To calculate the probability of a Type II error ($\beta$), which occurs when we fail to reject the null hypothesis ($H_0$) even though it is false, we need to go through the following steps:

Step 1: Set Up Hypotheses

1. Null Hypothesis ($H_0$): The unemployment rate has not decreased, so $p = 0.07$.
2. Alternative Hypothesis ($H_a$): The unemployment rate has decreased, so $p < 0.07$.

Step 2: Define Parameters

• Sample size ($n$) = 500
• Significance level ($\alpha$) = 0.032
• Observed sample proportion of unemployed, $\hat{p} = \frac{32}{500} = 0.064$
• Hypothesized unemployment rate under $H_0$, $p_0 = 0.07$
• Actual unemployment rate ($p$) under the alternative hypothesis = 6.5% or $p = 0.065$

Step 3: Determine the Critical Value

To determine the rejection region, we need the critical value for a one-tailed test at a 0.032 significance level.

Using the standard normal distribution table:

• The critical z-score corresponding to $\alpha = 0.032$ is approximately $z_{\alpha} = -1.85$.

Step 4: Compute the Critical Proportion

The critical proportion, $\hat{p}_{\text{crit}}$, is the sample proportion at which we would reject $H_0$. $\hat{p}_{\text{crit}} = p_0 + z_{\alpha} \cdot \sqrt{\frac{p_0 (1 - p_0)}{n}}$ Substitute the values: $\hat{p}_{\text{crit}} = 0.07 + (-1.85) \cdot \sqrt{\frac{0.07 \cdot (1 - 0.07)}{500}}$

Calculate the standard deviation under $H_0$: $\text{Standard deviation} = \sqrt{\frac{0.07 \cdot 0.93}{500}} \approx 0.0113$

Now, calculate $\hat{p}_{\text{crit}}$: $\hat{p}_{\text{crit}} = 0.07 - (1.85 \times 0.0113) \approx 0.0491$

Step 5: Find the Probability of Type II Error ($\beta$)

The Type II error occurs when the true unemployment rate is $p = 0.065$, but we fail to reject $H_0$. We need to calculate the probability that the sample proportion is above $0.0491$ when $p = 0.065$.

Convert this threshold into a z-score using $p = 0.065$: $z = \frac{\hat{p}_{\text{crit}} - p}{\sqrt{\frac{p(1 - p)}{n}}}$

Compute the standard deviation under $p = 0.065$: $\text{Standard deviation} = \sqrt{\frac{0.065 \cdot 0.935}{500}} \approx 0.0109$

Now calculate the z-score: $z = \frac{0.0491 - 0.065}{0.0109} \approx -1.46$

Using the z-score table, the probability corresponding to $z = -1.46$ is approximately 0.072. This is the probability of a Type II error ($\beta$).

Final Answer

The probability of a Type II error ($\beta$) is approximately 0.072.

Would you like further details on any of these steps or have any other questions?

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Here are some related questions to expand on the topic:

1. How would the Type II error probability change if the sample size were increased?
2. What is the impact of a lower significance level (e.g., 0.01) on the Type II error?
3. How would you interpret the result if the calculated Type II error was high?
4. Can you explain how to use power in hypothesis testing and its relationship to Type II error?
5. What are the implications of Type I and Type II errors in decision-making?

Tip: To reduce Type II error, consider increasing the sample size or choosing a more lenient significance level, depending on the study's requirements.