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.
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.
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.)

Explanation:

Step1: Recall Chebyshev's rule

Chebyshev's rule states that for any number \(k>1\), at least \(1 - \frac{1}{k^{2}}\) of the data lies within \(k\) standard - deviations of the mean. For \(k = 2\), \(1-\frac{1}{2^{2}}=1 - \frac{1}{4}=\frac{3}{4}=75\%\).

Step2: Recall the empirical rule

The empirical rule (for a normal distribution) states that approximately 95% of the data lies within 2 standard - deviations of the mean.

Step3: Compare the two rules for \(k = 2\)

Chebyshev's rule gives a lower - bound of 75% for any data set. The empirical rule gives an approximation of 95% for data that is normally distributed. The empirical rule is more precise for normal distributions, while Chebyshev's rule is more general and applies to all data sets.

Step4: Recall Chebyshev's rule for \(k = 3\)

For \(k = 3\), using Chebyshev's rule, \(1-\frac{1}{3^{2}}=1-\frac{1}{9}=\frac{8}{9}\approx88.89\%\) of the data lies within 3 standard - deviations of the mean.

Step5: Recall the empirical rule for \(k = 3\)

The empirical rule states that approximately 99.7% of the data lies within 3 standard - deviations of the mean for a normal distribution.

Step6: Compare the two rules for \(k = 3\)

Chebyshev's rule gives a lower - bound of about 88.89% for any data set. The empirical rule gives an approximation of 99.7% for data that is normally distributed. The empirical rule provides a much higher estimate for normal distributions, while Chebyshev's rule gives a conservative lower - bound for all data sets.

Answer:

a. Using Chebyshev's rule, at least 75% of the observations in any data set lie within 2 standard deviations to either side of the mean. Using the empirical rule, approximately 95% of the observations in a normally - distributed data set lie within 2 standard deviations to either side of the mean. The empirical rule gives a higher estimate for normal distributions, while Chebyshev's rule is a general lower - bound for all data sets.
b. Using Chebyshev's rule, at least \(\frac{8}{9}\approx88.89\%\) of the observations in any data set lie within 3 standard deviations to either side of the mean. Using the empirical rule, approximately 99.7% of the observations in a normally - distributed data set lie within 3 standard deviations to either side of the mean. The empirical rule provides a much higher estimate for normal distributions, while Chebyshev's rule gives a conservative lower - bound for all data sets.