A random sample of 838 births included 425 boys. Use a 0.01 significance level to test the claim that of newborn babies are boys. Do the results
support the belief that of newborn babies are boys?
Identify the null and alternative hypotheses for this test. Choose the correct answer below.
A.

D.

Identify the test statistic for this hypothesis test.
The test statistic for this hypothesis test is
(Round to two decimal places as needed.)

Question

A random sample of 838 births included 425 boys. Use a 0.01 significance level to test the claim that $$ 51.2 \% $$ of newborn babies are boys. Do the results support the belief that $$ 51.2 \% $$ of newborn babies are boys? Identify the null and alternative hypotheses for this test. Choose the correct answer below. A. $$ H_{0}: p=0.512 $$ $$ H_{1}: p>0.512 $$ $$ H_{0}: p=0.512 $$ $$ H_{1}: p 
eq 0.512 $$ $$ H_{0}: p=0.512 $$ $$ H_{1}: p<0.512 $$ D. $$ H_{0}: p 
eq 0.512 $$ $$ H_{1}: p=0.512 $$ Identify the test statistic for this hypothesis test. The test statistic for this hypothesis test is (Round to two decimal places as needed.)

UpStudy AI · Accepted Answer

**Step 1. Identify the Claim and Determine the Hypotheses**

The claim is that $$51.2\%$$ of newborn babies are boys, i.e., that the true proportion $$p$$ is equal to $$0.512$$.

In testing a claim about a proportion of a population, we set:
- The null hypothesis as the statement of equality:
  $$
  H_0: p = 0.512.
  $$
- The alternative hypothesis as the statement that $$p$$ is not equal to $$0.512$$:
  $$
  H_1: p \neq 0.512.
  $$

Thus, the correct choice is:  
$$ H_{0}: p=0.512 $$ and $$H_{1}: p \neq 0.512$$.

---

**Step 2. Compute the Test Statistic**

The test statistic for a hypothesis test for a proportion is calculated by
$$
z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}.
$$

Given:
- Sample size: $$ n = 838 $$
- Number of boys: 425, so sample proportion: 
  $$
  \hat{p} = \frac{425}{838} \approx 0.507.
  $$
- Claimed proportion: $$ p_0 = 0.512 $$.

**Step 2.1. Calculate the Standard Error**

First, compute $$ p_0(1-p_0) $$:
$$
0.512 \times 0.488 \approx 0.249856.
$$

Then divide by $$ n $$:
$$
\frac{0.249856}{838} \approx 0.000298.
$$

Take the square root:
$$
\sqrt{0.000298} \approx 0.01726.
$$

**Step 2.2. Calculate the $$ z $$-Statistic**

Now plug into the formula:
$$
z = \frac{0.507 - 0.512}{0.01726} = \frac{-0.005}{0.01726} \approx -0.29.
$$

(Rounding to two decimal places, $$ z \approx -0.29 $$.)

---

**Final Answers:**

- **Hypotheses:**  
  $$ H_0: p = 0.512 $$ and $$ H_1: p \neq 0.512 $$.

- **Test Statistic:**  
  $$ z \approx -0.29 $$.
**Hypotheses:** - $$ H_0: p = 0.512 $$ - $$ H_1: p \neq 0.512 $$ **Test Statistic:** $$ z \approx -0.29 $$

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