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(a) On the basis of a one-tailed test of Student s distribution at a 0.05 level, we would reject H0 if t t0.95 1.75 for 18 2 16 degrees of freedom. Therefore, we cannot reject H0 at a 0.05 level. (b) Since we cannot reject H0 at a 0.05 level, we certainly cannot reject it at a 0.01 level.

A pair of orthogonal axes, with the horizontal axis displaying Re z and the vertical axis j Im z, determine a complex plane in which each complex number is a unique point. Refer to Fig. A-1, on which six complex numbers are shown. Equivalently, each complex number is represented by a unique vector from the origin of the complex plane, as illustrated for the complex number z6 in Fig. A-1.

8.48. What is the minimum sample size necessary in order that we may conclude that a correlation coefficient of 0.32 is significantly greater than zero at a 0.05 level

12.2.4 METRIC 4: COUPLING BETWEEN OBJECT CLASSES (CBO)

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At a 0.05 level using a one-tailed test of Student s distribution, the minimum value of n must be such that 0.32 !n 2 !1 (0.32)2 For n For n For n 26, n 27, n 28, n 24, t0.95 25, t0.95 26, t0.95 28. t0.95 1.71, t 1.71, t 1.71, t for n 2 degrees of freedom (0.32)2 (0.32)2 (0.32)2 1.65. 1.69. 1.72.

Fig. A-1

0.32 !24> !1 0.32 !25> !1 0.32 !26> !1

Fig. A-2

8.49. A correlation coefficient based on a sample of size 24 was computed to be r 0.75. Can we reject the hypothesis that the population correlation coefficient is as small as (a) r 0.60, (b) r 0.50, at a 0.05 significance level

(a) Z 1.1513 log a 1 1 0.75 b 0.75 sZ The standardized variable is then z Z sZ mZ 0.9730 0.6932 0.2182 1.28 0.9730, 1 !n mZ 1.1513 log a 1 !21 0.2182 1 1 0.60 b 0.60 0.6932,

In addition to the de nition of j given in Section A1, it may be viewed as an operator which rotates any complex number (vector) A 908 in the counterclockwise direction. The case where A is a pure real number, x, is illustrated in Fig. A-2. The rotation sends A into jx, on the positive imaginary axis. Continuing, j 2 advances A 1808; j 3 , 2708; and j 4 , 3608. Also shown in Fig. A-2 is a complex number B in the rst quadrant, at angle . Note that jB is in the second quadrant, at angle 908.

At a 0.05 level of significance using a one-tailed test of the normal distribution, we would reject the hypothesis only if z were greater than 1.64. Therefore, we cannot reject the hypothesis that the population correlation coefficient is as small as 0.60. (b) If r 0.50, mZ 1.1513 log 3 0.5493 and z (0.9730 0.5493)>0.2182 reject the hypothesis that the population correlation coefficient is as small as r significance. 1.94. Therefore, we can 0.50 at a 0.05 level of

. Draw the simpli ed circuit and nd v for b 50; 100; 200. Note that v changes proportionally with b. Ans. v 0:5; 1; 2 V In the circuit of Fig. 3-25, R1 100

Coupling between modules has always been a concern (see Section 9.5). In objectoriented software, we can de ne coupling as the use of methods or attributes in another class. Two classes will be considered coupled when methods declared in one class use methods or instance variables de ned by the other class. Coupling is symmetric. If class A is coupled to class B, then B is coupled to A. The coupling between object classes (CBO) metric will be the count of the number of other classes to which it is coupled. This metric is calculated for each class.

8.50. The correlation coefficient between physics and mathematics final grades for a group of 21 students was computed to be 0.80. Find 95% confidence limits for this coefficient.

Since r 0.80 and n Z 21, 95% confidence limits for m2 are given by 1.1513 log a 1 1 r b r 1.96a 1 2n 3 b 1.0986 0.4620

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