t Distribution and Chi Squared Distribution

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t Distribution

The t Distribution is a probability distribution that arises when estimating the mean of a normally distributed population in situations where the sample size is small and/or the population standard deviation is unknown. It is similar to the normal distribution but has heavier tails, allowing for greater variability.

Formula:

Mathematically, if X is a t-distributed random variable with n degrees of freedom, the probability density function (PDF) is given by:

$f(x) = \frac{\Gamma\left(\frac{n+1}{2}\right)}{\sqrt{n\pi} \Gamma\left(\frac{n}{2}\right) \left(1 + \frac{x^2}{n}\right)^{\frac{n+1}{2}}}$

where:

$\Gamma(x)$ is the gamma function

n is the degrees of freedom

x is the random variable

Example:

Suppose you have a sample of 10 students, and you want to estimate the mean height of all students at your university. If the sample mean is 65 inches and the sample standard deviation is 3 inches, you can use the t distribution to calculate the probability that the true mean height is within a certain range.

Characteristics:

The t distribution is characterized by its degrees of freedom, which depend on the sample size. As the sample size increases, the t distribution approaches the normal distribution.

Chi-Squared Distribution

The Chi-Squared Distribution is a probability distribution that arises in statistical hypothesis testing and confidence interval estimation. It is used to model the sum of squared standard normal deviates and has applications in various fields, including physics, engineering, and finance.

Formula:

Mathematically, if X is a chi-squared random variable with k degrees of freedom, the probability density function (PDF) is given by:

$f(x) = \frac{1}{2^{\frac{k}{2}} \Gamma\left(\frac{k}{2}\right)} x^{\frac{k}{2} - 1} e^{-\frac{x}{2}}$

where:

$\Gamma(x)$ is the gamma function

k is the degrees of freedom

x is the random variable

Example:

In a study investigating the relationship between smoking and lung cancer, researchers want to test if there is a significant difference in lung cancer rates between smokers and non-smokers. They can use the chi-squared distribution to calculate the test statistic and determine the p-value for their hypothesis test.

Characteristics:

The chi-squared distribution is positively skewed and becomes more symmetric as the degrees of freedom increase. It is commonly used in goodness-of-fit tests and tests of independence in contingency tables.