Let's work through each part of this hypothesis test.

Part (a): Identifying the Type of Hypothesis Test

Since the claim is that $p > 0.5$, we are testing if the population proportion $p$ is greater than 0.5. This indicates a right-tailed test, as we are only interested in values greater than 0.5, which are on the right side of the distribution.

Part (b): Finding the P-value

To find the $P$-value, we will use the test statistic $z = 1.06$.

Since this is a right-tailed test, we will look for the probability that $Z > 1.06$:

• Using a standard normal table or a calculator, the $P$-value for $Z = 1.06$ is approximately $0.1446$.

Therefore, the $P$-value is $0.1446$.

Part (c): Decision Based on Significance Level $\alpha = 0.10$

With $\alpha = 0.10$:

1. We compare the $P$-value ($0.1446$) to $\alpha$ ($0.10$).
2. Since $0.1446 > 0.10$, we fail to reject $H_0$ (the null hypothesis).

Summary of Answers

• (a) This is a right-tailed test.
• (b) The $P$-value is approximately $0.1446$.
• (c) At $\alpha = 0.10$, we fail to reject $H_0$.

Would you like more details on any part of the solution?

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Here are five related questions to deepen understanding:

1. How does changing the significance level affect the hypothesis test decision?
2. What are the steps to calculate a $z$-score for proportions?
3. How would the $P$-value change if this were a two-tailed test?
4. What does it mean to "fail to reject" the null hypothesis in practical terms?
5. How is the direction of a hypothesis test determined?

Tip: Remember, the direction (left, right, or two-tailed) of a hypothesis test is defined by the alternative hypothesis.