Which of the following best describes a key difference between the $t$-distribution and the standard normal ($z$) distribution?

• A. The $t$-distribution is discrete, while the standard normal distribution is continuous.
• B. The $t$-distribution is always centered at a value greater than zero, while the standard normal distribution is centered at zero.
• C. The $t$-distribution has heavier tails than the standard normal distribution, especially for small sample sizes.
• D. The standard normal distribution is used only when the population variance is unknown, while the $t$-distribution is used when the population variance is known.

Answer: (C)

Verified step by step guidance

1. Step 1: Understand the nature of both distributions. The standard normal distribution (z-distribution) is a continuous probability distribution that is symmetric and centered at zero with a fixed variance of 1.
2. Step 2: Recognize that the t-distribution is also continuous and symmetric around zero, but it differs from the standard normal distribution in shape depending on the degrees of freedom, which relate to sample size.
3. Step 3: Note that the t-distribution has heavier tails compared to the standard normal distribution. This means it gives more probability to values far from the mean, which accounts for extra uncertainty when estimating the population variance from small samples.
4. Step 4: Understand that the t-distribution approaches the standard normal distribution as the sample size increases (degrees of freedom become large), because the estimate of the population variance becomes more precise.
5. Step 5: Conclude that the key difference is the heavier tails of the t-distribution, especially for small sample sizes, which is why it is used when the population variance is unknown and must be estimated from the sample.