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9.42. There is no significant difference in scholastic achievement due to differences in height, hair color, or birthplace. 9.43. There are significant differences in species and quantities of the first chemical but no other significant differences. 9.44. There are significant differences in types of cables but no significant differences in cable strengths due to operators, machines, or companies. 9.45. There is no significant difference in treatments at either level. 9.46. There is no significant difference in IQ scores at either level. 9.47. There are significant differences in examination scores due to both veteran status and IQ at the 0.05 level. 9.48. At the 0.01 level the differences in examination scores due to veteran status are not significant, but those due to IQ are significant. 9.49. There are no significant differences in test scores of students from different parts of the country, but there are significant differences in test scores due to IQ. 9.50. Same answer as Problem 9.49. 9.51. There is a significant difference due to chemicals or locations at the 0.05 level. 9.52. There are significant differences due to locations but not to fertilizers. 9.53. There are no significant differences due to locations or fertilizers. 9.54. There are no significant differences due to factor 1, factor 2, or treatments A, B, C. 9.55. There are no significant differences due to factors or treatments.

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We assume that there are only 365 days in a year and that all birthdays are equally probable, assumptions which are not quite met in reality. The first of the n people has of course some birthday with probability 365 > 365 1. Then, if the second is to have a different birthday, it must occur on one of the other 364 days. Therefore, the probability that the second person has a birthday different from the first is 364 > 365. Similarly the probability that the third person has a birthday different from the first two is 363 > 365. Finally, the probability that the nth person has a birthday different from the others is (365 n l) > 365. We therefore have P(all n birthdays are different) 365 364 363 c 365 n 365 365 365 365 a1 1 b a1 365 2 b c a1 365 1 n 1 b 365

! ! !0 Consequently,

1.46. Determine how many people are required in Problem 1.45 to make the probability of distinct birthdays less than 1 > 2.

212 14.3.4

Denoting the given probability by p and taking natural logarithms, we find (1) ln p ln a1 1 b 365 ln a1 2 b 365 c ln a1 n 1 b 365

2 The energy dissipated per cycle (in the resistor) is Wd Imax R =!. W 1=!CR ! !0 Q 2 s max !L=R ! ! !0 Wd

2, 3, . . . (Appendix A, formulas 1 and 2) (n 1) n(n 2 1) , 12 22 c (n 1)2 n(n 1)(2n 6 1)

Fig. 12-37 (b) For the parallel combination with applied voltage v t , 1 2 1 2 q Ws LIL 2 2C C and If v 0, then qC 0 and iL IL max Vmax !L dWs di q LiL L C iC v iL iC 0 dt dt C

For n small compared to 365, say, n 30, the second and higher terms on the right of (5) are negligible compared to the first term, so that a good approximation in this case is (6) In p n(n 1) 730

[&!ln!p *frac*{n(n-1)}{730}&]

2 Vmax 2L!2

1 2,

2 Vmax =R!.

OCL has many operations on collections: size, count(object), includes (object), sum, and includesall(collection).

n(n 1) 730

so that n 23. Our conclusion therefore is that, if n is larger than 23, we can give better than even odds that at least 2 people will have the same birthday.

A three-element series circuit contains R 10 , L 5 mH, and C 12:5 mF. Plot the magnitude and angle of Z as functions of ! for values of ! from 0.8 !0 through 1:2 !0 .

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