D.17 Answers: Sampling distributions

Answers to exercises in Sect. 17.12.

Answer to Exercise 17.1: 1. z=(88.8)/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 10.52379=0.4721, or about 47.2%. 3. The z-scores are z1=0.67 and z2=0.44; the probability is 0.67000.2514=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+(1.04×2.7)=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=(3940)/1.64=0.6097561, or z=0.61. Using tables: probability less than this value of z is 0.2709, so the answer is 10.2709=0.7291, or about 72.9%. 2. z=(3740)/1.64=1.83; probability is 0.0336, about 3.4%. 3. The two z-scores: z1=4.878 and z2=1.83. Drawing a diagram, probability is 0.03360=0.0336, or about 3.4%. 4. The z-score: 1.64 (or 1.65). Gestation length: x=40+(1.64×1.64)=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+(1.64×1.64)=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+(2.05×15)=130.75. An IQ greater than about 130 is required to join Mensa.
Answer to Exercise 17.4: An IQ score lower than about 80.8 leads to a rejection by the US military.
Answer to Exercise 17.5: 1: C; 2: A; 3: B; 4: D.
Answer to Exercise 17.6: 1: A; 2: C; 3: B; 4: D.

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 0.28×60=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=(2100)/136.8=1.54; probability: 10.9382=0.0618, or about 6.2%. 2. z=(55.5)/2.28=0.22; probability: 0.4129$, or about 41.3%. 3. z-scores are z1=0.22 and z2=0.22; probability: 0.58710.4129=0.1742, or about 17.4%. 4. z-score is 0.52; time is x=0+(0.52×136.8)=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+(1.04×136.8)=141.272, or 141.272 minutes before 5pm; about two hours and 21 minutes before 5:30pm, or 3:09pm.