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A property insurance company knows that fire insurance claims have a mean of $507 and a standard deviation of $4000. The insurance company randomly selects 300 fire insurance claims. Let us assume that the amount, X, of any fire insurance claim is independent of the amount of any other fire insurance claim.. Let M be the random variable representing the mean amount of the fire insurance claims for the 300 claims in the sample. Let T = the random variable representing the total amount of the 300 fire insurance claims. a) What theorem will let us treat T and M as approximately normal random variables? O Convolution Theorem O Central Limit Theorem O Monte Carlo Theorem O Chebychev's Theorem O 301 Theorem O Law of Large Numbers b) What is the expected value of T?[ c) What is the standard deviation of T? d) What is the approximate 95th percentile of the distribution of T? e) What is the approximate probability that T is greater than $125,000?

A property insurance company knows that fire insurance claims have a mean of $507 and a standard deviation of $4000. The insurance company randomly selects 300 fire insurance claims. Let us assume that the amount, X, of any fire insurance claim is independent of the amount of any other fire insurance claim.. Let M be the random variable representing the mean amount of the fire insurance claims for the 300 claims in the sample. Let T = the random variable representing the total amount of the 300 fire insurance claims. a) What theorem will let us treat T and M as approximately normal random variables? O Convolution Theorem O Central Limit Theorem O Monte Carlo Theorem O Chebychev's Theorem O 301 Theorem O Law of Large Numbers b) What is the expected value of T?[ c) What is the standard deviation of T? d) What is the approximate 95th percentile of the distribution of T? e) What is the approximate probability that T is greater than $125,000?

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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A property insurance company knows that fire insurance claims have a mean of $507 and a standard deviation of $4000. The insurance company randomly selects 300 fire insurance claims. Let us assume that the amount, X, of any fire insurance claim is independent of the amount of any other fire insurance claim. . Let M be the random variable representing the mean amount of the fire insurance claims for the 300 claims in the sample. Let T = the random variable representing the total amount of the 300 fire insurance claims. a) What theorem will let us treat T and M as approximately normal random variables? O Convolution Theorem O Central Limit Theorem O Monte Carlo Theorem O Chebychev's Theorem O 301 Theorem O Law of Large Numbers b) What is the expected value of T? c) What is the standard deviation of T? d) What is the approximate 95th percentile of the distribution of T? e) What is the approximate probability that T is greater than $125,000? f) What is the standard deviation of M? g) What is the approximate probability M is greater than $305? h) If the insurance company reserves $130,000 to pay the 300 selected claims, then what is the probability they do not have enough reserved to pay the claims?
Transcribed Image Text:A property insurance company knows that fire insurance claims have a mean of $507 and a standard deviation of $4000. The insurance company randomly selects 300 fire insurance claims. Let us assume that the amount, X, of any fire insurance claim is independent of the amount of any other fire insurance claim. . Let M be the random variable representing the mean amount of the fire insurance claims for the 300 claims in the sample. Let T = the random variable representing the total amount of the 300 fire insurance claims. a) What theorem will let us treat T and M as approximately normal random variables? O Convolution Theorem O Central Limit Theorem O Monte Carlo Theorem O Chebychev's Theorem O 301 Theorem O Law of Large Numbers b) What is the expected value of T? c) What is the standard deviation of T? d) What is the approximate 95th percentile of the distribution of T? e) What is the approximate probability that T is greater than $125,000? f) What is the standard deviation of M? g) What is the approximate probability M is greater than $305? h) If the insurance company reserves $130,000 to pay the 300 selected claims, then what is the probability they do not have enough reserved to pay the claims?
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