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 834 hours, with a standard deviation of 81 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 36% of the lifetimes lie between

hours

and

hours

. (Round your answer to the nearest integer.)

(b) According to Chebyshev's theorem, at least ?56%75%84%89% of the lifetimes lie between 672 hours and 996 hours.

(c) Suppose that the distribution is bell-shaped. According to the empirical rule, approximately ?68%75%95%99.7% of the lifetimes lie between 672 hours and 996 hours.

(d) Suppose that the distribution is bell-shaped. According to the empirical rule, approximately 68% of the lifetimes lie between

hours

and

hours

.

Answer 1

Answer:

Given,

sample size n = 1000

mean = 834

standard deviation = 81

a)

1 - (1/k^2) = 36% = 0.36

1/k^2 = 1 - 0.36 = 0.64

k^2 = 1/0.64 = 1.5625

k = 1.25

Now,

(x +/- ks) = 834 +/- 1.25*81

= 834 +/- 101.25

= (732.75 , 935.25)

= (733 , 935)

b)

Given,

x + 2s = 996

x - 2s = 672

Here k = 2

1 - (1/k^2) = 1 - (1/2^2)

= 1 - 0.25

= 0.75

= 75%

c)

95% values lies within 2*standard deviation from mean

d)

Here they will lie 1 std deviation away

i.e.,

834 - 81 , 834 + 81

= (753 , 915)