Questions: b. After surveying 100 adult Americans, the researcher finds that 81 people support legalizing gay marriage. Compute the test statistic. Round to two decimal places. c. What is the p-value? Round to 4 decimals.

Solution Steps

To compute the test statistic \( z \) for the sample proportion, we use the formula:

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

Where:

• \( \hat{p} = \frac{81}{100} = 0.81 \) (sample proportion)
• \( p_0 = 0.5 \) (hypothesized population proportion)
• \( n = 100 \) (sample size)

Substituting the values into the formula:

\[ z = \frac{0.81 - 0.5}{\sqrt{\frac{0.5(1 - 0.5)}{100}}} = \frac{0.31}{\sqrt{\frac{0.5 \cdot 0.5}{100}}} = \frac{0.31}{\sqrt{0.0025}} = \frac{0.31}{0.05} = 6.2 \]

Thus, the test statistic is:

\[ \text{Test Statistic (z)}: 6.2 \]

The p-value associated with the computed test statistic \( z = 6.2 \) is determined based on the standard normal distribution. Given the extreme value of the test statistic, the p-value is:

\[ \text{P-value} = 0.0 \]

Final Answer