Central Limit Theorem states that when numerous independent random variables with a known variance and mean are statistically analyzed, they will assume a normal distribution with a bell shape. This is true in real life since most events tend to be near the population mean with the number of events decreasing as you move away from the mean in both sides. Therefore, most occurrences in real life that appear random will assume a bell-shaped normal distribution when statistically analyzed .
Estimation and confidence interval are important in approximating the range of values of a population parameter using a sample from the population
Confidence interval represents the range of values of a population parameter within which we are confident that the parameter being considered lies.
The percentage of times will the mean, or population proportion, not be found within the confidence interval depends on the confidence level given . For example if the confidence level given is 95% then it will be 5 %( 100-95).
T-distribution takes into consideration that the population variance is unknown. However as the degrees of freedom increase due to an increase in the sample size, the population variance can be determined with accuracy. Central Limit Theorem states that sample distribution approximates a normal distribution when the sample size is large enough. The T-distribution therefore begins to approach the normal Z distribution when the sample size is greater than 30 .
A sample obtained from a normally distributed population will have a normal sample distribution.
Iversen, G. R., & Gergen, M. M. (2006). Statistics: the conceptual approach (revised ed.). Chicago: Springer.
Johnson, R. A., & Bhattacharyya, G. K. (2009). Statistics: Principles and Methods (2nd Edition ed.). New York: John Wiley and Sons.