Question

a random sample of 830 births included 430 boys. use a 0.01 significance level to test the claim that 50.5% of newborn babies are boys. do the results support the belief that 50.5% of newborn babies are boys?
identify the null and alternative hypotheses for this test. choose the correct answer below.
a. ( h_{0}: p<0.505 )
( h_{1}: p
eq 0.505 )
b. ( h_{0}: p=0.505 )
( h_{1}: p=0.505 )
c. ( h_{0}: p
eq 0.505 )
( h_{1}: p=0.505 )
d. ( h_{0}: p=0.505 )
( h_{1}: p>0.505 )
identify the test statistic for this hypothesis test.
the test statistic for this hypothesis test is
(round to two decimal places as needed.)

Explanation:

Step1: Identify Null and Alternative Hypotheses

The claim is about a proportion. The null hypothesis \(H_0\) is a statement of equality. The alternative hypothesis \(H_1\) is the claim we are testing. Since the claim is that \(p = 0.505\) (a two - tailed test, as we are just testing if the proportion is equal to \(0.505\) or not), \(H_0:p = 0.505\) and \(H_1:p
eq0.505\)

Step2: Calculate the Sample Proportion \(\hat{p}\)

The sample proportion \(\hat{p}=\frac{x}{n}\), where \(x = 430\) (number of boys) and \(n=830\) (sample size). So \(\hat{p}=\frac{430}{830}\approx0.518\)

Step3: Calculate the Test Statistic \(z\)

The formula for the test statistic \(z\) in a proportion test is \(z=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}\)
Substitute \(p = 0.505\), \(\hat{p}=0.518\), and \(n = 830\)

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Answer:

The correct null and alternative hypotheses are \(H_0:p = 0.505\) and \(H_1:p
eq0.505\) (Option A). The test statistic for this hypothesis test is approximately \(z = 0.75\)