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6.35. (a) at least 97 (b) at least 68 (c) at least 167 (d) at least 225 6.36. (a) at least 385 (b) at least 271 (c) at least 666 (d) at least 900 6.37. (a) 7.38 6.39. (a) 0.298 6.40. (a) 0.70 0.82 oz (b) 7.38 0.030 second 0.12, 0.69 1.16 oz 6.38. (a) 7.38 0.049 second 0.15, 0.68 0.15 (c) 0.70 0.18, 0.67 0.17 0.70 oz (b) 7.38 0.96 oz

.MODEL Statement The parameters of a passive element can be varied by using .MODEL statement. The syntax is :MODEL hnamei htypei hparameteri hvaluei For passive linear elements, htypei is

(b) 0.298 (b) 0.70

6.41. (a) at least 323 (b) at least 560 (c) at least 756 6.43. (a) 1.07 6.44. (a) 0.045 6.45. (a) 63.8 6.46. (a) 1800 0.09 hours (b) 1.07 0.12 hours 0.097 (c) 0.045 0.31 oz 328 lb (c) 1800 382 lb 0.112

We can sweep the parameter of the model though a desired range at desired steps by using the .STEP statement: :STEP LIN hnamei hinitial valuei hfinal valuei hstep sizei

0,1,0 *

0.073 (b) 0.045 0.24 oz (b) 63.8

As an example, the following two statements use .MODEL and .STEP commands to de ne a resistor called heater with the resistance parameter varying from 20 to 40

249 lb (b) 1800

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6.47. (a) at least 4802 (b) at least 8321 (c) at least 11,250 6.48. (a) 87.0 to 230.9 hours 6.49. (a) 95.6 to 170.4 hours 6.50. (a) 106.1 to 140.5 hours 6.51. (a) 0.269 to 7.70 (b) 78.1 to 288.5 hours (b) 88.9 to 190.8 hours (b) 102.1 to 148.1 hours 6.52. (a) 0.519 to 2.78 (b) 0.673 to 2.14

.MODEL .STEP heater RES RES(R 20 heater(R) 20 40 5

6.57. k

Fig. 15-16

n ln (x1 c xn) 3.06 (c) 2.79 (d) 2.75 (e) 2.70

4,4,4 *

6.58. (a)

CHAP. 15]

4.60 (b)

EXAMPLE 15.17 Use Probe to plot V in the circuit in Fig. 15-17(a) for f varying from 1 to 3 kHz in 100 steps. also, R from 500

6.59. (a) 2400

45 lb, 2400

. Using .MODEL command we create the resistor RLeak and sweep its value by .STEP in the following source le. The graph of the frequency response V versus f is plotted by using Probe and it is shown in Fig. 15-17(b). Parallel resonance with variable R, Fig. I 0 1 AC R 1 0 RLeak L 1 0 10 m C 1 0 1u .MODEL RLeak RES(R 1 .STEP LIN RES .AC LIN 100 1k .PROBE .END 15-17 1m 0 1

59 lb (b) 87.6%

Fig. 15-17

Very often in practice we are called upon to make decisions about populations on the basis of sample information. Such decisions are called statistical decisions. For example, we may wish to decide on the basis of sample data whether a new serum is really effective in curing a disease, whether one educational procedure is better than another, or whether a given coin is loaded.

* * * * * * * * * * * * * * * * * * *

In attempting to reach decisions, it is useful to make assumptions or guesses about the populations involved. Such assumptions, which may or may not be true, are called statistical hypotheses and in general are statements about the probability distributions of the populations. For example, if we want to decide whether a given coin is loaded, we formulate the hypothesis that the coin is fair, i.e., p 0.5, where p is the probability of heads. Similarly, if we want to decide whether one procedure is better than another, we formulate the hypothesis that there is no difference between the procedures (i.e., any observed differences are merely due to fluctuations in sampling from the same population). Such hypotheses are often called null hypotheses or simply hypotheses, are denoted by H0. Any hypothesis that differs from a given null hypothesis is called an alternative hypothesis. For example, if the null hypothesis is p 0.5, possible alternative hypotheses are p 0.7, p 2 0.5, or p 0.5. A hypothesis alternative to the null hypothesis is denoted by H1.

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