Final answer:
The statement is true; as the sample size n increases, the t-distribution approaches the standard normal distribution. This is because the variability of the t-distribution, which has thicker tails, decreases with increasing degrees of freedom, resulting in a shape more similar to the z-distribution.
Step-by-step explanation:
True. The t-distribution and the z-distribution are both used in statistics to conduct hypothesis testing and create confidence intervals. As the sample size n increases, the t-distribution approaches the standard normal distribution (z-distribution). This occurs because the t-distribution has thicker tails, meaning it accounts for more variability in smaller samples. As the sample size grows and the degrees of freedom increase (degrees of freedom are one less than the sample size), the excess variability in the t-distribution diminishes, making it resemble the standard normal distribution more closely.
The central limit theorem also supports this concept by stating that with a sufficiently large sample size, the sampling distribution of the means will approximate a normal distribution regardless of the initial distribution of the data.
It is important to note that the Family of t-distributions varies depending on the number of degrees of freedom, and as the degrees of freedom increase, the t-distribution graph becomes more like that of the standard normal distribution. However, with a smaller number of surveyed individuals (for example, 15), it may not be appropriate to use the normal distribution.