Question

Loretta, who turns eighty this year, has just learned about blood pressure problems in the elderly and is interested in how her blood pressure compare

Measurement
ANSWERED
asked 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 \(?36\% 56\% 75\% 84\%\ or\ 89\%\) of the measurements lie between 122.8 mmHg and 143.2 mmHg.
b) According to Chebyshev's theorem, at least \(8/9 (about\ 89\%)\) of the measurements lie between mmHg and mmHg. (Round your answer to 1 decimal place.)

Expert Answers (1)

2021-01-14

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 \(\frac{3}{4}\) 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 \((1-\frac{1}{k^{2}})\) 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:
\((\mu−3\sigma, \mu+3\sigma)=(133−3\times 5.1,133+3\times 5.1)=(117.7,148.3)\)
a.
The measurements 122.8 mmHg and 143.2 mmHg are two standard deviations away from the mean. According to Chebyshev’s rule, about \(\frac{3}{4}=75\%\) 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.

13
 
Best answer

expert advice

Have a similar question?
We can deal with it in 3 hours

Relevant Questions

asked 2021-05-14
Consider the accompanying data on flexural strength (MPa) for concrete beams of a certain type.
\(\begin{array}{|c|c|}\hline 11.8 & 7.7 & 6.5 & 6 .8& 9.7 & 6.8 & 7.3 \\ \hline 7.9 & 9.7 & 8.7 & 8.1 & 8.5 & 6.3 & 7.0 \\ \hline 7.3 & 7.4 & 5.3 & 9.0 & 8.1 & 11.3 & 6.3 \\ \hline 7.2 & 7.7 & 7.8 & 11.6 & 10.7 & 7.0 \\ \hline \end{array}\)
a) Calculate a point estimate of the mean value of strength for the conceptual population of all beams manufactured in this fashion. \([Hint.\ ?x_{j}=219.5.]\) (Round your answer to three decimal places.)
MPa
State which estimator you used.
\(x\)
\(p?\)
\(\frac{s}{x}\)
\(s\)
\(\tilde{\chi}\)
b) Calculate a point estimate of the strength value that separates the weakest \(50\%\) of all such beams from the strongest \(50\%\).
MPa
State which estimator you used.
\(s\)
\(x\)
\(p?\)
\(\tilde{\chi}\)
\(\frac{s}{x}\)
c) Calculate a point estimate of the population standard deviation ?. \([Hint:\ ?x_{i}2 = 1859.53.]\) (Round your answer to three decimal places.)
MPa
Interpret this point estimate.
This estimate describes the linearity of the data.
This estimate describes the bias of the data.
This estimate describes the spread of the data.
This estimate describes the center of the data.
Which estimator did you use?
\(\tilde{\chi}\)
\(x\)
\(s\)
\(\frac{s}{x}\)
\(p?\)
d) Calculate a point estimate of the proportion of all such beams whose flexural strength exceeds 10 MPa. [Hint: Think of an observation as a "success" if it exceeds 10.] (Round your answer to three decimal places.)
e) Calculate a point estimate of the population coefficient of variation \(\frac{?}{?}\). (Round your answer to four decimal places.)
State which estimator you used.
\(p?\)
\(\tilde{\chi}\)
\(s\)
\(\frac{s}{x}\)
\(x\)
asked 2021-06-07
Which of the following is NOT a conclusion of the Central Limit​ Theorem? Choose the correct answer below.
a) The distribution of the sample means x over bar x ​will, as the sample size​ increases, approach a normal distribution.
b) The distribution of the sample data will approach a normal distribution as the sample size increases.
c) The standard deviation of all sample means is the population standard deviation divided by the square root of the sample size.
d) The mean of all sample means is the population mean \(\mu\)
asked 2021-06-13
1. Who seems to have more variability in their shoe sizes, men or women?
a) Men
b) Women
c) Neither group show variability
d) Flag this Question
2. In general, why use the estimate of \(n-1\) rather than n in the computation of the standard deviation and variance?
a) The estimate n-1 is better because it is used for calculating the population variance and standard deviation
b) The estimate n-1 is never used to calculate the sample variance and standard deviation
c) \(n-1\) provides an unbiased estimate of the population and allows more variability when using a sample and gives a better mathematical estimate of the population
d) The estimate n-1 is better because it is use for calculation of both the population and sample variance as well as standard deviation.
\(\begin{array}{|c|c|}\hline \text{Shoe Size (in cm)} & \text{Gender (M of F)} \\ \hline 25.7 & M \\ \hline 25.4 & F \\ \hline 23.8 & F \\ \hline 25.4 & F \\ \hline 26.7 & M \\ \hline 23.8 & F \\ \hline 25.4 & F \\ \hline 25.4 & F \\ \hline 25.7 & M \\ \hline 25.7 & F \\ \hline 23.5 & F \\ \hline 23.1 & F \\ \hline 26 & M \\ \hline 23.5 & F \\ \hline 26.7 & F \\ \hline 26 & M \\ \hline 23.1 & F \\ \hline 25.1 & F \\ \hline 27 & M \\ \hline 25.4 & F \\ \hline 23.5 & F \\ \hline 23.8 & F \\ \hline 27 & M \\ \hline 25.7 & F \\ \hline \end{array}\)
\(\begin{array}{|c|c|}\hline \text{Shoe Size (in cm)} & \text{Gender (M of F)} \\ \hline 27.6 & M \\ \hline 26.9 & F \\ \hline 26 & F \\ \hline 28.4 & M \\ \hline 23.5 & F \\ \hline 27 & F \\ \hline 25.1 & F \\ \hline 28.4 & M \\ \hline 23.1 & F \\ \hline 23.8 & F \\ \hline 26 & F \\ \hline 25.4 & M \\ \hline 23.8 & F \\ \hline 24.8 & M \\ \hline 25.1 & F \\ \hline 24.8 & F \\ \hline 26 & M \\ \hline 25.4 & F \\ \hline 26 & M \\ \hline 27 & M \\ \hline 25.7 & F \\ \hline 27 & M \\ \hline 23.5 & F \\ \hline 29 & F \\ \hline \end{array}\)
asked 2021-08-06
2 points) Page Turner loves discrete mathematics. She has 8 "graph theory" books, 6 books about combinatorics, and 4 "set theory" books.
How many ways can she place her discrete mathematics books on the same shelf in a row if:
a) there are no restrictions.
b) graph theory books are next to each other but the others could be anywhere on the shelf.
c) books are organized by their topic (same kinds are next to each other).
...