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?

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.
do the results support the belief that 50.9% of newborn babies are boys?
a. the results do not support the belief that 50.9% of newborn babies are boys because there was sufficient evidence to show that the belief is untrue.
b. the results support the belief that 50.9% of newborn babies are boys because there was no evidence to show that the belief is untrue.
c. the results do not support the belief that 50.9% of newborn babies are boys; the results merely show that there is not strong evidence against the rate of 50.9%.
d. the results support the belief that 50.9% of newborn babies are boys because there was sufficient evidence to show that the belief is true.

Explanation:

Step1: Calculate the sample proportion

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

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 \(\hat{p}=0.49\), \(p = 0.509\), \(n = 868\)
\(z=\frac{0.49 - 0.509}{\sqrt{\frac{0.509\times(1 - 0.509)}{868}}}\)
\(=\frac{- 0.019}{\sqrt{\frac{0.509\times0.491}{868}}}\)
\(=\frac{-0.019}{\sqrt{\frac{0.25}{868}}}\)
\(=\frac{-0.019}{\frac{0.5}{29.46}}\)
\(=\frac{-0.019}{0.017}\approx - 1.12\)

Step4: Find the critical values

For a two - tailed test with \(\alpha=0.10\), the critical values are \(z=\pm1.645\)

Step5: Make a decision

Since \(-1.645<-1.12 < 1.645\), we fail to reject \(H_0\)

Answer:

C. Fail to reject \(H_0\). There is sufficient evidence to warrant rejection of the claim that \(50.9\%\) of newborn babies are boys.
C. The results do not support the belief that \(50.9\%\) of newborn babies are boys; the results merely show that there is not strong evidence against the rate of \(50.9\%\).