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.

Answer

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Answer

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

Steps

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 $\boxed{(0.4676, 0.5300)}$