D.17 Answers: Sampling distributions

Answers to exercises in Sect. 17.12.

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 $0.6700-0.2514=0.486$, 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 is 0.2709, or about 27.1%. 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 17.4: 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 $0.28\times 60=16.8$ 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.