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发表于 2011-3-1 14:40 | 只看该作者

Reading 7: Statistical Concepts and Market Returns-LOS h 习题

2011, interpret, specified, standard, falling
Session 2: Quantitative Methods: Basic Concepts
Reading 7: Statistical Concepts and Market Returns

LOS h: Calculate and interpret the proportion of observations falling within a specified number of standard deviations of the mean using Chebyshev's inequality.

 

 

Assume a sample of beer prices is negatively skewed. Approximately what percentage of the distribution lies within plus or minus 2.40 standard deviations of the mean?

A)
 95.5%.
B)
 58.3%.
C)
 82.6%.

 

Use Chebyshev’s Inequality to calculate this answer. Chebyshev’s Inequality states that for any set of observations, the proportion of observations that lie within k standard deviations of the mean is at least 1 – 1/k2. We can use Chebyshev’s Inequality to measure the minimum amount of dispersion whether the distribution is normal or skewed. Here, 1 – (1 / 2.42) = 1 ? 0.17361 = 0.82639, or 82.6%.

 

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发表于 2011-3-1 14:40 | 只看该作者

In a skewed distribution, what is the minimum proportion of observations between +/- two standard deviations from the mean?

A)
 95%.
B)
 84%.
C)
 75%.

For any distribution we can use Chebyshev’s Inequality, which states that the proportion of observations within k standard deviations of the mean is at least 1 – (1 / k2).

1 – (1 / 22) = 0.75, or 75%.

Note that for a normal distribution, 95% of observations will fall between +/- 2 standard deviations of the mean.

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发表于 2011-3-1 14:41 | 只看该作者

Regardless of the shape of a distribution, according to Chebyshev’s Inequality, what is the minimum percentage of observations that will lie within +/– two standard deviations of the mean?

A)
 68%.
B)
 89%.
C)
 75%.

According to Chebyshev’s Inequality, for any distribution, the minimum percentage of observations that lie within k standard deviations of the distribution mean is equal to:
1 – (1 / k2), with k equal to the number of standard deviations. If k = 2, then the percentage of distributions is equal to 1 – (1 / 4) = 75%.

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发表于 2011-3-1 14:41 | 只看该作者

In a skewed distribution, what is the minimum amount of observations that will fall between +/- 1.5 standard deviations from the mean?

A)
 44%.
B)
 56%.
C)
 95%.

Because the distribution is skewed, we must use Chebyshev’s Inequality, which states that the proportion of observations within k standard deviations of the mean is at least 1 – (1 / k2).

1 – (1 / 1.52) = 0.5555, or 56%.

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发表于 2011-3-1 14:41 | 只看该作者

According to Chebyshev’s Inequality, for any distribution, what is the minimum percentage of observations that lie within three standard deviations of the mean?

A)
 89%.
B)
 94%.
C)
 75%.

According to Chebyshev’s Inequality, for any distribution, the minimum percentage of observations that lie within k standard deviations of the distribution mean is equal to: 1 – (1 / k2). If k = 3, then the percentage of distributions is equal to 1 – (1 / 9) = 89%.

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