As degrees of freedom increase, the t-distribution becomes more like which distribution?

As degrees of freedom increase, the t-distribution becomes more like which distribution?

• A. Z-distribution
• B. t-distribution
• C. Chi-square
• D. Uniform

Answer: (A)

The limiting shape of the t distribution as degrees of freedom grow is the standard normal distribution (the Z-distribution). When you estimate the standard deviation from the sample, that extra uncertainty makes the t distribution have heavier tails for small degrees of freedom. As the sample size increases, the estimate becomes more accurate, reducing that extra variability, and the t distribution sharpens to match the standard normal. In the limit, with very large degrees of freedom, t converges to Z. This isn’t the Chi-square distribution, which is used for variance-related tests, nor the Uniform distribution, which has completely different properties. The t distribution itself is a family that approaches normality as df increases.

The limiting shape of the t distribution as degrees of freedom grow is the standard normal distribution (the Z-distribution). When you estimate the standard deviation from the sample, that extra uncertainty makes the t distribution have heavier tails for small degrees of freedom. As the sample size increases, the estimate becomes more accurate, reducing that extra variability, and the t distribution sharpens to match the standard normal. In the limit, with very large degrees of freedom, t converges to Z.

This isn’t the Chi-square distribution, which is used for variance-related tests, nor the Uniform distribution, which has completely different properties. The t distribution itself is a family that approaches normality as df increases.