答案和详解如下： Q1. 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%. Correct answer is C) 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%. Q2. In a skewed distribution, what is the minimum proportion of observations between +/- two standard deviations from the mean? A) 95%. B) 75%. C) 84%. Correct answer is B) 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. Q2. 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) 75%. C) 89%. Correct answer is B) 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%. Q3. 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) 95%. C) 56%. Correct answer is C) 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%. Q4. 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%. Correct answer is A) 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%. | 
发表于 2009-1-8 17:42
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