Question

The life expectancy of a brand of light bulbs is normally distributed with a mean of 1500 hours and a standard deviation of 75 hours. A. What is the probability that a bulb will last between 1500 and 1650 hours. The life expectancy of a brand of light bulbs is normally distributed with a mean of 1500 hours and a standard deviation of 75 hours. A. What is the probability that a bulb will last between 1500 and 1650 hours.

B. What percentage of the light bulbs will last between 1485 and 1500 hours.
C. What percentage of the light bulb will last between 1416 and 1677 hours.
D. What percentage of the light bulbs will last between 1563 and 1648 hours.
E. What percentage of the light bulbs will last less than 1410 hours.

Answer 1

Solution:

Let X be a random variable which represents the life expectancy of a brand of light bulbs.

Given that, X ~ N(1500, 75²)

i.e.  μ = 1500 hours and σ = 75 hours

A) We have to obtain P(1500 < X < 1650).

We know that if X ~ N(μ ,σ^{​​​​​​2}) then

Using "pnorm" function of R we get,

P(Z < 2) = 0.9772 and P(Z < 0) = 0.5000

Hence, the probability that a bulb will last between 1500 and 1650 hours is 0.4772.

B) We have to obtain P(1485 < X < 1500).

We know that if X ~ N(μ ,σ^{​​​​​​2}) then

Using "pnorm" function of R we get,

P(Z < 0) = 0.5000 and P(Z < -0.2) = 0.4207

0.0793= 7.93%

Hence, 7.93% of light bulb will last between 1485 and 1500 hours.

C) We have to obtain P(1416 < X < 1677).

We know that if X ~ N(μ ,σ^{​​​​​​2}) then

Using "pnorm" function of R we get,

P(Z < 2.36) = 0.9909 and P(Z < -1.12) = 0.1314

0.8595 = 85.95%

Hence, 85.95% of light bulb will last between 1416 and 1677 hours.

D) We have to obtain P(1563 < X < 1648).

We know that if X ~ N(μ ,σ^{​​​​​​2}) then

Using "pnorm" function of R we get,

P(Z < 1.9733) = 0.9758 and P(Z < 0.84) = 0.7995

0.1763 = 17.63%

Hence, 17.63% of light bulb will last between 1563 and 1648 hours.

E) We have to obtain P(X < 1410).

We know that if X ~ N(μ ,σ^{​​​​​​2}) then

Using "pnorm" function of R we get,

P(Z < -1.2) = 0.1151

0.1151 = 11.51%

Hence, 11.51% of light bulb will last less than 1410 hours.

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