Question

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 914 hours, with a standard deviation of 90 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 2 between 734 hours and 1094 hours. of the lifetimes lie o (b) According to Chebyshev's theorem, at least 2 between 644 hours and 1184 hours. of the lifetimes lie E х 6 (c) Suppose that the distribution is bell-shaped. According to the empirical rule, approximately 2 of the lifetimes lie between 734 hours and 1094 hours. (d) Suppose that the distribution is bell-shaped. According to the empirical rule, approximately 68% of the lifetimes lie between hours and I hours. Check Save For Later Submit Assignm 2000 Meducation. All Rights Reserved. Terms of Use Pumy Am

Answer 1

We are given here:
Mean = 914 and SD = 90 as the standard deviation

a) 734 = 914 - 2*90 that is 2 standard deviations less than the mean.
1094 = 914 + 2*90 that is 2 standard deviations more than the mean.

According to chebyshev's inequality theorem, at least 1 - 1/k² of the observations lies within k standard deviations of the mean.

Therefore 1 - 1/2² = 0.75

Therefore 0.75 or 75% is the required proportion here.

b) Similar to above part, 644 and 1184 are within 3 standard deviations of mean.

Therefore 1 - 1/3² = 8/9 = 0.8889

Therefore 0.8989 or 89.89% is the required proportion here.

c) Given that the distribution is bell shaped, according to empirical rule, 95% of the observations lies within 2 standard deviation so the mean. Therefore 95% is the correct answer here.

d) According to empirical rule, 68% of the observations lies within 1 standard deviation of the mean. Therefore the limits here are computed as:
L = Mean - SD = 914 - 90 = 824 and U = Mean + SD = 914 + 90 = 1004

Therefore 824 and 1004 are the required values here.