Which is true of a leptokurtic distribution?
I. It will be more peaked than the normal distribution.
II. It will have thinner tails than the normal distribution.
III. It will be less peaked than the normal distribution.
IV. It will have fatter tails than the normal distribution.
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A. B. C. D.B
I and IV are true of leptokurtic distributions. This is also called having "excess kurtosis." II and III are characteristic of platykurtic distributions.
A leptokurtic distribution is a term used in statistics to describe a distribution that has more kurtosis (i.e., the degree of peakedness or flatness of a distribution) than the normal distribution. It is characterized by a higher peak and heavier tails compared to the normal distribution. Let's analyze each statement in the question and determine which options are correct.
I. "It will be more peaked than the normal distribution." This statement is true. Leptokurtic distributions have a higher peak (i.e., more peaked) than the normal distribution. The excess kurtosis in a leptokurtic distribution is positive, indicating the presence of more data points around the mean.
II. "It will have thinner tails than the normal distribution." This statement is false. In fact, leptokurtic distributions have fatter tails than the normal distribution. The excess kurtosis in a leptokurtic distribution is positive, indicating the presence of outliers or extreme values that contribute to the heavier tails.
III. "It will be less peaked than the normal distribution." This statement is false. As mentioned earlier, leptokurtic distributions are more peaked than the normal distribution. Therefore, they are not less peaked.
IV. "It will have fatter tails than the normal distribution." This statement is true. Leptokurtic distributions have fatter tails than the normal distribution. The excess kurtosis in a leptokurtic distribution is positive, indicating the presence of extreme values or outliers that extend the tails of the distribution.
Based on the analysis above, the correct answers are: B. I and IV.
Therefore, a leptokurtic distribution is more peaked than the normal distribution (Statement I is true), and it has fatter tails than the normal distribution (Statement IV is true).