​As the sample size is increased, the distribution of t statistics becomes flatter and more spread out.


The statement you've made is incorrect when describing the behavior of the distribution of t-statistics as the sample size increases.

Actually, **as the sample size increases**, the distribution of t-statistics becomes **less spread out** and approaches the shape of the **standard normal distribution** (a bell curve). This happens because the degrees of freedom in the t-distribution increase with larger sample sizes, making it more closely resemble the normal distribution. The t-distribution has "fatter tails" for small sample sizes to account for the greater variability due to smaller sample size, but as the sample size grows, the tails thin out, and the distribution narrows.

### Key Points:
1. **Small sample sizes**: The t-distribution is wider and has fatter tails compared to the normal distribution because there is more uncertainty in estimating the population parameters.
2. **Larger sample sizes**: As the sample size increases, the t-distribution starts to resemble the standard normal distribution (with a mean of 0 and a standard deviation of 1).
3. **Degrees of freedom**: The shape of the t-distribution depends on the degrees of freedom (df), which increase with sample size. As df → ∞, the t-distribution converges to the normal distribution.
4. **Application in hypothesis testing**: With larger samples, the distinction between using the t-distribution and the normal distribution becomes negligible.

Would you like to go deeper into any specific aspect of this?

### Related Questions:
1. What is the relationship between the degrees of freedom and the shape of the t-distribution?
2. Why does the t-distribution have "fatter tails" than the normal distribution for small sample sizes?
3. How does the Central Limit Theorem relate to the behavior of the t-distribution for large sample sizes?
4. What are the practical implications of using a t-distribution versus a normal distribution in hypothesis testing?
5. How does increasing sample size impact the confidence interval of the mean?

**Tip**: Always consider using the t-distribution for small sample sizes (n < 30), especially when population variance is unknown.

As the sample size is increased, the distribution of t statistics becomes flatter and more spread out.

Explore how increasing sample size affects the t-distribution and its convergence to the normal distribution. This solution covers key statistical concepts such as degrees of freedom, hypothesis testing, and the Central Limit Theorem. Ideal for undergraduate statistics students.

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