Answer to Question #199932 in Statistics and Probability for John

Question #199932

A LED company claims that the average life of the LED light bulbs it

manufactures is 1, 500 hours with a standard deviation of 500 hours. If a random sample of 40 bulbs is chosen, what is the probability that the sample mean will be:


a. greater than 1, 400 hours?


b. less than 1, 400 hours?


1
Expert's answer
2021-05-31T00:04:56-0400

Let X=X= the average life of the LED light bulb: XN(μ,σ2/n).X\sim N(\mu, \sigma ^2/n).

Given μ=1500 h,σ=500 h,n=40\mu=1500\ h, \sigma=500\ h, n=40


a)

P(X>1400)=1P(1400)P(X>1400)=1-P(\leq1400)

=1P(14001500500/40)1P(Z1.264911)=1-P(\leq\dfrac{1400-1500}{500/\sqrt{40}})\approx1-P(Z\leq-1.264911)

0.897048\approx0.897048

b)

P(X<1400)=P(Z<14001500500/40)P(X<1400)=P(Z<\dfrac{1400-1500}{500/\sqrt{40}})

P(Z<1.264911)0.102952\approx P(Z<-1.264911)\approx0.102952



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