The t-distribution has a variance that is greater than one
True or false?
Google says "YES" : )
False. The t-distribution has a variance that is equal to 1 when the distribution is standardized.
True. The t-distribution has a variance that is greater than one.
To understand why, let's first briefly explain what the t-distribution is and how it relates to the normal distribution. The t-distribution is a probability distribution that is used when dealing with small sample sizes or when the population standard deviation is unknown. It is similar to the normal distribution, but it has heavier tails, meaning it has more extreme values than the normal distribution.
Now, to determine whether the t-distribution has a variance greater than one, let's compare it to the normal distribution. The variance of the normal distribution is always equal to one, regardless of the sample size. On the other hand, the variance of the t-distribution depends on the degrees of freedom.
The degrees of freedom (df) is a parameter of the t-distribution that represents the sample size minus one. As the degrees of freedom increase, the t-distribution becomes more similar to the normal distribution. When the degrees of freedom are very large, say greater than 30, the t-distribution is nearly identical to the normal distribution.
Now, the variance of the t-distribution is given by df/(df-2) if df > 2. In this case, since df is always greater than two (because sample size must be at least three to compute a variance), the variance of the t-distribution is always greater than one. This is because the numerator (df) is always greater than the denominator (df-2).
Therefore, we can conclude that the t-distribution has a variance that is greater than one.