The following data represent the age (in weeks) at which...

The following data represent the age (in weeks) at which babies first crawl based on a survey of 12 mothers conducted by Essential Baby. In Problem 34 from Section $9.2,$ we constructed a $95 \%$ confidence interval for the mean age at which a baby first crawls. Construct a $95 \%$ confidence interval for the mean age at which a baby first crawls using a bootstrap sample with 1000 resamples. Compare the bootstrap interval to the $t$ -interval constructed in Problem 34 of Section 9.2

Key Concepts

Confidence Interval

A confidence interval is a range of values, derived from sample data, that is likely to contain the true value of an unknown population parameter. It provides an estimation of the parameter, along with a specified level of confidence (such as 95%), which reflects the degree of certainty that the interval covers the true parameter value.

Bootstrap Resampling

Bootstrap resampling is a non-parametric technique that involves drawing repeated samples, with replacement, from the observed data. This method is used to approximate the sampling distribution of a statistic without relying on strong parametric assumptions. It is particularly useful when the theoretical distribution of the statistic is complex or unknown.

t-Distribution Based Confidence Interval

A t-distribution based confidence interval is constructed using the t-distribution, which is appropriate when estimating the mean of a normally distributed population, especially when the sample size is small and the variance is unknown. This interval takes into account the additional uncertainty due to estimating the population standard deviation from the sample.

Comparison of Methods

Comparing bootstrap confidence intervals with t-distribution based confidence intervals helps to assess the robustness and validity of the results. While the t-interval relies on the assumption of normality and may perform better with smaller samples, the bootstrap method offers a flexible, assumption-light alternative that can be more robust in the presence of non-normal data or small sample sizes.

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