Quantitative Methods - Quantitative Methods Section 2

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1. The return on equity for four hypothetical companies from a list of 100 companies is given below:

Little Wonder 10.5%
Genesis Ltd.16.25%
Moral Corp. 9.81%
Travis Ltd. 12.0%

  • Option : B
  • Explanation : Use the following keystrokes to calculate the sample standard deviation: [2nd] [DATA] [2nd] [CLR WRK] X01 = 10.5 X02 = 16.25 X03 = 9.81 X04 = 12 s represents the value of sample standard deviation = 2.88.
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2. Semivariance is defined as the average squared deviation:

  • Option : A
  • Explanation : Semivariance can be defined as the average squared deviations below the mean.
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3. According to Chebyshev’s inequality, in a population of 1000 what is the minimum proportion of observation that must lie within three standard deviations of the mean, regardless of the shape of the distribution?

  • Option : B
  • Explanation : Chebyshev's inequality holds for any distribution, regardless of shape, and states that the minimum proportion of observations located within k standard deviations of the mean is equal to 1– 1/k2. In this case, k = 3 and 1– 1/9 = 0.89 or 89%.
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4. A sample of 320 observations is randomly selected from a population. The mean of the sample is 144 and the standard deviation is 12. Based on Chebyshev’s inequality, the endpoints of the interval that must contain at least 75% of the observations are closest to:

  • Option : B
  • Explanation : According to Chebyshev‟s inequality, the proportion of the observations within k standard deviations of the arithmetic mean is at least 1– 1/k2 for all k > 1. For k = 2, that proportion is 1– 1/22, which is 75%. The lower endpoint is, therefore the mean (144) minus 2 times 12 (the standard deviation) and the upper endpoint is 144 plus 2 times 12. 144– (2 × 12) = 120; 144 + 2(12) = 168
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5. According to Chebyshev’s inequality, at least 88.89 percent of the observations for any distribution must lie within:

  • Option : C
  • Explanation : The formula for Chebyshev‟s inequality is: 1 – 1/k2 = % of distribution 1 – 1/k2 = 0.8889; solving for k, we get k = 3 88.89% of any distribution lies within 3 standard deviations.
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