Explain why the​ t-distribution has less spread as the number of degrees of freedom increases.

Question: Explain why the​ t-distribution has less spread as the number of degrees of freedom increases.

The t-distribution, often used when the population variance is unknown, tends to have a wider spread compared to the normal distribution, especially with fewer degrees of freedom. This wider spread reflects the added uncertainty inherent in estimating the population variance from a small sample size. As the number of degrees of freedom increases, which corresponds to a larger sample size, the sample variance becomes a more accurate estimate of the population variance. Consequently, the t-distribution becomes more peaked and less spread out, approaching the shape of the normal distribution. This transition occurs because, with more data, there's less variability and uncertainty, leading to a tighter confidence interval around the mean. In essence, as the degrees of freedom grow, the t-distribution converges on the normal distribution, reflecting increased precision in statistical estimates.

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