# How Does the Central Limit Theorem Apply to Processing Times?

• Millacol88
In summary, the problem involves determining the probability of processing a certain number of customers' orders within a given time frame, using the central limit theorem due to the lack of information about the distribution of processing times except for mean and standard deviation. The central limit theorem states that the sum (or average) of a large number of independent and identically distributed random variables will follow a normal distribution, regardless of the underlying distribution of the individual variables. Therefore, we can use the normal distribution and its properties to approximate the probabilities in this problem.
Millacol88

## 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?

## 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.

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?

## 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?

Last edited:

## 1. What is the Central Limit Theorem?

The Central Limit Theorem (CLT) is a statistical concept that states that the sample means of a large number of independent and identically distributed random variables will follow a normal distribution, regardless of the distribution of the original variables.

## 2. Why is the Central Limit Theorem important?

The CLT is important because it allows us to make accurate predictions and inferences about a population based on a relatively small sample size. It also forms the basis for many statistical tests and models used in data analysis.

## 3. What are the assumptions of the Central Limit Theorem?

The CLT assumes that the sample is random and independent, and that the sample size is large enough (usually at least 30) for the normal distribution to apply.

## 4. Can the Central Limit Theorem be applied to any type of data?

The CLT can be applied to any type of data, as long as the assumptions are met. However, it is most commonly used for continuous numerical data.

## 5. How is the Central Limit Theorem used in practice?

In practice, the CLT is used to estimate population parameters, such as the mean or standard deviation, based on a sample. It is also used to perform hypothesis testing and construct confidence intervals for these parameters.

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