Question

What happens to the T distribution if the sample size increases?

Answer 1

To know the answer of the given question you must know about the probability distribution and there various types such as normal distribution and T distribution. After getting the knowledge of these concepts you can get your answer. So try to understand these concepts.
Complete step-by-step solution:
To get the answer to this question we should understand the concept of the T distribution. The T distribution is also known by the other name and that is Student’s t-distribution. It is a technique to find or to calculate the mean of the given data.
The idea behind the T distribution is to check the hypothesis that whether the hypothesis is accepted or it gets rejected. T distribution is generally used when the sample size of the given data is small in size that is approximately not more than 20 samples.
To calculate the T value the formula used is
$t=\dfrac{(\bar{x}-\mu )}{s/\sqrt{N}}$
Where,
$t=$T value
$\bar{x}=$Mean of the sample
$\mu =$Mean of the population
$s=$Standard deviation of the sample
$N=$Sample size
There are so many different types of T distributions. But there is a concept of degree of freedom which is used to determine the form of T distribution. The degree of freedom is defined as the number of independent observations in a set of data. For estimating a mean value or to estimate a proportion from a single sample, the number of independent observations is always equal to the sample size minus one. Hence, the distribution of the T statistic from sample size of $5$ would be described by the T distribution having $5-1$ or we can say $4$ degrees of freedom. Similarly, if we say $21$ is the sample size then the degree of freedom of T distribution will be $20$.
The T distribution becomes very much similar to the normal distribution if the sample size increases.

Note: Normal distribution is a kind of probability distribution for a real valued random variable. Normal distribution is sometimes called a bell curve due to its shape. If for any sample mean is zero and variance of that sample is one then normal distribution is known as standard normal distribution.