Questions: Tammy, who turns eighty years old 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. She has uncovered an article in a scientific journal that reports that the mean systolic blood pressure measurement for women over seventy-five is 132.1 mmHg, with a standard deviation of 6.5 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 (Choose one) of the measurements lie between 119.1 mmHg and 145.1 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.)

Tammy, who turns eighty years old 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. She has uncovered an article in a scientific journal that reports that the mean systolic blood pressure measurement for women over seventy-five is 132.1 mmHg, with a standard deviation of 6.5 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 (Choose one) of the measurements lie between 119.1 mmHg and 145.1 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.)

Transcript text: Tammy, who turns eighty years old 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. She has uncovered an article in a scientific journal that reports that the mean systolic blood pressure measurement for women over seventy-five is 132.1 mmHg , with a standard deviation of 6.5 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 $\square$ (Choose one) of the measurements lie between 119.1 mmHg and 145.1 mmHg . (b) According to Chebyshev's theorem, at least $\frac{8}{9}$ (about $89 \%$ ) of the measurements lie between $\square$ mmHg and $\square$ mmHg . (Round your answer to 1 decimal place.)

Solution

Solution Steps

Step 1: Understand the given information

The mean systolic blood pressure for women over seventy-five is $ \mu = 132.1 $ mmHg, and the standard deviation is $ \sigma = 6.5 $ mmHg.

Step 2: Solve part (a) using Chebyshev's theorem

Chebyshev's theorem states that for any distribution, at least $ 1 - \frac{1}{k^2} $ of the data lies within $ k $ standard deviations of the mean. Here, the range is from 119.1 mmHg to 145.1 mmHg.

First, calculate the number of standard deviations ($ k $) from the mean for the lower and upper bounds:

Lower bound: $ \frac{119.1 - 132.1}{6.5} = -2 $
Upper bound: $ \frac{145.1 - 132.1}{6.5} = 2 $

Since $ k = 2 $, apply Chebyshev's theorem: \[ 1 - \frac{1}{k^2} = 1 - \frac{1}{2^2} = 1 - \frac{1}{4} = \frac{3}{4} \] Thus, at least $ \frac{3}{4} $ (75%) of the measurements lie between 119.1 mmHg and 145.1 mmHg.

Step 3: Solve part (b) using Chebyshev's theorem

We are given that at least $ \frac{8}{9} $ (about 89%) of the measurements lie within a certain range. Using Chebyshev's theorem: \[ 1 - \frac{1}{k^2} = \frac{8}{9} \] Solve for $ k $: \[ \frac{1}{k^2} = 1 - \frac{8}{9} = \frac{1}{9} \implies k^2 = 9 \implies k = 3 \] Now, calculate the range: \[ \text{Lower bound} = \mu - k\sigma = 132.1 - 3 \times 6.5 = 132.1 - 19.5 = 112.6 \, \text{mmHg} \] \[ \text{Upper bound} = \mu + k\sigma = 132.1 + 3 \times 6.5 = 132.1 + 19.5 = 151.6 \, \text{mmHg} \] Thus, at least $ \frac{8}{9} $ of the measurements lie between 112.6 mmHg and 151.6 mmHg.