Question #339626

 A bulb manufacturer claims that the lives of its bulbs are normally distributed with a mean of 6000 hours and a standard deviation of 400 hours. A random sample of 16 bulbs had an average life of 5850 hours. If the manufacturer’s claim is correct-

a. What is the sampling distribution of the sample mean?

b. what is the probability of finding a sample mean of 5850 or less?


1
Expert's answer
2022-05-12T04:08:28-0400

a. If the population is normally distributed, then the sampling distribution of Xˉ\bar{X} is normally distributed for any sample size n.n.

Let Xˉ=\bar{X}= the sample mean: XˉN(μ,σ2/n).\bar{X}\sim N(\mu, \sigma^2/n).

Given μ=600h,σ=400h,n=16\mu=600h, \sigma=400h, n=16


XˉN(6000,4002/16)\bar{X}\sim N(6000, 400^2/16)

b.


P(Xˉ5850)=P(Z58506000400/16)P(\bar{X}\le5850)=P(Z\le\dfrac{5850-6000}{400/\sqrt{16}})

=P(Z1.5)0.0668=P(Z\le-1.5)\approx0.0668


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