Step 1

Chebyshev’s inequality:

Chebyshev’s rule is appropriate for any distribution. That is, Chebyshev’s inequality applies to all distributions, regardless of shape. Moreover, it provides the minimum percentage of the observation that lies within k standard deviations of the mean. The Chebyshev’s rule states that, for any quantitative data set and any real number greater than k, at least \((1-\frac{1}{k^{2}})\) observations lie within k standard deviations to either side of the mean.

It is possible that very few measurements will fall within one standard deviation of the mean.

If \(k= 2\), at least 3/4 of the measurements lie within 2 standard deviations to either side of the mean.

If \(k = 3\), at least 8/9 of the measurements lie within 3 standard deviations to either side of the mean.

Generally, for any number k greater than 1, at least \((1-\frac{1}{k^{2}})\) of the measurements will fall within k standard deviations of the mean.

Step 2

It is given that the mean systolic blood pressure measurement for women over seventy-five is 133.0 mmHg, with a standard deviation of 5.1 mmHg.

That is, \(\mu=133\) and standard deviation \(\sigma=5.1\)

One standard deviation below and above the mean is as follows:

\((\mu−\sigma, \mu+\sigma)=(133−5.1,133+5.1)=(127.9,138.1)\)

Two standard deviation below and above the mean is as follows:

\((\mu−2\sigma, \mu+2\sigma)=(133−2\times 5.1,133+2\times 5.1)=(122.8,143.2)\)

Three standard deviation below and above the mean is as follows:

\((\mu−3\sigma, \mu+3\sigma)=(133−3times 5.1,133+3times 5.1)=(117.7,148.3)\)

a.

The measurements 122.8 mmHg and 143.2 mmHg are two standard deviations away from the mean. According to Chebyshev’s rule, about \(3/4(=75\%)\) of the measurements lie within 2 standard deviations to either side of the mean.

Thus, according to Chebyshev's theorem, at least 75\% of the measurements lie between 122.8 mmHg and 143.2 mmHg.

Chebyshev’s inequality:

Chebyshev’s rule is appropriate for any distribution. That is, Chebyshev’s inequality applies to all distributions, regardless of shape. Moreover, it provides the minimum percentage of the observation that lies within k standard deviations of the mean. The Chebyshev’s rule states that, for any quantitative data set and any real number greater than k, at least \((1-\frac{1}{k^{2}})\) observations lie within k standard deviations to either side of the mean.

It is possible that very few measurements will fall within one standard deviation of the mean.

If \(k= 2\), at least 3/4 of the measurements lie within 2 standard deviations to either side of the mean.

If \(k = 3\), at least 8/9 of the measurements lie within 3 standard deviations to either side of the mean.

Generally, for any number k greater than 1, at least \((1-\frac{1}{k^{2}})\) of the measurements will fall within k standard deviations of the mean.

Step 2

It is given that the mean systolic blood pressure measurement for women over seventy-five is 133.0 mmHg, with a standard deviation of 5.1 mmHg.

That is, \(\mu=133\) and standard deviation \(\sigma=5.1\)

One standard deviation below and above the mean is as follows:

\((\mu−\sigma, \mu+\sigma)=(133−5.1,133+5.1)=(127.9,138.1)\)

Two standard deviation below and above the mean is as follows:

\((\mu−2\sigma, \mu+2\sigma)=(133−2\times 5.1,133+2\times 5.1)=(122.8,143.2)\)

Three standard deviation below and above the mean is as follows:

\((\mu−3\sigma, \mu+3\sigma)=(133−3times 5.1,133+3times 5.1)=(117.7,148.3)\)

a.

The measurements 122.8 mmHg and 143.2 mmHg are two standard deviations away from the mean. According to Chebyshev’s rule, about \(3/4(=75\%)\) of the measurements lie within 2 standard deviations to either side of the mean.

Thus, according to Chebyshev's theorem, at least 75\% of the measurements lie between 122.8 mmHg and 143.2 mmHg.