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 z1=−0.67 and z2=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+(1.04×2.7)=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: z1=−4.878 and z2=−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+(1.64×1.64)=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=(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 z1=−0.22 and z2=0.22; probability: 0.5871−0.4129=0.1742, or about 17.4%. 4. z-score is 0.52; time is x=0+(0.52×136.8)=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.