I. As the sample size increases, the confidence interval decreases. II. If the sample size does not change and we want to increase the confidence level, the margin of error must increase. III. The smaller the confidence interval, the more confidence we have that it contains all of the data values.

When it comes to confidence intervals, understanding how sample size affects them is crucial. As you increase the sample size, you gather more information, which leads to a more precise estimate of the population parameter. This is why larger samples yield narrower confidence intervals, giving you a clearer picture and more precise predictions! In real-world scenarios, these concepts play a vital role, especially in fields like medicine or market research. For instance, when testing a new drug, researchers need large sample sizes to ensure that their confidence intervals are narrow enough to confidently say that the drug's effects are consistent across the population. If they want to be more certain (higher confidence level) without changing the sample size, they'll have to accept a wider interval, which means less precision but more reassurance that they are within the bounds of true effectiveness.