Variables with t and Chi-sq distributions

In summary, a variable with t distribution is a type of continuous probability distribution used for modeling random variables with small sample sizes and unknown population standard deviations. A variable with chi-squared distribution is used for modeling the sum of squares of independent random variables and for testing categorical data and population variances. The t distribution can be derived from the chi-squared distribution, and their degrees of freedom are used in the calculation of critical values and probabilities. T and chi-squared distributions have various applications in scientific research, particularly in psychology, biology, and social sciences, for hypothesis testing, confidence interval calculations, and modeling random variables.
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musicgold
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Hi,

I wish to know if there are any naturally occurring variables that have a t-distribution or chi-distribution.

I know that test statistics such as the mean or the sum of a sample has a t-distribution, where as the variance of a sample takes the chi-distribution. What I am trying to understand is why these statistics take a particular distribution.

Thanks,

MG.
 
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What is a variable with t distribution?

A variable with t distribution is a type of continuous probability distribution that is used to model the behavior of a random variable when the sample size is small and the population standard deviation is unknown. It is commonly used in hypothesis testing and confidence interval calculations.

What is a variable with chi-squared distribution?

A variable with chi-squared distribution is a type of continuous probability distribution that is used to model the behavior of a random variable when the variable is the sum of the squares of other independent random variables. It is commonly used in statistical tests for categorical data and in hypothesis testing for population variances.

What is the relationship between t and chi-squared distributions?

The t distribution can be derived from the chi-squared distribution by taking the square root of a chi-squared variable divided by its degrees of freedom. This relationship allows for the calculation of confidence intervals and hypothesis tests involving t distributions using chi-squared distributions.

How do you calculate the degrees of freedom for a t or chi-squared distribution?

The degrees of freedom for a t distribution is equal to the sample size minus one. For a chi-squared distribution, the degrees of freedom is equal to the number of categories or variables being tested minus one. These values are used in the calculation of critical values and probabilities for these distributions.

What are the applications of t and chi-squared distributions in scientific research?

T and chi-squared distributions are commonly used in hypothesis testing, confidence interval calculations, and modeling the behavior of random variables in scientific research. They are particularly useful for small sample sizes and for testing hypotheses about population parameters when the population standard deviation is unknown. These distributions are commonly used in fields such as psychology, biology, and social sciences.

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