The useful life of one type of projector lamps has a normal distribution with a

Question

The useful life of one type of projector lamps has a normal distribution with a mean of = 2250 hours and a standard deviation of =175 hours. Use X to denote the life of a projector lamp selected at random.

(A) The probability that the useful life of a projector lamp selected at random is shorter than or equal to 2500 hours is closest to

(B) The probability that the useful life of a projector lamp selected at random is shorter than or equal to 2100 hours is closest to

(C) The probability that the useful life of a projector lamp selected at random is longer than 2300 hours is closest to

(D) The probability that the useful life of a projector lamp selected at random is longer than 2200 hours is closest to

(E) The probability that the useful life of a projector lamp selected at random is between 2300 and 2500 hours is closest to

(F) The probability that the useful life of a projector lamp selected at random is between 2100 and 2200 hours is closest to

(G) The probability that the useful life of a projector lamp selected at random is between 2200 and 2350 hours is closest to

(H) The lamp manufacturer is planning to offer a warranty on the useful lamp life. If the useful life is less than the warrantied useful life, represented by xw , the manufacturer will offer a free replacement. If the manufacturer wants about 5% of the lamps to be replaced free, the warrantied life xw should be closest to

Is it possible to solve this without a graph? I cannot figure out how to use the graph. Is there a ti84 shortcut?

Explanation / Answer

A) We need to calculate Z value here

Z= (X'-)/

So.,

Z= (2500-2250)/175 =  1.428571

P(Z<1.428571) = 0.9234363 (From z table look for area till Z=1.43 we will get area under the curve as 0.9234363)

So., The probability that the useful life of a projector lamp selected at random is shorter than or equal to 2500 hours is 0.9234363

B) Same way The probability that the useful life of a projector lamp selected at random is shorter than or equal to 2100 hours is

Z= (X'-)/

So.,

Z= (2100-2250)/175 = -0.8571429

P(Z<1.428571) = 0.195683(From z table look for area till Z=-0.86 we will get area under the curve as 0.195683)

So., The probability that the useful life of a projector lamp selected at random is shorter than or equal to 2100 hours is 0.195683

C) Same way The probability that the useful life of a projector lamp selected at random is longer than 2300 hours

Z= (2300-2250)/175 =  0.2857143

(From z table look for area till Z=0.2857 we will get area under the curve as 0.6124515)

So., The probability that the useful life of a projector lamp selected at random is more than or equal to 2300 hours is 1-0.6124515 = 0.3875485

D) Same way

The probability that the useful life of a projector lamp selected at random is longer than 2200 hours is closest to

Z= (2200-2250)/175 = -0.2857143

(From z table look for area till Z=-0.2857 we will get area under the curve as 0.3875485)

So., The probability that the useful life of a projector lamp selected at random is more than or equal to 2200 hours is 1-0.3875485= 0.6124515

Due to time crunch and as per theChegg policy we are supposed to answer 1st 4 questions at a time here I have answered all 1st 4 questions. If you want the ans for the next que kindly post it seperately or repost the same que by specifically asking the reply for rest ques. Hope you understand.

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