Which of the following is a similarity between the $t$-distribution and the standard normal ($z$) distribution?

• A. Both distributions are symmetric and bell-shaped.
• B. Both distributions are only used when the population standard deviation is known.
• C. Both distributions have the same variance for all sample sizes.
• D. Both distributions are skewed to the right.

Answer: (A)

Verified step by step guidance

1. Step 1: Understand the characteristics of the standard normal (z) distribution. It is symmetric, bell-shaped, has a mean of 0, and a standard deviation of 1. It is used when the population standard deviation is known.
2. Step 2: Understand the characteristics of the t-distribution. It is also symmetric and bell-shaped but has heavier tails than the normal distribution. It is used when the population standard deviation is unknown and the sample size is small.
3. Step 3: Compare the shapes of both distributions. Both are symmetric and bell-shaped, meaning they have the same general shape centered around their mean.
4. Step 4: Consider the variance of both distributions. The standard normal distribution has a fixed variance of 1, while the t-distribution's variance depends on the degrees of freedom and is generally larger for smaller sample sizes.
5. Step 5: Evaluate the options given in the problem. The correct similarity is that both distributions are symmetric and bell-shaped, while the other options are incorrect based on the properties of these distributions.