Questions: In tests about a population proportion, po represents the (A) hypothesized population proportion (B) observed p-value (C) probability of p=p̄=0 (D) observed sample proportion

Solution Steps

In hypothesis testing for a population proportion, we define the null hypothesis \( H_0: p = p_0 \) and the alternative hypothesis \( H_a: p \neq p_0 \). Here, \( p_0 \) represents the hypothesized population proportion. For this test, we set \( p_0 = 0.5 \), a significance level \( \alpha = 0.05 \), and a sample proportion \( \hat{p} = 0.55 \) from a sample size \( n = 100 \).

The test statistic \( Z \) is calculated using the formula:

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

Substituting the values:

\[ Z = \frac{0.55 - 0.5}{\sqrt{\frac{0.5(1 - 0.5)}{100}}} = 1.0 \]

The P-value associated with the test statistic \( Z = 1.0 \) is calculated to be \( 0.3173 \). This value indicates the probability of observing a sample proportion as extreme as \( \hat{p} \) under the null hypothesis.

For a two-tailed test at the significance level \( \alpha = 0.05 \), the critical region is defined as:

\[ Z < -1.96 \quad \text{or} \quad Z > 1.96 \]

Since the calculated test statistic \( Z = 1.0 \) does not fall within the critical region, we fail to reject the null hypothesis. This means that there is not enough evidence to conclude that the population proportion differs from \( p_0 \).

Final Answer