A battery manufacturer claims that the lifetime of a certain type of battery has

Question

A battery manufacturer claims that the lifetime of a certain type of battery has a population mean of 40 hours and standard deviation of 5 hours. Let X represent the mean lifetime of the batteries in a simple random sample of size 100. a., If the claim is true, what is P (X lessthanorequalto 36.7)? Based on the answer to part a), if the claim is true, is a sample mean lifetime of 36.7 hours unusually short? If the claim is true. what is P(X lessthanorequalto 39.8)? Based on the answer to part c), if the claim is true, is a sample mean lifetime of 39.8 hours unusually short?

Explanation / Answer

a) mean = 40 , s = 5 , n =100 , x = 36.7

By normal distribution formula,

z =( x- mean)/ ( s / sqrt(n))

= ( 36.7 - 40) / ( 5 / sqrt(100))

= -6.6

Now,we need to find p(z < -6.6)

p(x < 36.7) = p(z < -6.6) = 0

c)

mean = 40 , s = 5 , n =100 , x = 39.8

By normal distribution formula,

z =( x- mean)/ ( s / sqrt(n))

= ( 39.8 - 40) / ( 5 / sqrt(100))

= -0.4

Now,we need to find p(z < - 0.4)

p(x < 39.8) = p(z < - 0.4) = 0.3446

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