Which of the following is not a property of the t distribution?

• A. The variance of the t distribution is always equal to $1$ for all degrees of freedom.
• B. The t distribution is symmetric about $0$.
• C. The t distribution has heavier tails than the normal distribution.
• D. As the degrees of freedom increase, the t distribution approaches the standard normal distribution.

Answer: (A)

Verified step by step guidance

1. Step 1: Understand the t distribution and its key properties. The t distribution is a probability distribution used in statistics, especially when dealing with small sample sizes and unknown population standard deviations.
2. Step 2: Recall that the t distribution is symmetric about zero, meaning its shape is mirrored on both sides of the mean (which is zero). This is a fundamental property.
3. Step 3: Recognize that the t distribution has heavier tails than the normal distribution. This means it gives more probability to values far from the mean, which accounts for extra uncertainty with small samples.
4. Step 4: Know that as the degrees of freedom (df) increase, the t distribution approaches the standard normal distribution. This is because with more data, the estimate of the population standard deviation becomes more accurate.
5. Step 5: Understand the variance of the t distribution depends on the degrees of freedom and is not always equal to 1. Specifically, the variance is given by \(\frac{\nu}{\nu - 2}\) for degrees of freedom \(\nu > 2\), which means it changes with \(\nu\) and is only equal to 1 in the limit as \(\nu\) approaches infinity.