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Solved: Chebyshev’s Inequality and the Empirical Rule
Chapter 10, Problem 5RE(choose chapter or problem)
Problem 5RE
Problem
Chebyshev’s Inequality and the Empirical Rule Suppose that a certain brand of light bulb has a mean life of 600 hours and a standard deviation of 53 hours.
(a) A histogram of the data indicates the sample data follow a bell-shaped distribution. According to the Empirical Rule, 99.7% of light bulbs have lifetimes between ______ and ______ hours.
(b) Assuming the data are bell shaped, determine the percentage of light bulbs that will have a life between 494 and 706 hours.
(c) Assuming the data are bell shaped, what percentage of light bulbs will last between 547 and 706 hours.
(d) If the company that manufactures the light bulb guarantees to replace any bulb that does not last at least 441 hours, what percentage of light bulbs can the firm expect to have to replace, according to the Empirical Rule?
(e) Use Chebyshev’s Inequality to determine the minimum percentage of light bulbs with a life within 2.5 standard deviations of the mean.
(f ) Use Chebyshev’s Inequality to determine the minimum percentage of light bulbs with a life between 494 and 706 hours.
Questions & Answers
QUESTION:
Problem 5RE
Problem
Chebyshev’s Inequality and the Empirical Rule Suppose that a certain brand of light bulb has a mean life of 600 hours and a standard deviation of 53 hours.
(a) A histogram of the data indicates the sample data follow a bell-shaped distribution. According to the Empirical Rule, 99.7% of light bulbs have lifetimes between ______ and ______ hours.
(b) Assuming the data are bell shaped, determine the percentage of light bulbs that will have a life between 494 and 706 hours.
(c) Assuming the data are bell shaped, what percentage of light bulbs will last between 547 and 706 hours.
(d) If the company that manufactures the light bulb guarantees to replace any bulb that does not last at least 441 hours, what percentage of light bulbs can the firm expect to have to replace, according to the Empirical Rule?
(e) Use Chebyshev’s Inequality to determine the minimum percentage of light bulbs with a life within 2.5 standard deviations of the mean.
(f ) Use Chebyshev’s Inequality to determine the minimum percentage of light bulbs with a life between 494 and 706 hours.
ANSWER:
Answer :
Step 1:
Chebyshev’s Inequality and the Empirical Rule Suppose that a certain brand of light bulb has a mean life of 600 hours and a standard deviation of 53 hours.
Here, mean and standard deviation
= 53.
- According to the Empirical Rule, 99.7% of light bulbs have lifetimes between ______ and ______ hours is
Three standard deviations =