Question

a random sample of 875 births included 427 boys. use a 0.10 significance level to test the claim that 51.3% of ne babies are boys. do the results support the belief that 51.3% of newborn babies are boys? identify the null and alternative hypotheses for this test. choose the correct answer below. a. $h_{0}:p = 0.513$ $h_{1}:p<0.513$ b. $h_{0}:p = 0.513$ $h_{1}:p>0.513$ c. $h_{0}:p
eq0.513$ $h_{1}:p = 0.513$ d. $h_{0}:p = 0.513$ $h_{1}:p
eq0.513$ 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: Calculate the sample proportion

The sample proportion $\hat{p}$ is calculated as $\hat{p}=\frac{x}{n}$, where $x = 427$ (number of boys) and $n=875$ (sample size).
$$\hat{p}=\frac{427}{875}\approx0.488$$

Step2: Calculate the test - statistic

The formula for the test - statistic $z$ in a hypothesis test for a proportion is $z=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}$, where $p = 0.513$ (the proportion in the null hypothesis).
Substitute the values:
$$z=\frac{0.488 - 0.513}{\sqrt{\frac{0.513\times(1 - 0.513)}{875}}}$$
First, calculate the denominator:
$$\sqrt{\frac{0.513\times0.487}{875}}=\sqrt{\frac{0.250831}{875}}\approx\sqrt{0.000286664}\approx0.0169$$
Then, calculate the numerator: $0.488−0.513=- 0.025$
$$z=\frac{-0.025}{0.0169}\approx - 1.48$$

Answer:

The test statistic for this hypothesis test is $-1.48$