When Brett Kavanaugh was nominated to be a Supreme Court justice, a survey of 1036 Americans showed that of them disapproved of Kavanaugh. A newspaper published an article with this headline:
“Majority of Americans Disapprove of Kavanaugh.” Use a 0.05 significance level to test the claim made in that headline. Use the P-value method. Use the normal distribution as an approximation to the binomial
distribution.
Let p denote the population proportion of all Americans who disapproved of Kavanaugh. Identify the null and alternative hypotheses.

(Type integers or decimals Do not round)
Identify the test statistic.

(Rouftd to two decimal places as needed.)

Ask by Matthews Bright. in the United States

Mar 21,2025

Question

When Brett Kavanaugh was nominated to be a Supreme Court justice, a survey of 1036 Americans showed that of them disapproved of Kavanaugh. A newspaper published an article with this headline:
“Majority of Americans Disapprove of Kavanaugh.” Use a 0.05 significance level to test the claim made in that headline. Use the P-value method. Use the normal distribution as an approximation to the binomial
distribution.
Let p denote the population proportion of all Americans who disapproved of Kavanaugh. Identify the null and alternative hypotheses.

(Type integers or decimals Do not round)
Identify the test statistic.

(Rouftd to two decimal places as needed.)

Ask by Matthews Bright. in the United States

Mar 21,2025

UpStudy AI · Accepted Answer

### Step 1. Define the Hypotheses

We have:
- The null hypothesis: $$ H_{0}: p = 0.5 $$
- The alternative hypothesis: $$ H_{1}: p > 0.5 $$

### Step 2. Identify the Test Statistic

The test statistic for a population proportion is given by

$$
z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}
$$

where
- $$\hat{p} = 0.514$$ is the sample proportion,
- $$p_0 = 0.5$$ is the hypothesized proportion,
- $$n = 1036$$ is the sample size.

### Step 3. Calculate the Standard Error

The standard error is

$$
\sqrt{\frac{p_0(1-p_0)}{n}} = \sqrt{\frac{0.5 \times 0.5}{1036}} = \sqrt{\frac{0.25}{1036}}
$$

$$
\sqrt{\frac{0.25}{1036}} \approx \sqrt{0.0002415} \approx 0.01555
$$

### Step 4. Compute the Test Statistic

Subtract the hypothesized proportion from the sample proportion:

$$
\hat{p} - p_0 = 0.514 - 0.5 = 0.014
$$

Now, plug into the formula:

$$
z = \frac{0.014}{0.01555} \approx 0.90
$$

### Final Answer

Test statistic:

$$
z = 0.90
$$
Test statistic: $$ z = 0.90 $$