Question

compare chebyshevs rule and the empirical rule.
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.
b. compare the estimates given by the two rules for the percentage of observations that lie within three standard deviations to either side of the mean. comment on the differences.
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.
approximately
for a data set that is roughly bell shaped and gives the percentage
the number of observations that could be included in the area. the
any quantitative data set and gives the percentage of the least
at most
of the observations in the area.
exactly
given by the two rules for the percentage of observations that lie
itions to either side of the mean. comment on the differences.
at least
d the estimate for the percentage of observations that lie within three
er side of the mean.
% of the observations in any data set lie within 3 standard deviations to either side of the mean.
(round to the nearest integer as needed.)

Explanation:

Chebyshev's rule applies to any quantitative data - set and gives the percentage of the least number of observations that are definitely included in the area within a certain number of standard - deviations from the mean. The empirical rule applies to roughly bell - shaped data sets and gives an approximation of the percentage of observations in the area within a certain number of standard - deviations from the mean. For two standard - deviations from the mean, the empirical rule says approximately 95% of observations in a bell - shaped data set lie within this range. For three standard - deviations from the mean, the empirical rule says approximately 99.7% of observations in a bell - shaped data set lie within this range. Chebyshev's rule for \(k = 2\) gives \(1-\frac{1}{k^{2}}=1 - \frac{1}{4}=75\%\) (at least) and for \(k = 3\) gives \(1-\frac{1}{k^{2}}=1-\frac{1}{9}\approx89\%\) (at least).

Answer:

a. Using Chebyshev's rule, at least \(75\%\) of the observations lie within two standard deviations of the mean. Using the empirical rule, approximately \(95\%\) of the observations lie within two standard deviations of the mean in a roughly bell - shaped data set. The empirical rule gives a higher percentage for bell - shaped data sets compared to Chebyshev's rule which is a more general result for any quantitative data set.
b. Using Chebyshev's rule, at least \(89\%\) of the observations lie within three standard deviations of the mean. Using the empirical rule, approximately \(99.7\%\) of the observations lie within three standard deviations of the mean in a roughly bell - shaped data set. The empirical rule gives a higher percentage for bell - shaped data sets compared to Chebyshev's rule.
c. 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.
d. Approximately \(99.7\%\) (using the empirical rule for bell - shaped data sets) and at least \(89\%\) (using Chebyshev's rule for any quantitative data set) of the observations lie within three standard deviations of the mean.