# Normal Distributions

## Exercises

1. A forest products company claims that the amount of usable lumber in its harvested trees averages 172 cubic feet and has a standard deviation of 12.4 cubic feet. Assume that these amounts have approximately a normal distribution.
1. What proportion of trees contain more than 150 cubic feet?
2. What proportion of trees contain between 175 and 190 cubic feet?
2. At two years of age, sardines inhabiting Japanese waters have a length distribution that is approximately normal with mean 20.20 cm and standard deviation 0.65 cm.
1. What proportion of these sardines are between 19.0 and 21.0 cm long?
2. How long are the longest 15% of all these sardines?
3. How short are the shortest 15% of all these sardines?
4. In what range do the middle 70% of all sardines fall?
3. The mean cholesterol level in children is 175 mg/dL with standard deviation 35 mg/dL. Assume this level varies from child to child according to an approximate normal distribution.
1. What percentage of children have a cholesterol level above 200 mg/dL?
2. How high are the levels for the highest 2% of all children?
4. Porphyrin is a pigment in blood protoplasm and other body fluids that is significant in body energy and storage. In healthy Alaskan brown bears, the amount of porphyrin in the bloodstream (in mg/dL) has approximate normal distribution with mean 47.5 and standard deviation 12.2.
1. What proportion of these bears have between 27.5 and 67.5 mg/dL porphyrin in their bloodstream? (That’s within 20 of the mean amount of 47.5.)
2. How low are the porphyrin levels for the lowest 3% of all the bears?
5. Healthy 10-week-old domesticated kittens have average weight 24.5 oz. with a standard deviation of 5.25 oz. The distribution is approximately normal.
1. A kitten is designated as dangerously underweight when, at 10 weeks, it weighs less than 10.0 oz. What proportion of healthy kittens will designated as dangerously underweight?
2. What is the median weight of the kittens?
3. What are the first and third quartiles of the kitten weights? (25% of the kittens weigh less than Q1; 75% weigh more. 75% of the kittens weigh less than Q3; 25% weigh more.) What is the interquartile range (IQR) for the kitten weights?

## Solutions

1. a) 0.9616; b) 0.3317
2. a) 0.8585; b) above 20.88 cm; c) below 12.52 cm; d) between 19.52 and 20.88 cm
3. a) 23.89%; b) above 246.75 mg/dL
4. a) 0.8990; b) below 24.564 mg/dL
5. a) 0.0029; b) 24.5 oz; c) Q1 = 20.98 oz, Q3 = 28.02 oz, IQR = 7.04 oz.