Questions: Question 7 The probability of rejecting the null hypothesis when, in fact, the null hypothesis is true is called the standard error p-value power of the test significance level Question 8 What is the statistic that compares the observed proportion to the expected proportion? the one-proportion t-test statistic the one-proportion z-test statistic the standard error

Solution Steps

We are conducting a hypothesis test for a population proportion. The null hypothesis $H_0$ states that the population proportion $p$ is equal to the hypothesized value $p_0 = 0.5$. The alternative hypothesis $H_a$ states that the population proportion is not equal to $p_0$.

The test statistic $Z$ is calculated using the formula:

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

where:

• $\hat{p} = 0.55$ (sample proportion),
• $p_0 = 0.5$ (hypothesized population proportion),
• $n = 100$ (sample size).

Substituting the values, we find:

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

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 $0.55$ under the null hypothesis.

For a two-tailed test with a significance level $\alpha = 0.05$, the critical regions are defined as:

$$Z < -1.96 \quad \text{or} \quad Z > 1.96$$

Since the calculated test statistic $Z = 1.0$ does not fall into the critical region, we fail to reject the null hypothesis. The P-value $0.3173$ is greater than $\alpha = 0.05$, further supporting this conclusion.

Final Answer