burning0spear

For the past three years, Acme Corp. has generated the following sample returns on equity (ROE): 4%, 10%, and 1%. What is the sample variance of the ROE over the last three years?

A) 21.0%.

B) 4.6%.

C) 21.0(%2).

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[(4 − 5)2 + (10 − 5)2 + (1 − 5)2] / (3 − 1) = 21(%2).

burning0spear

There is a 40% chance that an investment will earn 10%, a 40% chance that the investment will earn 12.5%, and a 20% chance that the investment will earn 30%. What is the mean expected return and the standard deviation of expected returns, respectively?

A) 15.0%; 5.75%.

B) 15.0%; 7.58%.

C) 17.5%; 5.75%.

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Mean = (0.4)(10) + (0.4)(12.5) + (0.2)(30) = 15%
Var = (0.4)(10 − 15)2 + (0.4)(12.5 − 15)2 + (0.2)(30 − 15)2 = 57.5
Standard deviation = √57.5 = 7.58

burning0spear

Cameron Ryan wants to make an offer on the condominium he is renting. He takes a sample of prices of condominiums in his development that closed in the last five months. Sample prices are as follows (amounts are in thousands of dollars): $125, $175, $150, $155 and $135. The sample standard deviation is closest to:

A) 370.00.

B) 38.47.

C) 19.24.

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Calculations are as follows:

• Sample mean = (125 + 175 + 150 + 155 + 135) / 5 = 148
• Sample Variance = [(125 – 148)2 + (175 – 148)2 + (150 – 148)2 + (155 – 148)2 + (135 – 148)2] / (5 – 1) = 1,480 / 4 = 370
• Sample Standard Deviation = 3701/2 = 19.24%.

burning0spear

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%.

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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%.

burning0spear

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

A) 95%.

B) 84%.

C) 75%.

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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.

burning0spear

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%.

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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%.

burning0spear

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%.

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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%.

burning0spear

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) 94%.

B) 89%.

C) 75%.

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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%.

burning0spear

Which of the following statements about statistical concepts is least accurate?

A) The coefficient of variation is useful when comparing dispersion of data measured in different units or having large differences in their means.

B) For a normal distribution, only 95% of the observations lie within ±3 standard deviations from the mean.

C) For any distribution, based on Chebyshev’s Inequality, 75% of the observations lie within ±2 standard deviations from the mean.

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For a normal distribution, 95% of the observations lie within ±2 standard deviations of the mean while 99% of the observations lie within plus or minus three standard deviations of the mean. Both remaining statements are true. Note that 75% of observations for any distribution lie within ±2 standard deviations of the mean using Chebyshev’s inequality.

burning0spear

A higher Sharpe ratio indicates:

A) a higher excess return per unit of risk.

B) lower volatility of returns.

C) a lower risk per unit of return.

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The Sharpe ratio is excess return (return − Rf) per unit of risk (defined as the standard deviation of returns).