Answer to Exercise 17.1: 1. $$z=(8 - 8.8)/2.7 = 0.2962$$, or $$z=-0.30$$. From tables, the probability is 0.3821, or about 38.2%. 2. $$z= 0.07$$; probability is $$1 - 0.52379 = 0.4721$$, or about 47.2%. 3. The $$z$$-scores are $$z_1 = -0.67$$ and $$z_2 = 0.44$$; the probability is $$0.6700 - 0.2514 = 0.4186$$, or about 41.9%. (Draw a diagram!) 4. Using the tables ‘backwards’: $$z$$-score is about 1.04; corresponding tree diameter is $$x = 8.8 + (1.04\times 2.7) = 11.608$$, or about 11.6 inches. About 15% of tress will have diameters larger than about 11.6 inches.
Answer to Exercise 17.2: 1. $$z = (39 - 40)/1.64 = -0.6097561$$, or $$z=-0.61$$. Using tables: probability less than this value of $$z$$ is 0.2709, so the answer is $$1 - 0.2709 =0.7291$$, or about 72.9%. 2. $$z = (37 - 40)/1.64 = -1.83$$; probability is 0.0336, about 3.4%. 3. The two $$z$$-scores: $$z_1=-4.878$$ and $$z_2 = -1.83$$. Drawing a diagram, probability is $$0.0336 - 0 = 0.0336$$, or about 3.4%. 4. The $$z$$-score: 1.64 (or 1.65). Gestation length: $$x= 40 + (1.64 \times 1.64) = 42.7$$ (same answer to one decimal place using $$z=1.65$$). 5% of gestation lengths longer than about 42.7 weeks. 5. $$z$$-score is -1.64 (or -1.65). Gestation length: $$x= 40 + (-1.64 \times 1.64) = 37.3$$ (same answer to one decimal place using $$z=-1.65$$). 5% of gestation lengths shorter than about 37.3 weeks.
Answer to Exercise 17.3: $$z$$-score: about $$z=2.05$$. Corresponding IQ: $$x = 100 + (2.05\times 15) = 130.75$$. An IQ greater than about 130 is required to join Mensa.
Answer to Exercise 17.7: Be very careful: work with the number of minutes from the mean, or from 5:30pm. The standard deviation already is in decimal, but converted to minutes, standard deviation is 120 minutes, plus $$0.28\times 60 = 16.8$$ minutes. The standard deviation is 136.8 minutes.
1. 9pm is 3 hours and 30 minutes from 5:30pm: 210 minutes. $$z$$-score: $$z=(210 - 0)/136.8 = 1.54$$; probability: $$1 - 0.9382 = 0.0618$$, or about 6.2%. 2. $$z = (5 - 5.5)/2.28 = -0.22$$; probability: 0.4129\$, or about 41.3%. 3. $$z$$-scores are $$z_1 = -0.22$$ and $$z_2 = 0.22$$; probability: $$0.5871 - 0.4129 = 0.1742$$, or about 17.4%. 4. $$z$$-score is $$0.52$$; time is $$x = 0 + (0.52\times 136.8) = 71.136$$ minutes after 5pm; about one hour and 11 minutes after 5:30pm, or 6:41pm. 5. $$z$$-score: $$-1.04$$; time is $$x = 0 + (-1.04\times 136.8) = -141.272$$, or 141.272 minutes before 5pm; about two hours and 21 minutes before 5:30pm, or 3:09pm.