The chief design engineer for Colonial Warning Systems wants to know the reliability of a two-lamp device intended to warn the operator of potential problems with a commercial steam kettle.One (either)lamp is to be lit if steam pressure drops below the desired level,and two (both)are to be lit if temperature drops below a desired level.His supplier estimates that a lamp has a failure rate of .2 per hour of operation,and a mean wear-out time of seven years with a standard deviation of eight months.What is the probability that a lamp will last at least until the expiration of the warranty,which is six years?
A).9772
B).9332
C).8664
D).8413
E).4332

Question

Quizplus · Accepted Answer

To find the probability that a lamp will last at least until the expiration of the warranty (six years), we use the Z-score formula in the context of a normal distribution, since we're given the mean and standard deviation. The mean wear-out time is seven years, and the standard deviation is eight months (or 2/3 year). We want to find the probability of a lamp lasting at least six years.First, convert the time to years: 8 months = 8/12 = 2/3 years.Next, calculate the Z-score: $Z = \frac{X - \mu}{\sigma}$, where $X$ is 6 years, $\mu$ is 7 years, and $\sigma$ is 2/3 years.$Z = \frac{6 - 7}{2/3} = \frac{-1}{2/3} = -1.5$Using the Z-table, a Z-score of -1.5 corresponds to a left-tail probability of approximately 0.0668. However, we're interested in the lamp lasting at least six years, so we look at the right-tail probability, which is $1 - 0.0668 = 0.9332$.Therefore, the probability that a lamp will last at least until the expiration of the warranty, which is six years, is approximately 0.9332.

The Chief Design Engineer for Colonial Warning Systems Wants to Know