Please include all steps and be 100% accurate Thank you so much Before burning o

Question

Please include all steps and be 100% accurate

Thank you so much

Before burning out, a light bulb gives X hours of light, where X is normally distributed with mean 100, and standard deviation 100. Namely X~N(100,1002). If we have 4 bulbs, what is the probability that they will give a total of at least 560 hours? (Critical point for standard normal distribution: z0 5 = 0.0, z0.55 = 0.1, z0.6 = 0.2, z0 65 = 0.3, z0 7 = 0.5, z0 75 = 0.6, z0.8 = 0.8, z0.85 = 1.0, z0.9 = 1.2, z0.95 = 1.6, Z0.99 = 2.3, z0.999 = 3.0. If you cannot find the exact number, Please use the closest number as an approximation.)

Explanation / Answer

Total burning hours x=x1+x2+x3+x4
since x1,x2,x3,x4 all follows normal (100, 1002 )

so x follows N(400,4*1002 )

so P(x560)=1-P(x<560)=1-P((x-400)/200<(560-400)/200)=1-P(z<0.8)

where z is the standard normal variable.

so from the list we find the P(z<0.8) is 0.8

required probability is 0.2 (ans)

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