2 Question 6, 7.2.7-T Part 3 of 4 HW Score: \( 57.14 \% \), 16 of 28 points Points: 0 of 1 Here are summary statistics for the weights of Pepsi in randomly selected cans: \( n=36, \bar{x}=0.82414 \mathrm{lb}, \mathrm{s}=0.00572 \mathrm{lb} \). Use a confidence level of \( 95 \% \) to complete parts (a) through (d) below a. Identify the critical value \( t_{\alpha / 2} \) used for finding the margin of error. \[ t_{\alpha / 2}=2.03 \] (Round to two decimal places as needed.) b. Find the margin of error. \[ \mathrm{E}=0.00194 \mathrm{lb} \] (Round to five decimal places as needed.) c. Find the confidence interval estimate of \( \mu \). \( \square \) \( \mathrm{lb}<\mu< \) \( \square \) lb (Round to five decimal places as needed.)

Understanding the concept of confidence intervals is rooted in a rich historical context. Statistics as a discipline began to take shape in the 18th century, and the development of confidence intervals was significantly influenced by the work of Sir Ronald A. Fisher in the early 20th century. Fisher's methods laid the groundwork for modern statistical inference, and his contributions were crucial in helping statisticians quantify uncertainty, leading to the widespread application of confidence intervals in various fields. For real-world applications, confidence intervals are essential tools in quality control, medical research, and social sciences. For instance, companies can use confidence intervals to ensure the average weight of a product (like that can of Pepsi) falls within acceptable limits, thus maintaining quality standards. In clinical trials, researchers apply confidence intervals to estimate the effectiveness of new drugs, allowing for informed decisions on treatment protocols. This practical use showcases how confidence intervals provide meaningful insights beyond mere numbers, guiding crucial decisions across multiple sectors!