We could narrow a 95 percent confidence interval by:
A. using 99 percent confidence.
B. using a larger sample.
C. raising the standard error.
The correct answer and explanation is :
Correct Answer: B. using a larger sample.
Explanation:
A 95% confidence interval (CI) is a statistical range used to estimate an unknown population parameter (such as a mean or proportion). It provides an interval of values that likely includes the true value, with 95% confidence. The width of the confidence interval depends primarily on three factors:
- The confidence level (e.g., 95% or 99%)
- The variability in the data (standard deviation or standard error)
- The sample size
To narrow a confidence interval, we want to reduce the margin of error. The margin of error (ME) in a confidence interval is calculated as:
[
\text{ME} = z \times \frac{\sigma}{\sqrt{n}}
]
Where:
- ( z ) is the z-score associated with the desired confidence level (e.g., 1.96 for 95%)
- ( \sigma ) is the population standard deviation
- ( n ) is the sample size
Let’s examine the choices:
A. Using 99 percent confidence
This is incorrect. Increasing the confidence level from 95% to 99% actually widens the confidence interval. A higher confidence level requires a larger margin of error to ensure the interval captures the true population parameter more often.
B. Using a larger sample
This is correct. Increasing the sample size ( n ) decreases the standard error (( \frac{\sigma}{\sqrt{n}} )), which in turn narrows the confidence interval. Larger samples provide more precise estimates, reducing variability.
C. Raising the standard error
This is incorrect. Increasing the standard error makes the confidence interval wider, not narrower. A higher standard error implies more variability and less precision in the estimate.
Conclusion:
If your goal is to narrow a 95% confidence interval, the most effective method is to increase the sample size. This reduces the standard error and makes your estimate of the population parameter more precise.