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.