Q1. 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%.

Q2. In a skewed distribution, what is the minimum proportion of observations between +/- two standard deviations from the mean?

A)     95%.

B)     75%.

C)     84%.

Q2. 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)   75%.

C)   89%.

Q3. 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)   95%.

C)   56%.

Q4. 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)   89%.

B)   94%.

C)   75%.

答案和详解如下：

Q1. 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%.

Correct answer is C)

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%.

Q2. In a skewed distribution, what is the minimum proportion of observations between +/- two standard deviations from the mean?

A)     95%.

B)     75%.

C)     84%.

Correct answer is B)

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.

Q2. 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)   75%.

C)   89%.

Correct answer is B)

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%.

Q3. 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)   95%.

C)   56%.

Correct answer is C)

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%.

Q4. 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)   89%.

B)   94%.

C)   75%.

Correct answer is A)

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%.

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