Question

a. compare the estimates given by the two rules for the percentage of observations that lie within two standard deviations to either side of the mean. comment on the differences. using chebyshevs rule, find the estimate for the percentage of observations that lie within two standard deviations to either side of the mean. at least 75% of the observations in any data set lie within 2 standard deviations to either side of the mean. (type an integer or a decimal. do not round.) using the empirical rule, find the estimate for the percentage of observations that lie within two standard deviations to either side of the mean. approximately 95% of the observations in any data set lie within 2 standard deviations to either side of the mean. (type an integer or a decimal. do not round.) select the best answer that describes the differences in the two rules. a. chebyshevs rule is for a data set that is roughly bell shaped and gives the percentage of the least number of observations that are definitely included in the area. the empirical rule is for any quantitative data set and gives the percentage that approximates the number of observations that could be included in the area. in this case, chebyshevs rule captures more of the observations in the area. b. chebyshevs rule is for any quantitative data set and gives the percentage that approximates the number of observations that could be included in the area. the empirical rule is for a data set that is roughly bell shaped and gives the percentage of the least number of observations that are definitely included in the area. in this case, the empirical rule captures more of the observations in the area. c. chebyshevs rule is for any quantitative data set and gives the percentage of the least number of observations that are definitely included in the area. the empirical rule is for a data set that is roughly bell shaped and gives the percentage that approximates the number of observations that could be included in the area. in this case, the empirical rule captures more of the observations in the area. d. chebyshevs rule is for a data set that is roughly bell shaped and gives the percentage that approximates the number of observations that could be included in the area. the empirical rule is for any quantitative data set and gives the percentage of the least number of observations that are definitely included in the area. in this case, chebyshevs rule captures more of the observations in the area.

Explanation:

Step1: Recall Chebyshev's rule

Chebyshev's rule applies to any quantitative data - set. It gives the percentage of the least number of observations that are definitely included in the area within \(k\) standard - deviations of the mean. For \(k = 2\), the formula is \(1-\frac{1}{k^{2}}\), so \(1-\frac{1}{2^{2}}=1 - \frac{1}{4}=0.75\) or \(75\%\).

Step2: Recall the empirical rule

The empirical rule applies to data sets that are roughly bell - shaped. For \(k = 2\) standard - deviations from the mean, approximately \(95\%\) of the data lies within this range.

Step3: Analyze the differences

Chebyshev's rule is for any quantitative data set and gives the minimum percentage of observations within \(k\) standard - deviations of the mean. The empirical rule is for bell - shaped data sets and gives an approximate percentage of observations within \(k\) standard - deviations of the mean. In the case of \(k = 2\), the empirical rule captures more of the observations (since \(95\%>75\%\)).

Answer:

C. Chebyshev's rule is for any quantitative data set and gives the percentage of the least number of observations that are definitely included in the area. The empirical rule is for a data set that is roughly bell shaped and gives the percentage that approximates the number of observations that could be included in the area. In this case, the empirical rule captures more of the observations in the area.