Verified Answer :
- The margin of error formula is \(ME = z^* \cdot \frac{\sigma}{\sqrt{n}}\).
- Increasing the sample size \(n\) reduces the standard error, resulting in a smaller \(ME\).
- Decreasing the confidence level lowers the critical value \(z^*\), which also leads to a smaller \(ME\).
- Both actions independently result in a narrower confidence interval, making the statement \(\boxed{\text{True}}\).
Explanation
- Analyze the margin of error formula
To determine if increasing the sample size and decreasing the confidence level leads to a narrower margin of error (ME), we must analyze the formula for the margin of error of a confidence interval. For a population mean with a known standard deviation \(\sigma\), the margin of error is given by:
\[ME = z^* \cdot \frac{\sigma}{\sqrt{n}}\]
where \(z^*\) is the critical value corresponding to the chosen confidence level and \(n\) is the sample size. The same logic applies to proportions, where the standard error depends on \(\sqrt{n}\).
- Effect of increasing sample size
First, let's examine the effect of increasing the sample size, \(n\). In the formula, \(n\) appears in the denominator within the square root. As the sample size \(n\) increases, the term \(\sqrt{n}\) increases, which means the fraction \(\frac{\sigma}{\sqrt{n}}\) (the standard error) decreases. Because the margin of error is directly proportional to this term, increasing the sample size will indeed result in a smaller margin of error, making the interval narrower.
- Effect of decreasing confidence level
Next, consider the effect of the confidence level. The critical value \(z^*\) is determined by the confidence level. A higher confidence level requires a larger \(z^*\) to encompass more of the distribution, resulting in a wider interval. Conversely, if we decrease the confidence level (e.g., from 95% to 90%), the critical value \(z^*\) decreases. Since the margin of error \(ME\) is the product of \(z^*\) and the standard error, a smaller \(z^*\) results in a smaller \(ME\), thereby producing a narrower confidence interval.
- Conclusion on the statement's validity
Since both increasing the sample size \(n\) and decreasing the confidence level (which lowers \(z^*\)) lead to a reduction in the margin of error, the statement is true. A narrower confidence interval provides higher precision (or a smaller range of uncertainty) at the cost of being less confident that the true population parameter lies within that interval when the confidence level is decreased.
Examples
When researchers conduct a survey, they often balance precision and confidence. If they need a narrower margin of error to be more precise about an estimate, they can increase their sample size. Alternatively, if they are willing to accept a slightly lower level of certainty (e.g., shifting from a 99% to a 95% confidence level), they can also achieve a narrower interval with the same amount of data. Understanding these tradeoffs is essential for designing efficient experiments and reporting results accurately.