Problem 90 - Chemistry: An Atoms-Focused Approach

Question

Which confidence interval is the largest for a given value of \(n: 50 \%, 90 \%,\) or \(95 \% ?\)

Step-by-Step Solution

Answer

Answer: The largest confidence interval among the given options is the 95% confidence interval.

Step 1: Understanding Confidence Intervals

Confidence intervals are ranges within which we can be reasonably sure that a population parameter (such as a mean or proportion) lies. The width of the confidence interval depends on the desired level of confidence, sample size (n), and the amount of variability in the data. In general, as the confidence level increases, the interval becomes wider.

Step 2: Comparing 50%, 90%, and 95% Confidence Intervals

To determine which confidence interval is the largest, recall that a higher level of confidence entails a wider interval. This implies that a 95% confidence interval would be wider than a 90% confidence interval, which in turn would be wider than a 50% confidence interval.

Step 3: Conclusion

Based on the comparison of confidence levels, the largest confidence interval for a given value of n is the 95% confidence interval, followed by the 90% confidence interval, and lastly the 50% confidence interval.

Key Concepts

Population Parameter

When working with statistics, a population parameter is a key value that describes a characteristic of the entire population.
Common examples of population parameters include the mean or proportion of a population.
However, it is often impractical to measure every single member of a population.
Therefore, we use samples to estimate what we believe the population parameter to be. Population parameters are at the heart of confidence intervals.

• The interval provides a range in which the true value of the population parameter is expected to lie.
• This is important because it gives an idea about the precision of our estimate.

The more accurate our estimate of the population parameter, the more confident we can be in our conclusions about the entire population.

Confidence Level

The confidence level tells us the degree of certainty with which we expect the true population parameter to lie within the confidence interval.
A higher confidence level implies greater certainty but results in a wider interval. For example:

• A 50% confidence level suggests moderate certainty, meaning we are 50% sure that the parameter is within this interval.
• A 90% confidence level indicates higher assurance about the interval containing the true value.
• With a 95% confidence level, we have even more confidence, but the interval becomes even wider to accommodate that certainty.

Each increase in confidence level enlarges the interval because it needs to encompass more possible values of the parameter to maintain reliability.

Sample Size

Sample size, denoted by the letter "n", plays a crucial role in determining the accuracy of confidence intervals.
Generally, a larger sample size leads to a narrower confidence interval, assuming the variability in the data remains constant. Larger samples provide more data, which:

• Increases the reliability of the estimate by reducing random errors.
• Provides a more precise estimate of the population parameter.

This means that with a larger sample, we can be more confident that the interval accurately reflects the true population parameter. However, larger sample sizes can also require more resources, so finding a balance between accuracy and feasibility is key.

Data Variability

Data variability refers to how spread out the data values are in a dataset.
It is a crucial factor because it affects the width of the confidence interval. High variability implies that the data points are more spread out, resulting in a wider confidence interval, since it needs to account for greater diversity in the data values. On the other hand:

• Low variability means data points are closer together, leading to a narrower confidence interval because less variance needs to be accounted for.

Understanding the level of variability helps in deciding how reliable an interval estimation might be for predicting the population parameter.
Therefore, assessing data variability is a critical step in statistical analyses.