Question

To create a confidence interval from a bootstrap distribution using percentiles, we keep the middle values and chop off a certain percent from each tail. Indicate what percent of values must be chopped off from each tail for each confidence level given. (a) \(95 \%\) (b) \(90 \%\) (c) \(98 \%\) (d) \(99 \%\)

Short answer

(a) For a 95% confidence level, 2.5% is cut off from each tail. (b) For a 90% confidence level, 5% is cut off from each tail. (c) For a 98% confidence level, 1% is cut off from each tail. (d) For a 99% confidence level, 0.5% is cut off from each tail.

Step-by-step solution

Understanding the Concept of Confidence Interval

A confidence interval is an interval estimate, derived from a statistical procedure, which is likely to contain the value of an unknown parameter. Confidence levels are expressed as a percentage and indicate how confident we can be that the parameter lies within the specified interval. A crucial thing to understand is that the confidence level pertains to the center of the distribution; the remainder will be equally divided between the two tails.

Calculation of Values for Each Tail in 95% Confidence Interval

To find the proportion of the values to be cut from each tail for a 95% confidence interval, subtract the given percentage from 100% to get the remaining percentage which is to be equally divided between both tails of the distribution. 100% - 95% gives 5%, so 2.5% is to be cut off from each tail of the distribution to result in the 95% confidence interval.

Calculation of Values for Each Tail in 90% Confidence Interval

Following the same process for a 90% confidence level: 100% - 90% = 10%, divided by 2 gives 5%. Hence, 5% of values from each end need to be cut off to result in a 90% confidence level.

Calculation of Values for Each Tail in 98% Confidence Interval

For the 98% confidence level: 100% - 98% = 2%. This 2% is divided by 2, yielding 1%. So, 1% of the values from each end need to be cut off to result in a 98% confidence level.

Calculation of Values for Each Tail in 99% Confidence Interval

Lastly, for the 99% confidence level: 100% - 99% = 1%. Dividing this 1% by 2 gives 0.5%. Therefore, 0.5% of the values from each end should be cut off to create a 99% confidence level.

Key concepts

Bootstrap Distribution

The concept of a bootstrap distribution is central to modern statistical analysis, especially when it comes to estimating the precision of sample statistics. It is a method used to approximate the sampling distribution of an estimator by resampling with replacement from the original sample.

A bootstrap distribution involves repeatedly taking random samples (with replacement) from a single sample dataset and computing the statistic of interest for each resampling. This process results in a distribution of the statistic that can aid in assessing its variability. Imagine rolling a dice multiple times and recording the results; the bootstrap method is analogously rolling the data numerous times to figure out potential outcomes.

For the exercise in question, the bootstrap distribution serves as a platform from which we can create confidence intervals by cutting off a specified percentage from each tail, thereby generating percentile-based intervals. By understanding the shape and spread of the bootstrap distribution, one can more accurately estimate confidence intervals for statistical inference.

Percentile-Based Intervals

Confidence intervals created using percentile-based methods capture the range within which a population parameter is expected to lie according to sample data. This method, originating from the bootstrap distribution, relies on the empirical percentiles to determine the endpoints of the interval.

When constructing such intervals, the key is to preserve the middle portion of the bootstrap distribution, representing the most plausible values of the parameter, while excluding the extremes, or 'tails', which are considered less likely. As seen in the task solution, the confidence level (like 95%, 90%, etc.) guides how much of the data is kept; for example, a 95% confidence interval retains the central 95% of the bootstrap distribution and removes 2.5% from each tail.

It's imperative to remember that these intervals are based on the data available and thus vary with different samples. The percentile-based approach is particularly appealing because it is non-parametric, meaning it does not rely on assumptions about the form of the underlying population distribution.

Statistical Significance

Statistical significance is a determining factor in assessing whether the results of an experiment or a study reflect a true effect or if they are likely due to chance. It’s a measure of the likelihood that the relationship observed in your data occurred without any real-world effect.

In the context of confidence intervals, statistical significance relates to whether a particular interval includes the null value (often zero, representing no effect). If a 95% confidence interval does not contain the null value, we might say the result is statistically significant at the 5% level. This is because there's only a 5% probability that this interval would have occurred if the null hypothesis were true.

The connection between confidence intervals and statistical significance is profound. Narrow intervals suggest a more precise estimate of the parameter, which could lead to stronger evidence against the null hypothesis. In terms of the exercise, the choice of confidence level impacts the conclusion about significance; higher confidence levels require more extreme sample data to reach statistical significance, as they represent a broader range of values.