Question

In estimating the mean score on a fitness exam, we use an original sample of size \(n=30\) and a bootstrap distribution containing 5000 bootstrap samples to obtain a \(95 \%\) confidence interval of 67 to \(73 .\) In Exercises 3.106 to 3.111 , a change in this process is described. If all else stays the same, which of the following confidence intervals \((A, B,\) or \(C)\) is the most likely result after the change: \(\begin{array}{ll}A .66 \text { to } 74 & B .67 \text { to } 73\end{array}\) C. 67.5 to 72.5 Using 1000 bootstrap samples for the distribution.

Short answer

Based on the solution steps, the confidence interval which is likely to result from dwindling the number of bootstrap samples is a wider one. Therefore, Interval \(A .66 \text { to } 74\) appears to be the most probable result after the change.

Step-by-step solution

Understanding Bootstrap Sampling

Bootstrapping is a resampling technique used for inferring about a population from sample data. It helps ascertain the accuracy of predictive models by assigning measures of accuracy (bias, variance, confidence intervals, prediction error, etc.) to sample estimates.

Effect of the Number of Bootstrap Samples

The variation in the number of bootstrap samples can affect the stability and accuracy of the estimates. The more bootstrap samples, the more stable and precise the estimates, hence a better confidence interval would be generated. Conversely, lesser bootstrap samples may lead to a less precise and stable confidence interval estimate.

Applying the Concept to the Given Scenario

In this situation, the number of bootstrap samples have been reduced from 5000 to 1000. This reduction in the number of bootstrap samples may decrease the precision and stability of the confidence interval. This means the confidence interval length may increase (become wider).

Key concepts

Confidence Interval Estimation

Confidence interval estimation is a critical tool used in statistics to indicate the reliability of an estimated range for a certain parameter, such as a mean or proportion.

When we talk about a 95% confidence interval, we mean 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. This does not mean that there's a 95% chance that a specific interval contains the population parameter, but rather that 95% of similarly constructed intervals from repeated sampling would capture the true value.

In the fitness exam score example, the confidence interval is estimated using bootstrap samples, which provides a range where the true mean is likely to fall. The range of 67 to 73 represents such an interval where the mean score is expected to lie with 95% confidence, based on the bootstrap samples taken from the original data.

Resampling Techniques

Resampling techniques are statistical methods that involve repeatedly drawing samples from a dataset and calculating a statistic for each sample. This process allows analysts to create a sampling distribution and make inferences about the overall population.

Bootstrap sampling, a common resampling technique, involves creating thousands of 'pseudo samples' by sampling with replacement from the original data. These pseudo samples are then used to calculate various statistics, such as the mean or standard deviation, to construct a distribution.

Why Resampling Matters

Resampling is powerful because it does not make strict assumptions about the data distribution, and it can provide more accurate measures in small or unconventional datasets. Reducing the number of bootstrap samples, from 5000 to 1000 for instance, can impact the results, as the confidence interval may become less precise. The more samples used, the closer the bootstrap distribution tends to mirror what would be achieved from an actual population.

Inferential Statistics

Inferential statistics allow us to draw conclusions and make predictions about a population based on samples of data. It takes information from a small part of the population (a sample) to make inferences about the larger group from which the sample was drawn.

Key to inferential statistics is the use of probability theory to estimate the likelihood of accuracy in the conclusions being made, hence the term 'statistical inference'. Using methods like hypothesis testing, confidence intervals, and regression analysis, researchers can apply sample data to generalizations about a population.

In the fitness exam scenario, even with a reduced number of bootstrap samples, inferential statistics still enable the estimation of the confidence interval. However, it is important to note that with fewer samples, the confidence level remains the same (95%) but the interval might become wider, signaling less precision in the estimation.