A random sample of 838 births included 425 boys. Use a 0.01 significance level to test the claim that \( 51.2 \% \) of newborn babies are boys. Do the results support the belief that \( 51.2 \% \) of newborn babies are boys? Identify the null and alternative hypotheses for this test. Choose the correct answer below. A. \( H_{0}: p=0.512 \) \( H_{1}: p>0.512 \) \( H_{0}: p=0.512 \) \( H_{1}: p \neq 0.512 \) \( H_{0}: p=0.512 \) \( H_{1}: p<0.512 \) D. \( H_{0}: p \neq 0.512 \) \( H_{1}: p=0.512 \) Identify the test statistic for this hypothesis test. The test statistic for this hypothesis test is (Round to two decimal places as needed.)

To test the claim that \( 51.2 \% \) of newborn babies are boys, we will establish our hypotheses clearly. The null hypothesis, \( H_{0}: p = 0.512 \), reflects the statement we want to test against, while the alternative hypothesis, \( H_{1}: p > 0.512 \), proposes that the proportion of boys is higher than \( 51.2\% \). The correct pairing from your options is: **A.** \( H_{0}: p=0.512 \) \( H_{1}: p>0.512 \) Now, let's calculate the test statistic. The formula for the test statistic \( z \) in this scenario is given by: \[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \] Where: \( \hat{p} = \frac{425}{838} \approx 0.507 \) (the sample proportion), \( p_0 = 0.512 \) (the claimed proportion), and \( n = 838 \) (the sample size). Substituting in the values gives: \[ z = \frac{0.507 - 0.512}{\sqrt{\frac{0.512 \times (1 - 0.512)}{838}}} \] Calculating the denominator: \[ \sqrt{\frac{0.512 \times 0.488}{838}} \approx \sqrt{\frac{0.249856}{838}} \approx \sqrt{0.000298} \approx 0.01726 \] Now substituting back into the formula for \( z \): \[ z = \frac{-0.005}{0.01726} \approx -0.29 \] When rounding to two decimal places, the test statistic is approximately: **Test Statistic:** \( z \approx -0.29 \) --- Historical Background: The study of birth ratios has fascinated statisticians and demographers for centuries. In many cultures, there's been a belief in favorable or unfavorable birth ratios that serve various social, economic, and cultural narratives. Initially, the natural sex ratio at birth is often viewed to be around \( 105 \) boys for every \( 100 \) girls, but factors like maternal age, health, and environmental influences can impact this ratio. Real-World Application: Understanding birth ratios is crucial in fields ranging from healthcare to social planning. For instance, if a region shows a significant anomaly in birth ratios, it may indicate underlying health issues, gender biases, or socio-economic factors that need addressing. Moreover, policies can be informed by these statistics to promote gender equality and resource allocation for neonatal care.