- The t distribution is a family of curves.
- The variance of the t distribution is not necessarily less than 1.
- As the degrees of freedom increase, the t distribution approaches the Chi distribution.
- The median, mean, and mode of the t distribution are not always equal to 1.
- The t distribution can touch or intersect the x-axis.
- The t distribution is bell-shaped and symmetric.
t distribution is similar to the standard normal distribution in some aspects, but there are also important differences. Let's go through each statement to determine which ones are true.
1. It is a family of curves: True. The t distribution is a family of curves, with each curve corresponding to a different degrees of freedom value.
2. The variance is less than 1: False. The variance of the t distribution depends on the degrees of freedom. As the degrees of freedom increase, the variance approaches 1, but it is not necessarily less than 1.
3. As degrees of freedom increase, the t distribution approaches the Chi distribution: True. As the degrees of freedom increase, the t distribution becomes closer to the standard normal distribution. In fact, when the degrees of freedom are very large, the t distribution approximates the standard normal distribution, which is equivalent to the Chi distribution with 1 degree of freedom.
4. Median, mean, and mode all equal to 1: False. The median, mean, and mode of the t distribution are not always equal to 1. They can vary depending on the degrees of freedom and the location parameter of the distribution.
5. Never touches the x-axis: False. The t distribution can touch or intersect the x-axis. In fact, for certain degrees of freedom, the t distribution can have values below zero.
6. Bell-shaped and symmetric: True. Like the standard normal distribution, the t distribution is bell-shaped and symmetric, with the peak at the mean and the tails extending indefinitely in both directions.