When making a decision on whether a sample mean is significantly different than a population mean given that you sampled a small sample (n), what is the correct reasoning for choosing the t-distribution over the standard normal distribution (z-distribution)?
https://www.google.com/search?client=safari&rls=en&q=statistics+t+vs+Z+distribution&ie=UTF-8&oe=UTF-8
When deciding whether to use the t-distribution or the standard normal distribution (z-distribution) to analyze a small sample mean, there are a few key factors to consider.
1. Sample size (n): The t-distribution is appropriate when the sample size is small (typically n < 30), whereas the standard normal distribution can be used when the sample size is large (typically n ≥ 30). This is because the t-distribution accounts for the uncertainty associated with estimating the population standard deviation based on a small sample size.
2. Population standard deviation (σ): If the population standard deviation is known, you can use the z-distribution regardless of the sample size. However, in most cases, the population standard deviation is unknown, and it needs to be estimated using the sample data. In this situation, the t-distribution is used to account for the additional uncertainty introduced by the estimation process.
3. Shape of the population: The t-distribution assumes that the population from which the sample is drawn follows a normal distribution. If the population distribution is not approximately normal, using the t-distribution may still be valid as long as the sample size is large enough (n ≥ 30) due to the Central Limit Theorem.
To summarize, the correct reasoning for choosing the t-distribution over the standard normal distribution when analyzing a small sample mean is based on the sample size, the knowledge of the population standard deviation, and the assumption of a normal population distribution. If the sample size is small (n < 30) or if the population standard deviation is unknown, the t-distribution is the appropriate choice.