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Standard deviation of normally distributed battery lifetimes from a percentage, then a symmetric value with an equal tail probability

Standard deviation of normally distributed battery lifetimes from a percentage, then a symmetric value with an equal tail probability

7 marks

The lifetimes of a brand of battery are normally distributed with mean \(410\) hours. Of these batteries, \(70\%\) have a lifetime of less than \(445\) hours.

  1. Find the standard deviation of the lifetimes.[3]

One battery is chosen at random. The probability that its lifetime is less than \(480\) hours is equal to the probability that its lifetime is more than \(b\) hours.

  1. [4]
    1. Find the probability that the lifetime of this battery is less than \(480\) hours.
    2. Find the value of \(b\).

Solution:

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Answer:(a) \(\sigma=66.7\) hours (b)(i) \(0.853\) (b)(ii) \(b=340\)

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