My Math Forum  

Go Back   My Math Forum > High School Math Forum > Probability and Statistics

Probability and Statistics Basic Probability and Statistics Math Forum


Thanks Tree1Thanks
  • 1 Post By romsek
Reply
 
LinkBack Thread Tools Display Modes
December 11th, 2017, 07:38 AM   #1
Member
 
Joined: Jan 2016
From: Blackpool

Posts: 95
Thanks: 2

Central limit theorem question

Times spent on processing orders are independent random variables
with mean 1.5 minutes and standard deviation 1 minute. Let n be the number of orders an operator is scheduled to process in 2 hours. Use the CLT to find the largest value of n which give at least a 95% chance of completion in that time.
Jaket1 is offline  
 
December 11th, 2017, 08:29 AM   #2
Senior Member
 
romsek's Avatar
 
Joined: Sep 2015
From: USA

Posts: 1,977
Thanks: 1026

CLT says that if each task duration $T_k$ has mean $\mu$ and standard deviation $\sigma$ then

$S_n = \dfrac 1 n \sum \limits_{k=1}^n T_k \to N\left(\mu, \dfrac{\sigma}{\sqrt{n}}\right)$

so we need to find $n \ni P\left[S_n < \dfrac {200}{n}\right] > 0.95$

This in turn means that

$\Phi\left(\dfrac{\frac{200}{n}-\mu}{\frac{\sigma}{\sqrt{n}}}\right)> 0.95$

if we let $p=\Phi^{-1}(0.95)$ then we can solve for $n$ as

$n = \left \lfloor \dfrac{400 \mu +p^2 \sigma ^2-\sqrt{p^4 \sigma ^4+800 \mu p^2 \sigma ^2}}{2 \mu ^2}\right \rfloor$

using $\mu=1.5,~\sigma=1,~p=1.645$

we get

$n = 121$
romsek is offline  
December 11th, 2017, 04:24 PM   #3
Member
 
Joined: Jan 2016
From: Blackpool

Posts: 95
Thanks: 2

Hi Romsek, where does the number 200 come from?
Jaket1 is offline  
December 11th, 2017, 06:49 PM   #4
Senior Member
 
romsek's Avatar
 
Joined: Sep 2015
From: USA

Posts: 1,977
Thanks: 1026

Quote:
Originally Posted by Jaket1 View Post
Hi Romsek, where does the number 200 come from?
jeeze.. from me multiplying 2x60 and getting 200 instead of 120.

Must have been before coffee.

Replace it with 120.
Thanks from Jaket1
romsek is offline  
December 12th, 2017, 03:01 AM   #5
Member
 
Joined: Jan 2016
From: Blackpool

Posts: 95
Thanks: 2

haha thank you!
Jaket1 is offline  
December 12th, 2017, 11:23 AM   #6
Senior Member
 
romsek's Avatar
 
Joined: Sep 2015
From: USA

Posts: 1,977
Thanks: 1026

$n = \left \lfloor \dfrac{240 \mu +p^2 \sigma ^2-\sqrt{p^4 \sigma ^4+480 \mu p^2 \sigma ^2}}{2 \mu ^2} \right \rfloor $

$n = 70$
romsek is offline  
Reply

  My Math Forum > High School Math Forum > Probability and Statistics

Tags
central, limit, question, theorem



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Central limit theorem phideon Advanced Statistics 1 November 28th, 2015 07:21 AM
Central limit theorem question? green21317 Probability and Statistics 1 July 4th, 2014 04:36 PM
the How To's of Central Limit Theorem NeedToLearn Advanced Statistics 1 June 21st, 2013 12:26 PM
Combinatorics + Central Limit Theorem Question hgrin123 Advanced Statistics 2 March 20th, 2009 12:05 PM
Central Limit Theorem...help dbcmjk17 Advanced Statistics 3 June 27th, 2008 01:08 PM





Copyright © 2018 My Math Forum. All rights reserved.