1、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)   56%.

B)   75%.

C)   89%.

D)   94%.

2、In a skewed distribution, what is the minimum amount of observations that will fall between +/- 1.5 standard deviations from the mean?

A)   95%.

B)   44%.

C)   25%.

D)   56%.

3、Regardless of the shape of a distribution, according to Chebyshev’s Inequality, what is the minimum percentage of observations that will lie within + or – two standard deviations of the mean?

A)   68%.

B)   34%.

C)   75%.

D)   89%.

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

A)   95%.

B)   84%.

C)   25%.

D)   75%.

5、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)   58.3%.

B)   17.36%.

C)   95.5%.

D)   82.6%.

答案和详解如下：

1、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)   56%.

B)   75%.

C)   89%.

D)   94%.

The correct answer was C)

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%.

2、In a skewed distribution, what is the minimum amount of observations that will fall between +/- 1.5 standard deviations from the mean?

A)   95%.

B)   44%.

C)   25%.

D)   56%.

The correct answer was D)

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) = .5555, or 56%

3、Regardless of the shape of a distribution, according to Chebyshev’s Inequality, what is the minimum percentage of observations that will lie within + or – two standard deviations of the mean?

A)   68%.

B)   34%.

C)   75%.

D)   89%.

The correct answer was C)

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%.

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

A)   95%.

B)   84%.

C)   25%.

D)   75%.

The correct answer was D)    

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.

5、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)   58.3%.

B)   17.36%.

C)   95.5%.

D)   82.6%.

The correct answer was D)

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%.