Sorry to ask another question but I keep struggling with this class.
If all else remains the same, which of these will make a confidence interval for a proportion
narrower?
I. Increase the confidence level.
II. Increase the sample size.
III. Increase the margin of error.
A. I only
B. II only
C. I and II
D. I and III
E. I, II, and III
To determine which option will make the confidence interval for a proportion narrower, let's first understand what a confidence interval is and how it is calculated.
A confidence interval is a range of values within which we can have a certain level of confidence (expressed as a percentage) that the true population parameter lies. In this case, we are discussing the confidence interval for a proportion.
To calculate a confidence interval for a proportion, we need three main components:
1. Confidence level: This determines the level of confidence we want to have in our interval. The most common choices are 90%, 95%, and 99% confidence levels.
2. Sample size: This is the number of observations or cases we have in our sample.
3. Margin of error: This is the maximum allowed deviation or error between the observed sample proportion and the true population proportion.
Now let's evaluate each option in the given question:
I. Increase the confidence level: Increasing the confidence level will actually make the confidence interval wider. This is because a higher confidence level requires a larger range of values to capture.
II. Increase the sample size: Increasing the sample size will make the confidence interval narrower. This is because a larger sample size reduces the uncertainty in estimating the true population proportion, resulting in a narrower range.
III. Increase the margin of error: Increasing the margin of error will make the confidence interval wider. The margin of error reflects the uncertainty in estimating the true population proportion, so a larger margin of error leads to a wider interval.
Based on these explanations, we can deduce that increasing the sample size (Option II) will make the confidence interval narrower.
Therefore, the correct answer is B. II only, as only increasing the sample size will make the confidence interval narrower.