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.
% of the observations in any data set lie within 2 standard deviations to
al. do not round.)
at least
at most
exactly
approximately

Explanation:

Step1: Recall Chebyshev's rule formula

Chebyshev's rule states that for any number \(k>1\), the proportion of the data within \(k\) standard - deviations of the mean is at least \(1-\frac{1}{k^{2}}\).

Step2: Calculate for \(k = 2\) in Chebyshev's rule

When \(k = 2\), we substitute \(k\) into the formula \(1-\frac{1}{k^{2}}\). So, \(1-\frac{1}{2^{2}}=1 - \frac{1}{4}=\frac{3}{4}=75\%\). So, at least \(75\%\) of the observations in any data - set lie within 2 standard deviations to either side of the mean according to Chebyshev's rule.

Step3: Recall the empirical rule for \(k = 2\)

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

Step4: Comment on the difference

The estimate from Chebyshev's rule is a lower - bound (\(\geq75\%\)) and is applicable to any data set. The empirical rule gives an approximation (\(\approx95\%\)) and is only valid for data sets that are approximately normally distributed.

Step5: Calculate for \(k = 3\) in Chebyshev's rule

When \(k = 3\), we substitute \(k\) into the formula \(1-\frac{1}{k^{2}}\). So, \(1-\frac{1}{3^{2}}=1-\frac{1}{9}=\frac{8}{9}\approx88.89\%\). So, at least \(88.89\%\) of the observations in any data - set lie within 3 standard deviations to either side of the mean according to Chebyshev's rule.

Step6: Recall the empirical rule for \(k = 3\)

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

Step7: Comment on the difference for \(k = 3\)

The estimate from Chebyshev's rule is a lower - bound (\(\geq88.89\%\)) and is applicable to any data set. The empirical rule gives an approximation (\(\approx99.7\%\)) and is only valid for data sets that are approximately normally distributed.

Answer:

a. Chebyshev's rule: At least \(75\%\) of the observations lie within 2 standard deviations of the mean. Empirical rule: Approximately \(95\%\) of the observations lie within 2 standard deviations of the mean. The difference is that Chebyshev's rule gives a lower - bound for any data set, while the empirical rule gives an approximation for normally distributed data.
b. Chebyshev's rule: At least \(88.89\%\) of the observations lie within 3 standard deviations of the mean. Empirical rule: Approximately \(99.7\%\) of the observations lie within 3 standard deviations of the mean. The difference is that Chebyshev's rule gives a lower - bound for any data set, while the empirical rule gives an approximation for normally distributed data.