How Does the Central Limit Theorem Apply to Processing Times?

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The Central Limit Theorem (CLT) applies to the processing times of cashiers, as the times are independent random variables with known mean and standard deviation. To determine the probability of processing over 4 hours for 100 customers, the CLT can be used to approximate the distribution of the sum of processing times. For part b, the number of orders that can be processed in 5 hours with 95% probability can also be estimated using the CLT. Additionally, the probability of processing larger orders within 5.5 hours can be calculated by incorporating the adjusted mean and standard deviation for those orders. Understanding the CLT is crucial for solving these types of problems effectively.
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Homework Statement



The amounts of time that a cashier spends processing individual customers' orders are independent random variables with mean 2.5 minutes and standard deviation 2 minutes.

a) What is the approximate probability that it will take more than 4 hours to process orders of 100 people?

b)How many orders, at least, will be processed in 5 hours with probability 0.95?

c)Some orders are bigger and their mean processing time is 5 minutes with standard deviation of 3 minutes. If the probability of processing bigger orders is 0.2, what is the approximate probability that it will not take more than 5.5 hours to process orders of 100 customers?

Homework Equations

The Attempt at a Solution


This seems like it would involve using the central limit theorem, since no information is given about the distribution except for mean and standard deviation. I'm unsure how to apply it though.
 
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Millacol88 said:

Homework Statement



The amounts of time that a cashier spends processing individual customers' orders are independent random variables with mean 2.5 minutes and standard deviation 2 minutes.

a) What is the approximate probability that it will take more than 4 hours to process orders of 100 people?

b)How many orders, at least, will be processed in 5 hours with probability 0.95?

c)Some orders are bigger and their mean processing time is 5 minutes with standard deviation of 3 minutes. If the probability of processing bigger orders is 0.2, what is the approximate probability that it will not take more than 5.5 hours to process orders of 100 customers?

Homework Equations

The Attempt at a Solution


This seems like it would involve using the central limit theorem, since no information is given about the distribution except for mean and standard deviation. I'm unsure how to apply it though.

What do YOU think the Central Limit Theorem (not Central Limiting Theorem) says? Why do you think you can use it in this problem?
 
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