Here are summary statistics for randomly selected weights of newborn girls: \( \mathrm{n}=36, \overline{\mathrm{x}}=3197.2 \mathrm{~g}, \mathrm{~s}=692.6 \mathrm{~g} \). Use a confidence level of \( 90 \% \) to complete parts (a) through (d) below. a. Identify the critical value \( \mathrm{t}_{\alpha / 2} \) used for finding the margin of error. \[ t_{\alpha / 2}=1.69 \] (Round to two decimal places as needed.) b. Find the margin of error. \[ \mathrm{E}=\square \mathrm{g} \] (Round to one decimal place as needed.)

To find the margin of error (E), we use the formula: \[ E = t_{\alpha/2} \times \frac{s}{\sqrt{n}} \] Plugging in the values: \[ E = 1.69 \times \frac{692.6}{\sqrt{36}} \] Calculating: \[ E = 1.69 \times \frac{692.6}{6} \approx 1.69 \times 115.4333 \approx 195.5 \, \text{g} \] Thus, rounding to one decimal place, we have: \[ \mathrm{E} \approx 195.5 \, \text{g} \] --- To enhance your understanding, let's dive into a bit of history and practical application! The use of confidence intervals dates back to the work of statisticians in the 20th century who wanted a better way to express uncertainty in estimations. Fisher and Neyman were pivotal in developing these concepts, and their contributions have made statistical inference a whole lot more robust! In real-world applications, confidence intervals are critical in fields like medicine when determining the average weight of newborns can influence health policies. For example, knowing the average weight can help pediatricians anticipate potential health risks or developmental issues in infants, guiding preventative measures and treatments effectively!