burning0spear

Which of the following statements about the median is least accurate? It is:

A) equal to the 50th percentile.

B) more affected by extreme values than the mean.

C) equal to the mode in a normal distribution.

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Median is less influenced by outliers since the median is computed as the “middle” observation. On the other hand, all of the data including outliers are used in computing the mean. Both remaining statements are true regarding the median.

burning0spear

What are the median and the third quintile of the following data points, respectively?

9.2%, 10.1%, 11.5%, 11.9%, 12.2%, 12.8%, 13.1%, 13.6%, 13.9%, 14.2%, 14.8%, 14.9%, 15.4%

A) 13.1%; 13.6%.

B) 13.1%; 13.7%.

C) 12.8%; 13.6%.

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The median is the midpoint of the data points. In this case there are 13 data points and the midpoint is the 7th term.
The formula for determining quantiles is: Ly = (n + 1)(y) / (100). Here, we are looking for the third quintile (60% of the observations lie below) and the formula is: (14)(60) / (100) = 8.4. The third quintile falls between 13.6% and 13.9%, the 8th and 9th numbers from the left. Since L is not a whole number, we interpolate as: 0.136 + (0.40)(0.139 − 0.136) = 0.1372, or 13.7%.

burning0spear

What does it mean to say that an observation is at the sixty-fifth percentile?

A) 65% of all the observations are below that observation.

B) 65% of all the observations are above that observation.

C) The observation falls within the 65th of 100 intervals.

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If the observation falls at the sixty-fifth percentile, 65% of all the observations fall below that observation.

burning0spear

Consider the following set of stock returns: 12%, 23%, 27%, 10%, 7%, 20%,15%. The third quartile is:

A) 23%.

B) 21.5%.

C) 20.0%.

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The third quartile is calculated as: Ly = (7 + 1) (75/100) = 6. When we order the observations in ascending order: 7%, 10%, 12%, 15%, 20%, 23%, 27%, “23%” is the sixth observation from the left.

burning0spear

The following data points are observed returns.  

4.2%, 6.8%, 7.0%, 10.9%, 11.6%, 14.4%, 17.0%, 19.0%, 22.5%, 28.1%

What return lies at the seventh decile (70% of returns lie below this return)?

A) 17.0%.

B) 18.4%.

C) 19.0%.

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The formula for the seventh decile is Ly = (n + 1)(7 / 10) = 7.70 or between the seventh and eighth return from the left. The seventh return is 17%, while the eighth return is 19%. Interpolating, we find that the seventh decile is 17% + 0.7(19% – 17%) = 18.4%.

burning0spear

When creating intervals around the mean to indicate the dispersion of outcomes, which of the following measures is the most useful? The:

A) variance.

B) median.

C) standard deviation.

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The standard deviation is more useful than the variance because the standard deviation is in the same units as the mean. The median does not help in creating intervals around the mean.

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%.