[14.65 (modified)] An achievement test is said to be reliable if a student takes the test several times will consistently get high (or low) scores. One way of checking the reliability of a test is to divide it into two parts, usually the even-numbered problems and the odd-numbered problems, and observe the correlation between the scores that students gets in both halves of the test. Thus, the following data represent the grades, x and y, that 20 students obtained for the even-numbered problems and the odd-numbered problems ( = even-numbered problems and y = odd-numbered problems) of a new objective test designed to test eighth grade achievement in general science: 27 29 33 42 36 14 39 31 44 49 38 38 32 27 24 22 27 35 33 34 41 33 32 37 38 29 37 38 44 40 33 35 30 27 34 32 27 38 39 43 (a) Compute the least squares estimates of o and B. (b) Using the result from part (a), what score would we expect a student to obtain on the odd-numbered problems if they scored 36 on the even-numbered problems? (c) Construct a 95% confidence interval for the regression coefficient B. (d) Calculate and interpret, 72. (e) Use the formula from Exercise 14.31 to construct a 95% confidence interval for p. Does this result indicate that the test is reliable?The following formulas may help you complete the exercises: (n 2)5IF B + tax/ 2, n 2 0 t.= 8-Bo (n 2)5IF TI o=V.(Syy - BSIV) R = {t. : t. tan i) R = {t. : It.| >t x/2, n2 1) Pr(T > 1.812) = 0.05 if T ~ $10 Pr(T > 2.101) = 0.025 if T ~ t18