Confidence Levels
Given specific sample data, such as the data given in Exercise 1, which confidence interval is wider: the 95% confidence interval or the 80% confidence interval? Why is it wider?

Verified step by step guidance

1. Step 1: Understand the concept of confidence intervals. A confidence interval provides a range of values within which the true population parameter (e.g., mean or proportion) is expected to lie, based on the sample data. The confidence level (e.g., 95% or 80%) indicates the probability that the interval contains the true parameter.
2. Step 2: Recognize the relationship between confidence level and interval width. Higher confidence levels (e.g., 95%) require a wider interval to ensure that the true parameter is captured within the range. Lower confidence levels (e.g., 80%) result in narrower intervals because there is less certainty about capturing the true parameter.
3. Step 3: Recall the formula for a confidence interval: \( \text{Confidence Interval} = \bar{x} \pm z \cdot \frac{s}{\sqrt{n}} \), where \( \bar{x} \) is the sample mean, \( z \) is the critical value corresponding to the confidence level, \( s \) is the sample standard deviation, and \( n \) is the sample size. The critical value \( z \) increases as the confidence level increases, leading to a wider interval.
4. Step 4: Compare the 95% and 80% confidence intervals. The 95% confidence interval is wider because it uses a larger \( z \)-value (critical value) to ensure a higher probability of capturing the true parameter. The 80% confidence interval uses a smaller \( z \)-value, resulting in a narrower range.
5. Step 5: Conclude that the width of a confidence interval is directly related to the confidence level. A higher confidence level (e.g., 95%) requires a wider interval to account for greater uncertainty, while a lower confidence level (e.g., 80%) allows for a narrower interval due to reduced certainty.

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 95% or 80%, indicating the probability that the interval will capture the true value if the experiment were repeated multiple times.

Confidence Level

The confidence level represents the degree of certainty that the true population parameter lies within the confidence interval. A higher confidence level, such as 95%, means a wider interval because it accounts for more variability and uncertainty in the data, while a lower level, like 80%, results in a narrower interval.

Margin of Error

The margin of error is the amount of error that is allowed in the estimation of a population parameter. It is influenced by the confidence level and sample size; a higher confidence level increases the margin of error, leading to a wider confidence interval, while a lower confidence level decreases it, resulting in a narrower interval.