A random sample of 858 births in a state included 422 boys. Construct a \( 95 \% \) confidence interval estimate of the proportion of boys in all births. It is believed that among all births, If Do these sample results provide strong evidence against that belief? Construct a \( 95 \% \) confidence interval estimate of the proportion of boys in all births. \( 0.458

Ask by Norris Evans. in the United States

Mar 15,2025

We start with the data: a sample of \( n = 858 \) births includes \( x = 422 \) boys. The sample proportion is \[ \hat{p}=\frac{x}{n}=\frac{422}{858}\approx0.491. \] Next, we estimate the standard error (SE) of \(\hat{p}\): \[ \text{SE}=\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}=\sqrt{\frac{0.491(1-0.491)}{858}}=\sqrt{\frac{0.491 \times 0.509}{858}}. \] Calculating the product in the numerator: \[ 0.491 \times 0.509\approx0.250. \] Thus, \[ \text{SE}\approx\sqrt{\frac{0.250}{858}}\approx\sqrt{0.000291}\approx0.0171. \] For a \( 95\% \) confidence interval, the critical value (from the standard normal distribution) is approximately \( z^*=1.96 \). The margin of error (ME) is then \[ \text{ME}=z^*\times\text{SE}\approx1.96\times0.0171\approx0.0335. \] Now, the \( 95\% \) confidence interval for the proportion \( p \) is given by \[ \hat{p}\pm\text{ME} \quad \Longrightarrow \quad 0.491\pm0.0335. \] This gives us \[ \text{Lower bound} \approx 0.491 - 0.0335 = 0.4575, \] \[ \text{Upper bound} \approx 0.491 + 0.0335 = 0.5245. \] Rounded to three decimal places, the confidence interval is approximately \[ 0.458 < p < 0.525. \] In the problem statement, the interval is given as \( 0.458 < p < 0.526 \); the slight difference in the upper bound is due to rounding differences. Next, we compare this interval with the claimed value \( p = 0.513 \). Since \[ 0.458 < 0.513 < 0.525 \quad (\text{or } 0.526), \] the value \( 0.513 \) is contained within the \( 95\% \) confidence interval. Therefore, these sample results do not provide strong evidence against the belief that the proportion of boys among all births is \( 0.513 \). The correct response is: B. There is not strong evidence against \( 0.513 \) as the value of the proportion of boys in all births because \( 0.513 \) is contained within the \( 95\% \) confidence interval.