Question

A Christmas tree light has an expected life of 220 hr and a standard deviation of 2 hr. (a) Find a bound on the probability that one of these Christmas tree lights will require replacement between 210 hr and 230 hr at least 0.96 x% ts Caristm (b) Suppose a large city uses 180,000 of these Christmas tree lights as part of its Christmas decorations. Estimate the number of lights that are likely to require replacement between 200 hr and 240 hr of use 1800 lights leodx ihste lkey to require repac

Answer 1

(a)

Now X is less than 210 or greater than 230 if and only if,

$| X - \mu| \geq k$

where, $\mu$ = 220 and k = 10. So the probability that X <= 210 or X >= 230 is,

$P(|X-\mu| \geq k) \leq \frac{\sigma^2}{k^2} = \frac{2^2}{10^2} = \frac{4}{100} = \frac{1}{25}$

So X will be between 210 and 230 with probability = 1 - (1/25) = 24/25 = 96% (not 0.96% as written in the answer box)

(b)

Now X is less than 200 or greater than 240 if and only if,

$| X - \mu| \geq k$

where, $\mu$ = 220 and k = 20. So the probability that X <= 200 or X >= 240 is,

$P(|X-\mu| \geq k) \leq \frac{\sigma^2}{k^2} = \frac{2^2}{20^2} = \frac{4}{400} = \frac{1}{100}$

So X will be between 200 and 240 with probability = 1 - (1/100) = 99/100 = 0.99

Number of lights that are likely to require replacement between 200 hr and 240 hr is = 0.99 * 180000 = 178200

** If the answer does not match please comment