
Concept explainers
The length of a Colorado brook trout is
a.

Find the probability that the length of a brook trout exceeds the mean.
Answer to Problem 84CE
Theprobability that the length of a brook trout exceeds the meanis 0.5.
Explanation of Solution
Calculation:
It is given that the lengths of a population of brook trout are normally distributed.
Normal distribution:
A continuous random variable X is said to follow normal distribution if the probability density function of X is,
Denote X as the length of a randomly selected brook trout. It is taken from a population of brook trout that are normally distributed.
Denote
The normal distribution is symmetrically distributed about its mean. From the concept of symmetry of a random variable X about
Thus, the probability that the length of a brook trout exceeds the mean is:
Hence, the probability that the length of a brook trout exceeds the mean is 0.5.
b.

Find the probability that the length of a brook trout exceeds the mean by at least 1 standard deviation.
Answer to Problem 84CE
Theprobability that the length of a brook trout exceeds the mean by at least 1 standard deviationis 0.1587.
Explanation of Solution
Calculation:
Empirical Rule:
The Empirical Rule for a Normal model states that:
- • Within 1 standard deviation of mean, 68.26% of all observations will lie.
- • Within 2 standard deviations of mean, 95.44% of all observations will lie.
- • Within 3 standard deviations of mean, 99.73% of all observations will lie.
Consider the property regarding 1 standard deviation difference from mean of theEmpirical Rule. It indicates that:
Hence, the probability that the length of a brook trout exceeds the mean by at least 1 standard deviationis 0.1587.
c.

Find the probability that the length of a brook trout exceeds the mean by at least 2 standard deviations.
Answer to Problem 84CE
Theprobability that the length of a brook trout exceeds the mean by at least 2 standard deviations is 0.0228.
Explanation of Solution
Calculation:
Consider the property regarding 2 standard deviations difference from mean of theEmpirical Rule. It indicates that:
Hence, the probability that the length of a brook trout exceeds the mean by at least 2 standard deviations is 0.0228.
d.

Find the probability that the length of a brook trout is within 2 standard deviations of the mean.
Answer to Problem 84CE
Theprobability that the length of a brook trout is within2 standard deviations of the mean is 0.9544.
Explanation of Solution
Calculation:
Consider the property regarding 2 standard deviations difference from mean of theEmpirical Rule. It indicates that:
Hence, the probability that the length of a brook trout is within 2 standard deviations of the mean is 0.9544.
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Chapter 7 Solutions
APPLIED STAT.IN BUS.+ECONOMICS
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- Glencoe Algebra 1, Student Edition, 9780079039897...AlgebraISBN:9780079039897Author:CarterPublisher:McGraw Hill
