Question

Normal distribution. We expect an LED light bulb to last nine years with a standard deviation of two years. It follows a normal distribution. Find the probability that a light bulb will last between five and seven years.

Answer 1

X :Life of a LED light bulb

X follows normal distribution with mean 9 years and standard deviation 2 years

probability that a light bulb will last between five and seven years = P(5<X<7) = P(X<7)-P(X<5)

Z-score for 7 = (7-Mean)/standard deviation = (7-9)/2 = -2/2=-1

From standard normal tables , P(Z<-1) = 0.1587

P(X<7) = P(Z<-1) = 0.1587

Z-score for 5 = (5-Mean)/standard deviation = (5-9)/2 = -4/2=-2

From standard normal tables , P(Z<-2) = 0.0228

P(X<5) = P(Z<-2) = 0.0228

P(5<X<7) = P(X<7)-P(X<5) = 0.1587- 0.0228=0.1359

probability that a light bulb will last between five and seven years = 0.1359

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Using Emperical rule

The empirical rule states that for a normal distribution, nearly all of the data will fall within three standard deviations of the mean. The empirical rule can be broken down into three parts:

• 68% of data falls within the first standard deviation from the mean.
• 95% fall within two standard deviations.
• 99.7% fall within three standard deviations.

i.e

5 : mean - 2 standard deviations

7 : mean - 1 standard deviation

9 : mean

P(5<x<7) = 13.5% =0.135