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 % 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 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 \(k = 2\)

When \(k = 2\), we substitute \(k\) into the formula \(1-\frac{1}{k^{2}}\). So, \(1-\frac{1}{2^{2}}=1 - \frac{1}{4}=\frac{3}{4}=0.75\) or \(75\%\).

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: Compare the two estimates

The estimate from Chebyshev's rule is \(75\%\) (a lower - bound), while the estimate from the empirical rule is approximately \(95\%\). Chebyshev's rule is more conservative as it applies to any distribution, while the empirical rule applies only to normal distributions.

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

When \(k = 3\), we substitute \(k\) into the formula \(1-\frac{1}{k^{2}}\). So, \(1-\frac{1}{3^{2}}=1-\frac{1}{9}=\frac{8}{9}\approx0.8889\) or \(88.89\%\).

Step6: 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.

Step7: Compare the two estimates for \(k = 3\)

The estimate from Chebyshev's rule is approximately \(88.89\%\) (a lower - bound), while the estimate from the empirical rule is approximately \(99.7\%\). Again, Chebyshev's rule is more conservative as it is applicable to any distribution.

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. The empirical rule gives an estimate of approximately \(95\%\) for a normal distribution. Chebyshev's rule is more conservative as it applies to any distribution.
b. Using Chebyshev's rule, at least approximately \(88.89\%\) of the observations in any data set lie within 3 standard deviations to either side of the mean. The empirical rule gives an estimate of approximately \(99.7\%\) for a normal distribution. Chebyshev's rule is more conservative as it applies to any distribution.