Question

a random sample of 828 births included 427 boys. use a 0.01 significance level to test the claim that 50.9% of newborn babies are boys. do the results support the belief that 50.9% of newborn babies are boys? identify the conclusion for this hypothesis test. a. fail to reject ( h_0 ). there is not sufficient evidence to warrant rejection of the claim that 50.9% of newborn babies are boys. b. 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 the sample proportion

The sample proportion \(\hat{p}=\frac{427}{828}\approx0.516\)

Step2: State the null and alternative hypotheses

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

Step3: Calculate the test statistic

The formula for the test statistic \(z=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}\)
Substitute \(\hat{p}=0.516\), \(p = 0.509\), \(n = 828\)
\(z=\frac{0.516-0.509}{\sqrt{\frac{0.509\times(1 - 0.509)}{828}}}\)
\(=\frac{0.007}{\sqrt{\frac{0.509\times0.491}{828}}}\)
\(=\frac{0.007}{\sqrt{\frac{0.25}{828}}}\)
\(=\frac{0.007}{\sqrt{0.000302}}\)
\(=\frac{0.007}{0.0174}\approx0.40\)

Step4: Find the critical value

For a significance level \(\alpha=0.01\) (two - tailed test), the critical values are \(z=\pm2.58\)

Step5: Make a decision

Since \(|z| = 0.40<2.58\), we fail to reject \(H_0\)

Answer:

A. Fail to reject \(H_0\). There is not sufficient evidence to warrant rejection of the claim that \(50.9\%\) of newborn babies are boys.