A random sample of 856 births in a state included 426 boys. Construct a $95 \%$ confidence interval estimate of the proportion of boys in al births it is believed that among at tirths, the proportion of boys is 0.509 . Do these sample results provide strong evidence against that belier? Construct a $95 \%$ confidence interval estimate of the proportion of boys in al births.

Step 1 ：Calculate the sample proportion of boys: \(\text{Sample Proportion} = \frac{426}{856} = 0.4988\)

Step 2 ：Calculate the standard error: \(\text{Standard Error} = \sqrt{\frac{0.4988 \cdot (1 - 0.4988)}{856}} = 0.0159\)

Step 3 ：Calculate the margin of error: \(\text{Margin of Error} = 1.96 \cdot 0.0159 = 0.0312\)

Step 4 ：Construct the confidence interval: \(\text{Confidence Interval} = 0.4988 \pm 0.0312 = (0.4676, 0.5300)\)

Step 5 ：The 95% confidence interval estimate of the proportion of boys in all births is \(\boxed{(0.4676, 0.5300)}\)