Question 1161136: BIG Corporation produces just about everything but is currently interested in the lifetimes of its batteries, hoping to obtain its share of a market boosted by the popularity of portable CD and MP3 players. To investigate its new line of Ultra batteries, BIG randomly selects 1000 Ultra batteries and finds that they have a mean lifetime of 913 hours, with a standard deviation of 94 hours. Suppose that this mean and standard deviation apply to the population of all Ultra batteries. Complete the following statements about the distribution of lifetimes of all Ultra batteries.
A) According to Chebyshev’s theorem, at least 84% of the lifetimes lie between BLANK hours and BLANK hours. (round your answer to the nearest integer.)
B) According to Chebyshev’s theorem, at least BLANK of the lifetimes lie between 725 hours and 1101 hours.
C) Suppose that the distribution is bell-shaped. According to the empirical rule, approximately BLANK of the lifetimes lie between 725 hours and 1101 hours.
D) Suppose that the distribution is bell-shaped. According to the empirical rule, approximately 68% of the lifetimes lie between BLANK hours and BLANK hours.
Answer by CPhill(2285)   (Show Source):
You can put this solution on YOUR website! Here are the completed statements based on Chebyshev's theorem and the Empirical Rule:
### Given Information:
* Population Mean ($\mu$) = $913\text{ hours}$
* Population Standard Deviation ($\sigma$) = $94\text{ hours}$
---
### A) According to Chebyshev’s theorem, at least 84% of the lifetimes lie between **678** hours and **1148** hours.
* **Finding $k$:**
$$1 - \frac{1}{k^2} = 0.84 \implies \frac{1}{k^2} = 0.16 \implies k^2 = 6.25 \implies k = 2.5$$
* **Lower limit:** $\mu - 2.5\sigma = 913 - 2.5(94) = 913 - 235 = \mathbf{678}$
* **Upper limit:** $\mu + 2.5\sigma = 913 + 2.5(94) = 913 + 235 = \mathbf{1148}$
---
### B) According to Chebyshev’s theorem, at least **75%** (or **3/4**) of the lifetimes lie between 725 hours and 1101 hours.
* **Finding $k$:**
$$725 = 913 - k(94) \implies k(94) = 188 \implies k = 2$$
* **Applying Chebyshev's formula:**
$$1 - \frac{1}{k^2} = 1 - \frac{1}{2^2} = 1 - \frac{1}{4} = 0.75 = \mathbf{75\%}$$
---
### C) Suppose that the distribution is bell-shaped. According to the empirical rule, approximately **95%** of the lifetimes lie between 725 hours and 1101 hours.
* As calculated in Part B, $725$ and $1101$ hours correspond to $k = 2$ standard deviations from the mean ($\mu \pm 2\sigma$).
* The Empirical Rule states that approximately **95%** of observations in a bell-shaped distribution fall within 2 standard deviations of the mean.
---
### D) Suppose that the distribution is bell-shaped. According to the empirical rule, approximately 68% of the lifetimes lie between **819** hours and **1007** hours.
* The Empirical Rule states that approximately 68% of the data falls within 1 standard deviation of the mean ($\mu \pm 1\sigma$).
* **Lower limit:** $\mu - 1\sigma = 913 - 1(94) = \mathbf{819}$
* **Upper limit:** $\mu + 1\sigma = 913 + 1(94) = \mathbf{1007}$
| |
|
