The management of KSmall Industries is considering a new method of assembling a computer. The current assembling method requires a mean time of 68 minutes with a standard deviation of 3.1 minutes. Using the new method, the mean assembly time for a random sample of 24 computers was 62 minutes.

What is the probability of a Type II error?
Note: Round "Critical values" to 2 decimal places and round your final answer to 4 decimal places.

To calculate the probability of a Type II error (\(\beta\)), we need to go through the following steps:

### Step 1: Hypotheses
- **Null Hypothesis (\(H_0\))**: The mean assembly time with the new method is the same as the current method, i.e., \(\mu = 68\) minutes.
- **Alternative Hypothesis (\(H_1\))**: The mean assembly time with the new method is less than the current method, i.e., \(\mu < 68\) minutes (left-tailed test).

### Step 2: Critical value for \(\alpha\)
Given no significance level (\(\alpha\)) in the question, we will assume a common \(\alpha = 0.05\).

Since this is a left-tailed test, we will find the z-score corresponding to \(\alpha = 0.05\):
\[
z_{\alpha} = \text{inverse cumulative probability for 0.05}.
\]
Using a standard normal table or a calculator, we find:
\[
z_{\alpha} = -1.645.
\]

### Step 3: Test statistic
Now, calculate the test statistic for the sample mean using the following formula:
\[
z = \frac{\bar{X} - \mu_0}{\frac{\sigma}{\sqrt{n}}},
\]
where:
- \(\bar{X} = 62\) (sample mean),
- \(\mu_0 = 68\) (population mean),
- \(\sigma = 3.1\) (population standard deviation),
- \(n = 24\) (sample size).

Plugging in the values:
\[
z = \frac{62 - 68}{\frac{3.1}{\sqrt{24}}} = \frac{-6}{\frac{3.1}{4.899}} = \frac{-6}{0.632653} \approx -9.48.
\]
Thus, the calculated \(z\)-statistic is approximately -9.48.

### Step 4: Find the critical value of \(\mu\) that separates rejection and non-rejection regions

The critical value of \(\mu\) can be found by using the formula:
\[
\mu_c = \mu_0 + z_{\alpha} \times \left(\frac{\sigma}{\sqrt{n}}\right).
\]
Substitute the values:
\[
\mu_c = 68 + (-1.645) \times \left(\frac{3.1}{\sqrt{24}}\right) = 68 + (-1.645) \times 0.632653 \approx 68 - 1.04 \approx 66.96.
\]
Thus, \(\mu_c \approx 66.96\). This is the critical mean assembly time that separates rejection from non-rejection.

### Step 5: Calculate the z-score under the alternative hypothesis (\(\mu_1 = 62\))

Now, we calculate the z-score for the sample mean under the alternative hypothesis:
\[
z_{\beta} = \frac{66.96 - 62}{\frac{3.1}{\sqrt{24}}} = \frac{66.96 - 62}{0.632653} = \frac{4.96}{0.632653} \approx 7.84.
\]

### Step 6: Find the probability of a Type II error

Finally, the probability of a Type II error (\(\beta\)) is the area to the left of \(z_{\beta}\) in the standard normal distribution. Using standard normal tables or a calculator, the probability corresponding to \(z_{\beta} = 7.84\) is practically 1, as the value is far in the tail of the distribution.

Thus, \(\beta \approx 1\).

### Final Answer:
The probability of a Type II error is approximately \( \boxed{1.0000} \).

Do you want further details or have any questions about the steps?

---
Here are 5 questions to further understand this concept:
1. How does the choice of significance level (\(\alpha\)) affect the probability of a Type II error?
2. What is the difference between Type I and Type II errors in hypothesis testing?
3. How can increasing the sample size reduce the probability of a Type II error?
4. Why do we use the z-distribution instead of the t-distribution in this case?
5. What is the role of the critical value in hypothesis testing?

**Tip:** Increasing the sample size or reducing variability can reduce the chances of a Type II error (\(\beta\)), improving the test's power.

The management of KSmall Industries is considering a new method of assembling a computer. The current assembling method requires a mean time of 68 minutes with a standard deviation of 3.1 minutes. Using the new method, the mean assembly time for a random sample of 24 computers was 62 minutes. What is the probability of a Type II error? Note: Round 'Critical values' to 2 decimal places and round your final answer to 4 decimal places.

Calculate the probability of a Type II error in a hypothesis test comparing assembly times for computers. Learn how to apply z-scores, hypothesis testing, and standard normal distribution. Suitable for advanced high school or college students.

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