Question

a random sample of 857 births included 427 boys. use a 0.01 significance level to test the claim that 51.1% of newbor babies are boys. do the results support the belief that 51.1% of newborn babies are boys?
the p - value for this hypothesis test is (round to three decimal places as needed.)
identify the conclusion for this hypothesis test.
a. fail to reject ( h_0 ). there is sufficient evidence to warrant rejection of the claim that 51.1% of newborn babies are boys.
b. reject ( h_0 ). there is suffient evidence to warrant rejection of the claim that 51.1% of newborn babies are boys.
c. reject ( h_0 ). there is not sufficient evidence to warrant rejection of the claim that 51.1% of newborn babies are boys.
d. fail to reject ( h_0 ). there is not sufficient evidence to warrant rejection of the claim that 51.1% of newborn babies are boys.

Explanation:

Step1: Calculate the sample proportion

The sample proportion $\hat{p}=\frac{427}{857}\approx0.498$.

Step2: State the null and alternative hypotheses

$H_0:p = 0.511$, $H_1:p
eq0.511$.

Step3: Calculate the test statistic

The formula for the test statistic $z=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}$.
Substitute $\hat{p}=0.498$, $p = 0.511$, $n = 857$ into the formula:

$$
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$$

Step4: Calculate the p - value

Since it is a two - tailed test, $p - value=2\times P(Z\lt|z|)$.
$P(Z\lt0.76)=0.7764$ (from standard normal table), so $p - value=2\times(1 - 0.7764)=2\times0.2236 = 0.447$.

Step5: Make a decision

Since the significance level $\alpha = 0.01$ and $p - value=0.447\gt0.01$, we fail to reject $H_0$.

Answer:

D. Fail to reject $H_0$. There is not sufficient evidence to warrant rejection of the claim that $51.1\%$ of newborn babies are boys.