For the same sample statistics, which level of confidence would produce the widest confidence interval? Explain your reasoning.
a. 90%
b. 95%
c. 98%
d. 99%

Verified step by step guidance

1. Understand the concept of a confidence interval: A confidence interval provides a range of values within which we expect the true population parameter (e.g., mean or proportion) to lie, based on a sample statistic. The level of confidence indicates the probability that the interval contains the true parameter.
2. Recognize the relationship between the confidence level and the width of the confidence interval: Higher confidence levels require a wider interval to ensure that the true parameter is captured within the range. This is because higher confidence levels demand greater certainty, which necessitates accounting for more variability.
3. Recall the critical value associated with each confidence level: The critical value (z* or t*) increases as the confidence level increases. For example, the critical value for a 99% confidence level is larger than that for a 90% confidence level. This directly impacts the width of the interval.
4. Examine the formula for a confidence interval: For a population mean, the formula is typically \( \text{CI} = \bar{x} \pm z^* \frac{\sigma}{\sqrt{n}} \), where \( z^* \) is the critical value. A larger \( z^* \) results in a wider margin of error, and thus a wider confidence interval.
5. Compare the given confidence levels (90%, 95%, 98%, 99%): Since 99% is the highest confidence level among the options, it will produce the widest confidence interval because it requires the greatest certainty and accounts for the most variability.

Key Concepts

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter. It is expressed with a certain level of confidence, such as 90%, 95%, or 99%, indicating the probability that the interval includes the parameter. Wider intervals suggest more uncertainty about the parameter's exact value.

Level of Confidence

The level of confidence represents the degree of certainty that the confidence interval contains the true population parameter. Common levels include 90%, 95%, and 99%. A higher level of confidence means that the interval is constructed to be wider, as it accounts for more variability and uncertainty in the data.

Critical Value

The critical value is a factor used to calculate the margin of error in a confidence interval. It is derived from the standard normal distribution (Z-distribution) or t-distribution, depending on the sample size and whether the population standard deviation is known. As the level of confidence increases, the critical value also increases, leading to a wider confidence interval.