As a sample size approaches infinity, how does the student’s t distribution compare to the normal z distribution?
To understand how the student's t-distribution compares to the normal z-distribution as the sample size approaches infinity, we need to discuss the concept of the central limit theorem and its implications.
The Central Limit Theorem states that as the sample size increases, the sampling distribution of the sample mean approaches a normal distribution, regardless of the shape of the original population distribution. This theorem holds under certain assumptions, such as the random sampling of observations and the independence of each observation within the sample.
So, as the sample size approaches infinity, both the student's t-distribution and the normal z-distribution converge towards a standard normal distribution. In other words, they become increasingly similar.
The difference between the student's t-distribution and the normal z-distribution lies in the fact that the t-distribution takes into account the uncertainty in estimating the population standard deviation. This uncertainty is due to the fact that the population standard deviation is estimated using the sample standard deviation, which is subject to variation.
For small sample sizes, the t-distribution is more spread out and has heavier tails compared to the normal z-distribution. This means that there is more uncertainty in estimating the population mean when working with a small sample size.
However, as the sample size becomes larger, the t-distribution approaches the normal distribution because the sample standard deviation becomes a better estimate of the population standard deviation. Consequently, the student's t-distribution becomes very similar to the standard normal distribution (z-distribution) as the sample size increases.
Therefore, as the sample size approaches infinity, the student's t-distribution becomes indistinguishable from the normal z-distribution in practice.