In order to achieve increased confidence for a given sample size, what must you do?

To achieve increased confidence for a given sample size, one must widen the interval. The confidence interval reflects the range within which we expect the true population parameter to lie. A wider interval is necessary when we want to increase the level of confidence because it accounts for more variability in the data and reduces the chance of failing to capture the true parameter.

When the confidence level is increased (e.g., from 95% to 99%), the corresponding confidence interval must also be widened to ensure that we can maintain that higher level of confidence. This is due to the statistical properties associated with confidence intervals, where a higher confidence percentage corresponds to a larger margin of error in the estimation process.

Narrowing the interval, reducing the sample size, or keeping the interval constant would not increase confidence; rather, they would potentially undermine the reliability of the estimate by either constraining the range or reducing the data points needed to make a robust inference. In summary, widening the interval is essential to achieve increased confidence while maintaining the specificity of the sample size.