 Kaycee Roche

2021-01-13

Loretta, who turns eighty this year, has just learned about blood pressure problems in the elderly and is interested in how her blood pressure compares to those of her peers. Specifically, she is interested in her systolic blood pressure, which can be problematic among the elderly. She has uncovered an article in a scientific journal that reports that the mean systolic blood pressure measurement for women over seventy-five is 133.0 mmHg, with a standard deviation of 5.1 mmHg.
Assume that the article reported correct information. Complete the following statements about the distribution of systolic blood pressure measurements for women over seventy-five.
a) According to Chebyshev's theorem, at least of the measurements lie between 122.8 mmHg and 143.2 mmHg.
b) According to Chebyshev's theorem, at least of the measurements lie between mmHg and mmHg. (Round your answer to 1 decimal place.) tabuordg

Expert

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 $\left(1-\frac{1}{{k}^{2}}\right)$ 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 $\frac{3}{4}$ of the measurements lie within 2 standard deviations to either side of the mean.
If $k=3$, at least $\frac{8}{9}$ of the measurements lie within 3 standard deviations to either side of the mean.
Generally, for any number k greater than 1, at least $\left(1-\frac{1}{{k}^{2}}\right)$ 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:
$\left(\mu -\sigma ,\mu +\sigma \right)=\left(133-5.1,133+5.1\right)=\left(127.9,138.1\right)$
Two standard deviation below and above the mean is as follows:
$\left(\mu -2\sigma ,\mu +2\sigma \right)=\left(133-2×5.1,133+2×5.1\right)=\left(122.8,143.2\right)$
Three standard deviation below and above the mean is as follows:
$\left(\mu -3\sigma ,\mu +3\sigma \right)=\left(133-3×5.1,133+3×5.1\right)=\left(117.7,148.3\right)$
a.
The measurements 122.8 mmHg and 143.2 mmHg are two standard deviations away from the mean. According to Chebyshev’s rule, about $\frac{3}{4}=75\mathrm{%}$ of the measurements lie within 2 standard deviations to either side of the mean.
Thus, according to Chebyshev's theorem, at least $75\mathrm{%}$ of the measurements lie between 122.8 mmHg and 143.2 mmHg.

Do you have a similar question?