The width of a confidence interval for µ is not affected by:
A. the sample size.
B. the confidence level.
C. the standard deviation.
D. the sample mean.
The correct answer and explanation is :
The correct answer is D. the sample mean.
Explanation:
A confidence interval (CI) is a statistical range, derived from sample data, that is used to estimate the true population parameter (in this case, the population mean, µ). The width of the confidence interval indicates the degree of uncertainty in the estimate: a larger width suggests more uncertainty, while a narrower width suggests more precision in the estimate of µ.
Let’s break down how each factor affects the width of the confidence interval:
- Sample size (A): The width of the confidence interval is inversely proportional to the square root of the sample size. That is, as the sample size increases, the confidence interval becomes narrower. This happens because a larger sample provides more information and leads to a more accurate estimate of the population parameter. The standard error (which is part of the formula for the confidence interval) decreases as the sample size increases, thus reducing the interval width.
- Confidence level (B): The confidence level (such as 90%, 95%, or 99%) refers to the probability that the confidence interval will contain the true population parameter. A higher confidence level (e.g., 99% instead of 95%) means a wider confidence interval because a higher level of confidence requires a larger range to ensure that the true parameter is likely to fall within the interval. Conversely, a lower confidence level results in a narrower interval.
- Standard deviation (C): The width of the confidence interval is directly related to the standard deviation of the population (or sample). A larger standard deviation indicates more variability in the data, which leads to a wider confidence interval. This is because greater variability in the data increases uncertainty about the true population mean.
- Sample mean (D): The sample mean is the point estimate around which the confidence interval is constructed. However, the sample mean does not affect the width of the confidence interval. The sample mean determines the center of the interval but does not influence its width. The width is determined by the variability (standard deviation) and the sample size.
Thus, the correct answer is D. the sample mean, as it does not affect the width of the confidence interval.