manchester88

Consider the following graph of a distribution for the prices for various bottles of California-produced wine. Which of the following statements about this distribution is least accurate?

A) Approximately 68% of observations fall within one standard deviation of the mean.

B) The graph could be of the sample $16, $12, $15, $12, $17, $30 (ignore graph scale).

C) The distribution is positively skewed.

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This statement is true for the normal distribution. The above distribution is positively skewed. Note: for those tempted to use Chebyshev’s inequality to determine the percentage of observations falling within one standard deviation of the mean, the formula is valid only for k > 1.
The other statements are true. When we order the six prices from least to greatest: $12, $12, $15, $16, $17, $30, we observe that the mode (most frequently occurring price) is $12, the median (middle observation) is $15.50 [(15 + 16)/2], and the mean is $17 (sum of all prices divided by number in the sample). Time-Saving Note: Just by ordering the distribution, we can see that it is positively skewed (there are large, positive outliers). By definition, mode < median < mean describes a positively skewed distribution.

manchester88

A distribution with a mode of 10 and a range of 2 to 25 would most likely be:

A) positively skewed.

B) normally distributed.

C) negatively skewed.

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The distance to the left from the mode to the beginning of the range is 8. The distance to the right from the mode to the end of the range is 15. Therefore, the distribution is skewed to the right, which means that it is positively skewed.

manchester88

Which of the following statements regarding skewness is least accurate?

A) A distribution that is not symmetrical has skew not equal to zero.

B) A positively skewed distribution is characterized by many small losses and a few extreme gains.

C) In a skewed distribution, 95% of all values will lie within plus or minus two standard deviations of the mean.

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For a normal distribution, the mean will be equal to its median and 95% of all observations will fall within plus or minus two standard deviations of the mean. For a skewed distribution, because it is not symmetrical, this may not be the case. Chebyshev’s inequality tells us that at least 75% of observations will lie within plus or minus two standard deviations from the mean.

manchester88

If a distribution is skewed:

A) the magnitude of positive deviations from the mean is different from the magnitude of negative deviations from the mean.

B) it will be more or less peaked reflecting a greater or lesser concentration of returns around the mean.

C) each side of a return distribution is the mirror image of the other.

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Skewness is caused by the magnitude of positive deviations from the mean being either larger or smaller than the magnitude of negative deviations from the mean. Each side of a skewed distribution is not a mirror image of the other. Peakedness of a distribution is measured by kurtosis.

manchester88

Which of the following statements concerning skewness is least accurate? A distribution with:

A) a distribution with skew equal to 1 is not symmetrical.

B) positive skewness has a long left tail.

C) negative skewness has a large number of outliers on its left side.

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A distribution with positive skewness has long right tails.

manchester88

Which of the following statements concerning kurtosis is least accurate?

A) A distribution that is more peaked than a normal distribution is leptokurtic.

B) A leptokurtic distribution has fatter tails than a normal distribution.

C) A leptokurtic distribution has excess kurtosis less than zero.

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A leptokurtic distribution is more peaked than normal and has fatter tails. However, the excess kurtosis is greater than zero.

manchester88

Which of the following statements about kurtosis is least accurate? Kurtosis:

A) measures the peakedness of a distribution reflecting a greater or lesser concentration of returns around the mean.

B) is used to reflect the probability of extreme outcomes for a return distribution.

C) describes the degree to which a distribution is not symmetric about its mean.

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The degree to which a distribution is not symmetric about its mean is measured by skewness. Excess kurtosis which is measured relative to a normal distribution, indicates the peakedness of a distribution, and also reflects the probability of extreme outcomes.

manchester88

Which of the following statements concerning a distribution with positive skewness and positive excess kurtosis is least accurate?

A) It has a lower percentage of small deviations from the mean than a normal distribution.

B) The mean will be greater than the mode.

C) It has fatter tails than a normal distribution.

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A distribution with positive excess kurtosis has a higher percentage of small deviations from the mean than normal. So it is more “peaked” than a normal distribution. A distribution with positive skew has a mean > mode.

manchester88

A distribution of returns that has a greater percentage of small deviations from the mean and a greater percentage of large deviations from the mean compared to a normal distribution:

A) is positively skewed.

B) has positive excess kurtosis.

C) has negative excess kurtosis.

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A distribution that has a greater percentage of small deviations from the mean and a greater percentage of large deviations from the mean will be leptokurtic and will exhibit positive excess kurtosis. The distribution will be taller (more peaked) with fatter tails than a normal distribution.

manchester88

A distribution that is more peaked than normal is:

A) skewed.

B) leptokurtic.

C) platykurtic.

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A distribution that is more peaked than normal is leptokurtic. A distribution that is flatter than normal is platykurtic.