Question

use the information from section 3.3 to answer the following questions. a. a quantitative data set has mean 67.1 and standard deviation 6.6. what can be said about the percentage of the observations that lie between 47.3 and 86.9? a. at least 75% of the data values lie between 47.3 and 86.9. b. approximately 95% of the data values lie between 47.3 and 86.9. c. approximately 99.7% of the data values lie between 47.3 and 86.9. d. at least 89% of the data values lie between 47.3 and 86.9. b. if you are given the additional information that the distribution is approximately bell - shaped, what can be said about the percentage of the observations that lie between 47.3 and 86.9? a. approximately 95% of the data values lie between 47.3 and 86.9. b. at least 89% of the data values lie between 47.3 and 86.9. c. at least 75% of the data values lie between 47.3 and 86.9. d. approximately 99.7% of the data values lie between 47.3 and 86.9.

Explanation:

Step1: Calculate the number of standard - deviations away from the mean

First, find the number of standard - deviations $k$ for the values 47.3 and 86.9 from the mean $\mu = 67.1$ with standard deviation $\sigma=6.6$.
For $x = 47.3$, $k_1=\frac{\mu - x}{\sigma}=\frac{67.1 - 47.3}{6.6}=\frac{19.8}{6.6}=3$.
For $x = 86.9$, $k_2=\frac{x-\mu}{\sigma}=\frac{86.9 - 67.1}{6.6}=\frac{19.8}{6.6}=3$.

Step2: Apply the Empirical Rule (for bell - shaped distributions) and Chebyshev's Theorem

The Empirical Rule states that for a bell - shaped (normal) distribution:

• Approximately 68% of the data lies within 1 standard deviation of the mean ($\mu\pm\sigma$).
• Approximately 95% of the data lies within 2 standard deviations of the mean ($\mu\pm2\sigma$).
• Approximately 99.7% of the data lies within 3 standard deviations of the mean ($\mu\pm3\sigma$).

Chebyshev's Theorem states that for any data set, at least $1-\frac{1}{k^{2}}$ of the data lies within $k$ standard deviations of the mean. When $k = 3$, $1-\frac{1}{k^{2}}=1-\frac{1}{9}\approx89\%$.

a. Since the values 47.3 and 86.9 are 3 standard deviations away from the mean ($\mu = 67.1$ and $\sigma = 6.6$), by Chebyshev's Theorem, at least $89\%$ of the data values lie between 47.3 and 86.9. So the answer for part a is D.
b. If the distribution is approximately bell - shaped, and the values 47.3 and 86.9 are 3 standard deviations away from the mean, approximately 99.7% of the data values lie between 47.3 and 86.9. So the answer for part b is D.

Answer:

a. D. At least 89% of the data values lie between 47.3 and 86.9.
b. D. Approximately 99.7% of the data values lie between 47.3 and 86.9.