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In a t-distribution, the scale parameter, also known as the standard error, is inversely related to the square root of the sample size. It adjusts the spread of the distribution to accurately reflect the precision of the sample mean estimate.

The shape of the t-distribution is heavily influenced by the sample size, due to its dependency on the degrees of freedom. Smaller samples yield a spread-out t-distribution with heavier tails, while larger samples result in a shape closer to normal distribution.

A t-distribution is a type of probability distribution that is symmetric and bell-shaped, similar to the normal distribution, but with heavier tails. It arises when estimating the mean of a normally distributed population in situations where the sample size is small and population standard deviation is unknown.

The t-distribution is extensively used in hypothesis testing, especially for tests involving means such as the one-sample t-test and two-sample t-test. It helps determine whether the observed sample mean significantly differs from the hypothesized or true mean.

The tail probability in a t-distribution refers to the likelihood of occurrence of values at the extreme ends of the distribution (the tails). It is higher compared to a normal distribution, allowing for more extreme values.

While both t-distribution and normal distribution are symmetric and bell-shaped, the t-distribution has heavier tails, which means it has a higher probability of values further from the mean. As the degrees of freedom increase, the t-distribution approaches the normal distribution.

The degrees of freedom in a t-distribution refer to the number of independent values in a dataset that are allowed to vary when estimating a statistical parameter. It is calculated as the sample size minus one ($n - 1$).