Question 1197979
**1. Hypothesis Testing for Each Sample**

* **Hypotheses:**
    * **H0:** μ = 12 (The process mean is 12)
    * **H1:** μ ≠ 12 (The process mean is not 12) 

* **Test Statistic:** 
    * Since the population standard deviation (σ) is known, we use the z-test.
    * z = (x̄ - μ) / (σ / √n) 
        * where:
            * x̄ is the sample mean
            * μ is the population mean (12)
            * σ is the population standard deviation (0.21)
            * n is the sample size (30)

* **Calculate z-statistic and p-value for each sample.** 
    * **Sample 1:**
        * Calculate x̄ for Sample 1.
        * Calculate z = (x̄ - 12) / (0.21 / √30)
        * Use a z-table or statistical software to find the p-value associated with the calculated z-statistic.
    * **Repeat for Sample 2, Sample 3, and Sample 4.**

* **Decision Rule:**
    * If p-value ≤ 0.01, reject H0. 
    * If p-value > 0.01, fail to reject H0.

* **Action:**
    * If H0 is rejected, the process is likely out of control. Take corrective action.
    * If H0 is not rejected, the process is likely in control.

**2. Sample Standard Deviations**

* Calculate the standard deviation for each of the four samples.
* Compare these sample standard deviations to the population standard deviation (0.21). 
    * If the sample standard deviations are consistently close to 0.21, the assumption of a population standard deviation of 0.21 appears reasonable.
    * If the sample standard deviations are significantly different from 0.21, the assumption may not be valid.

**3. Control Limits**

* **Calculate the standard error of the mean:** 
    * Standard Error (SE) = σ / √n = 0.21 / √30

* **Calculate the control limits:**
    * Upper Control Limit (UCL) = μ + (z-value * SE) 
        * Use the z-value corresponding to the desired level of significance (e.g., for a 99% confidence level, z-value ≈ 2.576)
    * Lower Control Limit (LCL) = μ - (z-value * SE)

* **Monitoring:**
    * Continuously monitor the process by collecting new samples.
    * Calculate the sample mean for each new sample.
    * If the sample mean falls outside the control limits (above UCL or below LCL), investigate and take corrective action to bring the process back in control.

**4. Implications of Changing the Level of Significance**

* **Increasing the level of significance (e.g., from 0.01 to 0.05) increases the probability of rejecting the null hypothesis when it is actually true.** 
* This is known as a **Type I error** (false positive). 
* In this context, a Type I error would mean incorrectly concluding that the process is out of control when it is actually operating satisfactorily. 
* This could lead to unnecessary investigations, adjustments, and potential disruptions to the production process.

**Note:**

* This analysis requires the actual data from the "Quality" data set to perform the calculations and draw conclusions.
* Statistical software like Excel, R, or statistical packages can be used to efficiently perform the calculations and generate the necessary reports.

This framework provides a foundation for implementing a statistical process control system. By continuously monitoring the process and taking corrective action when necessary, you can improve product quality, reduce waste, and increase efficiency.