Central Limit Theorem: Fisheries Management

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SUMMARY

The Central Limit Theorem (CLT) is a fundamental statistical principle stating 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 standardized average age of the sampled fish will approximate a normal distribution, particularly when the sample size exceeds 30. The formula \(\frac{\sqrt{n}}{\sigma}(\frac{1}{n}\sum_{i=1}^{n}Y_i-\mu) \rightarrow N(0,1)\) illustrates this concept, emphasizing the importance of sample size in achieving reliable results.

PREREQUISITES
  • Understanding of statistical concepts such as mean and variance
  • Familiarity with independent and identically distributed (i.i.d.) random variables
  • Knowledge of the normal distribution and t-distribution
  • Basic proficiency in sampling techniques in fisheries management
NEXT STEPS
  • Study the implications of the Central Limit Theorem in ecological statistics
  • Learn about sampling methods specific to fisheries management
  • Explore the differences between the normal distribution and t-distribution
  • Investigate statistical software tools for analyzing fish population data
USEFUL FOR

This discussion is beneficial for statisticians, fisheries biologists, and researchers involved in fisheries management who seek to apply statistical principles to analyze fish populations effectively.

majin
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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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