Final answer:
The average of independent random variables from any distribution approximates a normal distribution as the sample size becomes large, due to the Central Limit Theorem. The correct answer is B) n is large.
Step-by-step explanation:
The student has posed a question related to the distribution of the average value of independent instances of random variables. Specifically, the student is asking which option correctly states when the average of these instances approximates a t-distribution. The correct answer is B) n is large. However, to provide a more detailed explanation, we'll need to clarify that the central tendency of the averages approaches a normal distribution as the sample size (n) gets larger, due to the Central Limit Theorem. The t-distribution comes into play when sample sizes are small and especially when the population standard deviation is unknown. In such cases, we use the t-distribution, which is more spread out relative to the normal distribution, with heavier tails to account for the added uncertainty and variability in the estimate. The t-distribution converges to the normal distribution as the sample size increases.
Historically, statisticians defaulted to the normal distribution for large sample sizes, typically over 30, and utilized the t-distribution for smaller sample sizes. With modern computational tools, the t-distribution can be used more frequently as a more precise model when estimating population parameters from sample statistics. The number of degrees of freedom in the t-distribution is one less than the sample size, highlighting that the distribution becomes more normal as the number of data points increases. Also, for sums of random variables from sufficiently large samples, they are normally distributed. For practical applications such as the binomial and Poisson distributions, different rules apply for approximations. For instance, Poisson can be an approximation of the binomial given a large number of trials and a low probability of success.