Questions: For a given sample size, when we increase the probability of a Type I error, the probability of a Type II error decreases. remains unchanged. is impossible to determine without more information. increases.

Solution Steps

We performed a hypothesis test for a population proportion with the following parameters:

• Hypothesized population proportion \( p_0 = 0.5 \)
• Sample proportion \( \hat{p} = 0.55 \)
• Sample size \( n = 100 \)
• Significance level \( \alpha = 0.05 \)

The test statistic was calculated as follows:

\[ Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} = \frac{0.55 - 0.5}{\sqrt{\frac{0.5(1 - 0.5)}{100}}} = 1.0 \]

The corresponding p-value was found to be \( 0.1587 \). The critical region for this one-tailed test is defined as \( Z > 1.6449 \).

Next, we increased the significance level to \( \alpha = 0.10 \) and repeated the hypothesis test. The test statistic remained the same:

\[ Z = 1.0 \]

The p-value also remained unchanged at \( 0.1587 \). The new critical region for this test is defined as \( Z > 1.2816 \).

From the results, we observe that increasing the significance level (alpha) did not change the test statistic or the p-value. This indicates that the relationship between Type I error (α) and Type II error (β) is not straightforward in this case.

Final Answer