Question

Use R code to answer questions Use the CLT to approximate probabilities. Use the CLT to answer these questions.

(a) Suppose that you have a sample of 100 values from a population with meanμ= 500 andwith standard deviationσ= 80. i. What is the (approximate) probability that the sample mean will be in the interval(490,510)? ii. Give an interval that covers (approximately) the middle 95% of the distribution ofthe sample mean.

(b) A large freight elevator can transport a maximum of 9800 pounds. Suppose a load ofcargo containing 49 boxes must be transported via the elevator. Experience has shownthat the weight of boxes of this type of cargo follows a distribution with meanμ= 205pounds and standard deviationσ= 15 pounds. Based on this information, what is the(approximate) probability that all 49 boxes can be safely loaded onto the freight elevatorand transported?

(c) A batch of 100 steel rods passes inspection if the average of their diameters falls between0.495 cm and 0.505 cm. Letμandσdenote the mean and standard deviation, respec-tively, of the diameter of a randomly selected rod. Assuming thatμ= 0.503 cm andσ= 0.03 cm, what is the (approximate) probability the batch will pass inspection?

(d) Two towers are independently constructed, each by stacking 30 segments of concretevertically. The height (in inches) of a randomly selected segment is uniformly distributedbetween 35.5 in and 36.5 in. A roadway can be laid across the 2 towers provided theheights of the 2 towers are within 4 inches of each other. Find the probability that theroadway can be laid. State whether your probability is approximate or exact.Hint: IfX1∼N(μ1,σ21),X2∼N(μ2,σ22), andX1andX2are independent, thenX1−X2∼N(μ1−μ2,σ21+σ22).

Answer 1