MHB Central Limit Theorem & Gamma Distribution

AI Thread Summary
The discussion focuses on applying the Central Limit Theorem (CLT) to estimate the probability that the average completion time of 64 independent projects falls within 15 minutes of the true mean, given an exponential distribution with a parameter of β=2 hours. Participants highlight that for large sample sizes, the distribution of the sample mean approximates a normal distribution with specific mean and variance. The calculations involve determining the mean and variance for the exponential distribution, leading to a probability calculation using the error function (erf). The final probability is approximated to be around 0.999925, indicating a high likelihood that the average completion time will meet the specified criteria. Understanding these statistical concepts is crucial for accurately applying the CLT in practical scenarios.
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The time it takes to complete a project is a random variable Y with the exponential distribution with parameter β=2 hours.

Apply the central limit theorem to obtain an approximation for the probability that the average project completion time of a sample of n=64 projects undertaken independently over the last year will be within 15 minutes of the true mean completion time.

Any ideas on even how to start this? :\
 
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Hi there,

Welcome to MHB :)

The central limit theorem states that for a sufficiently large $n$ the value of [math]\frac{S_n-n \mu}{\sigma \sqrt{n}}[/math] is approximately normal. So if you plug in the given information plus make some inferences from the fact that you have an exponential distribution, you can figure this out.

Have you done anything like this already? If you haven't seen it done it's not a process that I think many would just be able to do through intuition.

Jameson
 
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Jameson said:
Hi there,

Welcome to MHB :)

The central limit theorem states that for a sufficiently large $n$ the value of [math]\frac{S_n-n \mu}{\delta \sqrt{n}}[/math] is approximately normal. So if you plug in the given information plus make some inferences from the fact that you have an exponential distribution, you can figure this out.

Have you done anything like this already? If you haven't seen it done it's not a process that I think many would just be able to do through intuition.

Jameson

Thanks for the quick reply. We've done a few similar examples, but like most of our homework, none of the questions match the practice problems.
 
dcht said:
The time it takes to complete a project is a random variable Y with the exponential distribution with parameter β=2 hours.

Apply the central limit theorem to obtain an approximation for the probability that the average project completion time of a sample of n=64 projects undertaken independently over the last year will be within 15 minutes of the true mean completion time.

Any ideas on even how to start this? :\

A good starting point may be to read the posts...

http://www.mathhelpboards.com/f23/unsolved-statistics-questions-other-sites-932/index9.html#post7118

http://www.mathhelpboards.com/f23/unsolved-statistics-questions-other-sites-932/index9.html#post7147

... where is explained that for n 'large enough' the p.d.f. pf the mean of n r.v. each ot them with mean $\mu$ and variance $\sigma^{2}$ is a normal distribution with mean $\mu$ and variance $\displaystyle \sigma^{2}_{n}=\frac{\sigma^{2}}{n}$. In Your case is $\displaystyle \mu=\sigma=\frac{1}{2}$, so that is $\displaystyle \sigma_{n}= \frac{1}{16}$ and the requested probability is...$\displaystyle P= \text{erf} (\frac{4}{\sqrt{2}})= .99993666575...$

$\chi$ $\sigma$
 
The effective computation of erf(x) for x 'large enough' [say x>2.5...] may be a difficult task and in these cases the identity erf(x)= 1-erfc(x) may be sucessfully used. Several years ago I created the annexed table of the erfc(x) function. In this case is $\displaystyle x=\frac{4}{\sqrt{2}} \sim 2.82$ so that is $\displaystyle \text {erfc}\ (x) \sim 7.5\ 10^{-5} \implies \text{erf}\ (x) \sim .999925$...

Kind regards

$\chi$ $\sigma$ View attachment 448
 

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