As the degrees of freedom for the t-distribution increase, the distribution approaches
the value of zero for the mean.
the t-distribution.
the normal distribution.
the F distribution.
The correct answer is the normal distribution.
To understand why, let's first explain what degrees of freedom (df) are and how they relate to the t-distribution. In statistics, degrees of freedom refers to the number of independent observations in a statistical analysis. In the case of the t-distribution, the degrees of freedom are associated with the sample size and the estimation of population parameters.
The t-distribution is a probability distribution that takes into account the uncertainty associated with estimating population parameters based on sample data. It is similar to the normal distribution but has thicker tails, which make it appropriate for smaller sample sizes or when the population standard deviation is unknown.
As the degrees of freedom increase, the t-distribution approaches the shape of the normal distribution. This implies that the t-distribution becomes less distinct from the normal distribution and the differences between them diminish. In other words, when the degrees of freedom are large enough (typically around 30 or more), the t-distribution closely approximates the normal distribution.
Therefore, as the degrees of freedom for the t-distribution increase, the distribution approaches the normal distribution.