(b) Compute the probability of a type II error for the test of part (a) when 𝜇1 − 𝜇2 = 1.

To compute the probability of a type II error (denoted as \( \beta \)) for the given test when \( \mu_1 - \mu_2 = 1 \), follow these steps:

### Step 1: Define the Given Data
- From part (a):
  - \( \bar{x} = 18.18 \) kgf/cm²
  - \( m = 42 \)
  - \( \sigma_1 = 1.6 \)
  - \( \bar{y} = 16.89 \)
  - \( n = 30 \)
  - \( \sigma_2 = 1.3 \)
  - \( \alpha = 0.01 \)
  - Calculated standard error (SE) = 0.3436

### Step 2: Determine the Critical Value for the Test
The critical value \( z_{\alpha} \) for a one-tailed test at \( \alpha = 0.01 \) can be found using standard normal distribution tables or statistical software. For \( \alpha = 0.01 \), the critical value \( z_{\alpha} \) is:
\[ z_{\alpha} \approx 2.33 \]

### Step 3: Calculate the Non-Central Parameter
When \( \mu_1 - \mu_2 = 1 \), the non-central parameter \( \delta \) is:
\[ \delta = \frac{(\mu_1 - \mu_2) - 0}{SE} \]
\[ \delta = \frac{1}{0.3436} \]
\[ \delta \approx 2.91 \]

### Step 4: Find the Corresponding \( z \)-value for Type II Error
We need to find the probability that the test statistic falls below the critical value when \( \mu_1 - \mu_2 = 1 \). This requires adjusting the critical value by the non-central parameter:

\[ z_{\beta} = z_{\alpha} - \delta \]
\[ z_{\beta} = 2.33 - 2.91 \]
\[ z_{\beta} \approx -0.58 \]

### Step 5: Calculate the Probability of a Type II Error
The probability of a type II error \( \beta \) is the probability that the \( z \)-value is less than \( z_{\beta} \) when the true difference is \( \mu_1 - \mu_2 = 1 \):

Using standard normal distribution tables or statistical software, we find:
\[ \beta = P(Z < -0.58) \]
\[ \beta \approx 0.2810 \]

### Conclusion
The probability of a type II error \( \beta \) when \( \mu_1 - \mu_2 = 1 \) is approximately 0.2810.

Do you want details or have any questions?

Here are 5 numbered questions for you to ask:

1. How do you interpret the result of the type II error probability?
2. Can you explain the concept of power in hypothesis testing?
3. How would you calculate the probability of a type II error if the standard deviations were unknown?
4. What is the relationship between type I and type II errors?
5. Can you demonstrate a power analysis for this hypothesis test?

**Tip:** The type II error probability (\( \beta \)) decreases as the sample size increases, as larger samples provide more precise estimates of the population parameters.

Learn how to compute the probability of a type II error in hypothesis testing when the true difference between population means is 1. This detailed solution covers calculating the standard error, determining critical values, and using non-central parameters in the context of statistical testing.

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