Question

a random sample of 868 births included 425 boys. use a 0.10 significance level to test the claim that 50.9% of new babies are boys. do the results support the belief that 50.9% 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. reject ( h_0 ). there is not sufficient evidence to warrant rejection of the claim that 50.9% of newborn babies are boys.
b. fail to reject ( h_0 ). there is not sufficient evidence to warrant rejection of the claim that 50.9% of newborn babies are boys.
c. fail to reject ( h_0 ). there is sufficient evidence to warrant rejection of the claim that 50.9% of newborn babies are boys.
d. reject ( h_0 ). there is sufficient evidence to warrant rejection of the claim that 50.9% of newborn babies are boys.

Explanation:

Step1: Calculate sample proportion

The sample proportion $\hat{p}=\frac{425}{868}\approx0.49$.

Step2: Set up hypotheses

$H_0:p = 0.509$ (claim), $H_1:p
eq0.509$.

Step3: Calculate test statistic

The formula for the $z$-test statistic in a proportion test is $z=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}$.
Substitute $\hat{p}=0.49$, $p = 0.509$, $n = 868$:

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Step4: Calculate P - value

Since it is a two - tailed test, $P - value=2\times P(Z\lt|z|)$.
From the standard normal table, $P(Z\lt1.12)=0.8686$.
So $P - value=2\times(1 - 0.8686)=0.263$.

Step5: Make a decision

Since $P - value=0.263>0.10$ (significance level $\alpha$), we fail to reject $H_0$.

Answer:

The P - value is $0.263$.
B. Fail to reject $H_0$. There is not sufficient evidence to warrant rejection of the claim that $50.9\%$ of newborn babies are boys.