Question

Why is a \(99 \%\) confidence interval wider than a \(95 \%\) confidence interval?

Short answer

A 99% confidence interval is wider because it requires a higher critical value to ensure more certainty that the interval contains the true parameter.

Step-by-step solution

Understand Confidence Intervals

A confidence interval is a range of values, derived from the sample data, that is likely to contain the value of an unknown population parameter. It is expressed with a confidence level that gives us the 'degree' of certainty about the probability of the true parameter lying within the interval.

Define Confidence Level

The confidence level refers to the proportion of times that an estimated confidence interval would contain the true population parameter if the same sampling procedure were repeated multiple times. A 95% confidence level indicates we expect the interval to contain the true parameter 95% of the time, while a 99% confidence level indicates 99% of the time.

Compare Confidence Levels

Because a 99% confidence interval needs to ensure that the true parameter is within the interval 99% of the time, it requires a larger range to account for more of the possible sample variations. A 95% confidence interval needs to account for less of the variation, resulting in a narrower interval.

Theoretical Justification

The width of a confidence interval is influenced by the multiplier of the standard error, known as the 'critical value'. A higher confidence level, like 99%, has a higher critical value from the Z or t-distribution tables than a 95% confidence interval, causing it to widen. This larger critical value accounts for more data being included to ensure a higher degree of confidence.

Key concepts

Confidence Level

The confidence level reflects how often the true population parameter will fall within the calculated confidence interval in repeated trials of the same experiment. For example, a 95% confidence level means that if we were to take 100 different samples and compute a confidence interval for each sample, we would expect about 95 of those intervals to contain the true population parameter. Confidence levels are pivotal in statistical inference because they provide a degree of certainty about the accuracy of our estimate. The higher the confidence level, the more "confident" we can be about our results. However, this increase in confidence comes at a cost: confidence intervals become wider as the level of confidence increases, reflecting a trade-off between precision and certainty. In summary, while higher confidence levels provide more assurance that the interval contains the true parameter, they also result in less precise estimates.

Population Parameter

A population parameter is a numerical value that describes a characteristic of a population. Population parameters, unlike sample statistics, represent the entire group being studied. They can include measures such as the mean, proportion, or standard deviation of the population. In practice, population parameters are often unknown because it is usually impractical or even impossible to study an entire population. Hence, researchers use sample data to make inferences about these parameters. This is where confidence intervals come into play. They help estimate where the true population parameter lies based on sample data. Understanding population parameters is crucial because they form the baseline for inferential statistics. All statistical estimates from sample data aim to approximate the unknown population parameters. This bridge between sample statistics and population parameters is what allows researchers to make educated guesses about population characteristics.

Critical Value

The critical value is a crucial element in calculating confidence intervals. It represents the number of standard deviations a sample statistic is from the mean of the sampling distribution. This value is determined by the desired confidence level and dictates how "far out" we need to extend the confidence interval to account for uncertainty. Critical values are derived from statistical tables, like the Z-distribution or t-distribution, depending on the sample size and known or unknown population variance. For example, at a 95% confidence level, the Z critical value typically used is approximately 1.96, whereas for a 99% confidence level, it is about 2.576. This higher critical value at a 99% level indicates more variability needs to be accounted for, hence a larger interval. Ultimately, the critical value plays a vital role in shaping the width of the confidence interval: higher critical values result in wider intervals, reflecting a higher degree of confidence.