Question

a random sample of 875 births included 427 boys. use a 0.10 significance level to test the claim that 51.3% of newb babies are boys. do the results support the belief that 51.3% of newborn babies are boys? are boys. c. fail to reject ( h_0 ). there is not sufficient evidence to warrant rejection of the claim that 51.3% of newborn babies are boys. d. reject ( h_0 ). there is sufficient evidence to warrant rejection of the claim that 51.3% of newborn babies are boys. do the results support the belief that 51.3% of newborn babies are boys? a. the results support the belief that 51.3% of newborn babies are boys because there was sufficient evidence to show that the belief is true. b. the results do not support the belief that 51.3% of newborn babies are boys because there was sufficient evidence to show that the belief is untrue. c. the results support the belief that 51.3% of newborn babies are boys because there was no evidence to show that the belief is untrue. d. the results do not support the belief that 51.3% of newborn babies are boys; the results merely show that there is not strong evidence against the rate of 51.3%.

Explanation:

Step1: Determine the null and alternative hypotheses

The null hypothesis \(H_0:p = 0.513\) (claim that 51.3% of newborn babies are boys), and the alternative hypothesis \(H_1:p
eq0.513\)

Step2: Calculate the sample proportion \(\hat{p}\)

\(\hat{p}=\frac{427}{875}\approx0.488\)

Step3: Calculate the test - statistic \(z\)

The formula for the test - statistic in a proportion test is \(z=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}\)
Substitute \(p = 0.513\), \(\hat{p}=0.488\), and \(n = 875\)
\(z=\frac{0.488 - 0.513}{\sqrt{\frac{0.513\times(1 - 0.513)}{875}}}\)
First, calculate the denominator: \(\sqrt{\frac{0.513\times0.487}{875}}\approx\sqrt{\frac{0.250}{875}}\approx\sqrt{0.000286}\approx0.0169\)
Then, \(z=\frac{- 0.025}{0.0169}\approx - 1.48\)

Step4: Find the critical values and p - value

For a two - tailed test with \(\alpha=0.10\), the critical values are \(z=\pm1.645\)
The p - value for \(z=-1.48\) (using a standard normal table or calculator) is \(P(Z\lt - 1.48)+P(Z\gt1.48)=2\times(1 - P(Z\lt1.48))\)
\(P(Z\lt1.48)=0.9306\), so \(p - value=2\times(1 - 0.9306)=0.1388\)
Since \(|z|=1.48\lt1.645\) (critical value) and \(p - value = 0.1388\gt0.10\) (significance level), we fail to reject \(H_0\)

Answer:

C. The results support the belief that 51.3% of newborn babies are boys because there was no evidence to show that the belief is untrue.