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

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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.
The confidence interval for the proportion of boys is $$ 0.458 < p < 0.526 $$. Since $$ 0.513 $$ is within this interval, the sample results do not provide strong evidence against the belief that the proportion of boys is $$ 0.513 $$.

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