Lightbulbs A factory produces lightbulbs that have a lifespan which is left-skew

Question

Lightbulbs

A factory produces lightbulbs that have a lifespan which is left-skewed with a mean of 1000 hours and a standard deviation of 39 hours. The factory needs to keep an eye on the production process to ensure that everything is working properly and that lightbulbs produced meet the advertised standard. This is done by testing the lifespans of a random sample of 36 in order to test the process.

a.) Describe the sampling distribution of the sample mean lifetime of lightbulbs (.

b.) If the process is working correctly, what is the probability that the sample will produce a mean lifetime (of less than 990 hours?

c.) What are the upper and lower boundaries of the middle 80% of the sample mean lifetime (

d.) Explain why we cannot determine the probability that a random sample of 10 lightbulbs has a mean lifespan less than 990 hours.

Explanation / Answer

here as sample size is greater then 30 from central limit theory  sampling distribution will follow normal distribution with

mean =1000 Hours and and std error of mean =std deviation/(n)1/2 =6.5

b) P(X<990)=P(Z<(990-1000)/6.5)=P(Z<-1.5385)=0.0620

c)for 80% CI, z=-/+1.2816

hence lower boundary =1000-1.2816*6.5= 991.67

and upper boundary =1000+1.2816*6.5=1008.33

d) as sample size is very small and shape of population distribution is know we can not claim that sampling distribution will be normal; therefore cannot determine the probability that a random sample of 10 lightbulbs has a mean lifespan less than 990 hours

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