If you put an ordinary incandescent bulb into a light xture designed for a high

Question

If you put an ordinary incandescent bulb into a light xture designed for a high eciency (low heat) bulb, it is possible{but not certain{that the contact point will melt and the light bulb will become stuck in the socket. Extra labor is then required to remove it and clean the xture's contact point. An electrician estimates that these extra tasks must be performed 0.751 of the time when replacing an incorrect bulb. The electrician has been called to a house on another matter, and notices they have used the wrong light bulb type in four of their downlights. (a) If the electrician points out the hazard and is employed to replace the bulbs, what distribution would best be used to model the number of bulbs that will be stuck in the socket? Explain your selection of distribution, including any assumptions made. (b) If the electrician is working his way through a large apartment block, and each apart- ment has seven incorrect bulbs that must be replaced, what is the long run proportion of times at least ve of the bulbs will be stuck and require the extra steps? (c) What is the mean and variance of the distribution of the number of stuck bulbs?

Explanation / Answer

If you put an ordinary incandescent bulb into a light xture designed for a high eciency (low heat) bulb, it is possible{but not certain{that the contact point will melt and the light bulb will become stuck in the socket. Extra labor is then required to remove it and clean the xture's contact point. An electrician estimates that these extra tasks must be performed 0.751 of the time when replacing an incorrect bulb. The electrician has been called to a house on another matter, and notices they have used the wrong light bulb type in four of their downlights.

(a) If the electrician points out the hazard and is employed to replace the bulbs, what distribution would best be used to model the number of bulbs that will be stuck in the socket? Explain your selection of distribution, including any assumptions made.

Solution:

If the electrician points out the hazard and is employed to replace the bulbs, the binomial distribution would be the best model for the number of bulbs that will be stuck in the socket as we get the binomial response as the bulb stuck in the socket or bulb does not stuck in the socket. Also, we are given a probability for the stuck of the bulb as the 0.751. So, for this scenario, we would use the binomial distribution.

(b) If the electrician is working his way through a large apartment block, and each apart- ment has seven incorrect bulbs that must be replaced, what is the long run proportion of times at least ve of the bulbs will be stuck and require the extra steps?

Solution:

Here, we are given sample size as n = 7 and p = 0.751

For the long run proportion, the estimate for the population proportion is given as the sample proportion.

Here, we have to find P(X5)

P(X5) = 1 – P(X<5)

Where, P(X<5) = P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4)

P(X<5) = 0.241518 by using binomial table or excel binomial command =binomdist(4,7,0.751,1)

P(X5) = 1 – P(X<5) = 1 – 0.241518 = 0.758482

Required answer = 0.7585

(c) What is the mean and variance of the distribution of the number of stuck bulbs?

Solution:

Here, we are given

Sample size = n = 7 and p = 0.751

Mean = np = 7*0.751 = 5.257

Standard deviation = sqrt(npq)

where q = 1 – p = 1 – 0.751 = 0.249

Standard deviation = sqrt(7*0.751*0.249)

Standard deviation = 1.1441

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