BIG Corporation produces just about everything but is curren

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 Ultra batteries and finds that they have a mean lifetime of hours, with a standard deviation of 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.

Solution

According to Chebyshev\'s theorem , if a data set has an approximately bell-shaped relative frequency histogram, then

3. approximately 99.7% of the data lies within three standard deviations of the mean, that is, in the interval with endpoints x±3sfor samples

Chebyshev\'s theorem applies to any data set, i.e. also to other than normally distributed data sets ( i.e. bell shaped data sets). His theorem states that the portion of any set of data within K standard deviations of the mean is always at least 1-1/K², where K may be any number greater than 1.

The answers to the questions are as under:

(a) the lifetimes lying between 623 hours and 1043 hours are within 2.5 times the standard deviation of the mean. Therefore, the correct answer is [1- 1/ (2.5)² } * 100 % = ( 1 -0.16)* 100 % = 84 %

(b) the lifetimes lying between 665 hours and 1001 hours are within 2 standard deviations of the mean.Therefore, the correct answer is ( 1 - 1/4) * 100 % = 75 %

(c)   If the distribution is bell-shaped, the lifetimes lying between 665 hours and 1001 hours are within 2 standard deviations of the mean. Therefore, the correct answer is 95 %.

(d) If the distribution is bell-shaped, approximately 99.7 % of the lifetimes lie within 3 standard deviations of the mean, i.e. within 631 and 1135 hours.