic is 17.158 16 4.531 1.807 50 t −= = . 5. Conclusion: Since t=4.531 > tc=2.01 we reject the null hypothesis. The sample information we have is incompatible with the hypothesis that β = 16, or, that the population mean hip size is 16 inches. Equivalently, for this test the p-value is p=.000038 < α=.05 and on this basis we reject the null hypothesis. (5.2) A Relationship Between Hypothesis Testing and Interval Estimation There is an algebraic relationship between two-tailed hypothesis tests and confidence interval estimates that is sometimes useful. Suppose that we are testing the null hypothesis 0 :H cβ = against the alternative 1 :H cβ ≠ . If we fail to reject the null hypothesis at the α level of significance, then the value c will fall within a (1−α)×100% confidence interval estimate of β. Conversely, if we reject the null hypothesis, then c will fall outside the (1−α)×100% confidence interval estimate of β. This algebraic relationship is true because we fail to reject the null hypothesis when − ≤ ≤t t tc c , or when ˆ c c b c t t T −− ≤ ≤ σ which, when rearranged becomes ˆ ˆ c cb t c b t T T σ σ− ≤ ≤ + The endpoints of this interval are the same as the endpoints of a (1−α)×100% confidence interval estimate of β. Thus for any value of c within the interval we do not reject 0 :H cβ = against the alternative
