Central Limit Theorem: Fisheries Management

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The Central Limit Theorem (CLT) states that under certain conditions, the distribution of the sample mean of independent and identically distributed random variables approaches a normal distribution as the sample size increases. In fisheries management, this means that if fish ages are sampled, the average age of a large sample will tend to follow a normal distribution, allowing for better statistical analysis. Specifically, the formula \(\frac{\sqrt{n}}{\sigma}(\frac{1}{n}\sum_{i=1}^{n}Y_i-\mu) \rightarrow N(0,1)\) illustrates this concept. For smaller sample sizes (n < 30), the t-distribution should be used instead of the normal distribution for more accurate results. Understanding the CLT is crucial for effective fisheries management and population assessments.
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:rolleyes: just a query... does anyone have a general definition of the central limit theorom. I've been looking on the internet and all I've got is a whole lot of complex crap

P.S it would help if it was related to fisheries management
 
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simply put: under certain regularity conditions (finiteness of mean and variance), given Y_i are independent and identically distributed random variables.

\frac{\sqrt{n}}{\sigma}(\frac{1}{n}\sum_{i=1}^{n}Y_i-\mu) \rightarrow N(0,1)

You can think of the Y_i as fish that you are sampling from a population of fish who's mean age and variance you know. Then if you sample a large amount of fish. The average age of your sampled fish standardized as above will be approximately normal(0,1).
 
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the one we were taught is that sqrt(n)(X-u)/(s) - >N(0,1) if s>30 otherwise you got to use the t distribution where s is the sample std deviation
 
ya so if your n is not large enough, the approximation is not comfortable enough to use the normal. that's fine.
 
yeh so as n->infinity the t distrubution tends to the normal
 

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