## Equivalence of Test Statistics### Problem StatementDemonstrate that test statistics (2.17) and (2.87) are equivalent.### BackgroundFor a test statistic \( (b_i - \beta_i) / s(b_i) \), with \( n - 2 \) degrees of freedom, tests concerning \( \beta_i \) use the t distribution.#### Two-Sided TestA cost analyst at the Toluca Company aims to explore if there's a linear relationship between work hours and lot size, i.e., whether \( \beta_1 = 0 \). The test alternatives are:- \( H_0: \beta_1 = 0 \)- \( H_a: \beta_1 \neq 0 \) \[ (2.16) \]The analyst intends to control the risk of a Type I error at \( \alpha = .05 \). The conclusion \( H_0 \) may be reached by examining the 95% confidence interval for \( \beta_1 \) constructed previously, since this test does not require new computations.An explicit alternative test (2.16) based on test statistic:\[ t^* = \frac{b_i}{s(b_i)} \] \[ (2.17) \]The decision rule with this statistic to maintain the significance level at \( \alpha \):- If \( |t^*| \leq t(1 - \alpha/2; n - 2) \), conclude \( H_0 \)- If \( |t^*| > t(1 - \alpha/2; n - 2) \), conclude \( H_a \) \[ (2.18) \]### Point Estimator of \( \rho_{12} \)The maximum likelihood estimator of \( \rho_{12} \), denoted by \( r_{12} \), is given by:\[r_{12} = \frac{\sum (Y_{1i} - \bar{Y_1})(Y_{2i} - \bar{Y_2})}{[\sum (Y_{1i} - \bar{Y_1})^2 \sum(Y_{2i} - \bar{Y_2})^2]^{1/2}}\] \[ (2.84) \]### Test for \( \rho

t∗=b1sb1...............(2.17)t∗=r12n−21−r122........(2.87)

2.60. Show that test statistics (2.17) and (2.87) are equivalent. Since (b₁-B₁)/s(b) is distributed as t with n-2 degrees of freedom, tests concerning B₁ can be set up in ordinary fashion using the distribution. Two-Sided Test A cost analyst in the Toluca Company is interested in testing, using regression model (2.1), whether or not there is a linear association between work hours and lot size, i.e., whether or not = 0. The two alternatives then are: B₁ Ho: B₁=0 He: Br 0 (2.16) The analyst wishes to control the risk of a Type I error at ar= .05. The conclusion H, could be reached at once by referring to the 95 percent confidence interval for B, constructed earlier, since this interval does not include 0. An explicit test of the alternatives (2.16) is based on the test statistic: b₁ r (2.17) s(bi) The decision rule with this test statistic for controlling the level of significance at a is: If Ir st(1-a/2;n -2), conclude Ho If Ir">t(1-a/2;n-2), conclude He (2.18) Point Estimator of P12. The maximum likelihood estimator of P12, denoted by 12, is given by: 712 Σ(Υ - Pi)(Yaz - Pa) [Σ(Υ -- 8,2 Σ(Υz - Pa)|a Test whether P12=0. When the population is bivariate normal, it is frequently desired to test whether the coefficient of correlation is zero: Ho: P12-0 H₂: P120 (2.84) (2.86) The reason for interest in this test is that in the case where Y, and Y₂ are jointly normally distributed, P12 = 0 implies that Y, and Y₂ are independent. We can use regression procedures for the test since (2.80b) implies that the following alternatives are equivalent to those in (2.86): Ho: B12-0 He: B12 #0 (2.86a) and (2.82b) implies that the following alternatives are also equivalent to the ones in (2.86): (2.86b) It can be shown that the test statistics for testing either (2.86a) or (2.86b) are the same and can be expressed directly in terms of 12: Ho: B₂10 He: B21 40 12√√/M-2 VI- (2.87) If He holds, r* follows the 1(n-2) distribution. The appropriate decision rule to control the Type I error at or is: