Question 1167739
```python?code_reference&code_event_index=2
import pandas as pd

# Data provided
data = {
    'Hour': [1, 2, 3, 4, 5, 6, 7, 8, 9, 10],
    'Sample Values': [
        [1.7, 2.2, 1.9, 1.2],
        [0.8, 1.5, 2.1, 0.9],
        [1.0, 1.4, 1.0, 1.3],
        [0.4, -0.6, 0.7, 0.2],
        [1.4, 2.3, 2.8, 2.7],
        [1.8, 2.0, 1.1, 0.1],
        [1.6, 1.0, 1.5, 2.0],
        [2.5, 1.6, 1.8, 1.2],
        [2.9, 2.0, 0.5, 2.2],
        [1.1, 1.1, 3.1, 1.6]
    ]
}

# Create a DataFrame for easier manipulation
df = pd.DataFrame(data)

# Subgroup size
n = 4

# --- Step 1 & 2: Calculate subgroup means (X_bar) and ranges (R) ---
df['X_bar'] = df['Sample Values'].apply(lambda x: sum(x) / n)
df['R'] = df['Sample Values'].apply(lambda x: max(x) - min(x))

# --- Step 3 & 4: Calculate grand mean (X_double_bar) and average range (R_bar) ---
X_double_bar = df['X_bar'].mean()
R_bar = df['R'].mean()

print("Subgroup Means and Ranges:")
print(df[['Hour', 'X_bar', 'R']])
print(f"\nGrand Mean (X_double_bar): {X_double_bar:.3f}")
print(f"Average Range (R_bar): {R_bar:.3f}")

# --- Step 5 & 6: Calculate Control Limits ---
# Control chart constants for n=4
A2 = 0.729
D3 = 0
D4 = 2.282

# R-chart control limits
UCL_R = D4 * R_bar
LCL_R = D3 * R_bar

# X-bar chart control limits
UCL_Xbar = X_double_bar + A2 * R_bar
LCL_Xbar = X_double_bar - A2 * R_bar

print("\nControl Limits for R-chart:")
print(f"  UCL_R: {UCL_R:.3f}")
print(f"  LCL_R: {LCL_R:.3f}")

print("\nControl Limits for X-bar chart:")
print(f"  UCL_Xbar: {UCL_Xbar:.3f}")
print(f"  LCL_Xbar: {LCL_Xbar:.3f}")

# --- Step 7 & 8: Determine if the process is in statistical control ---

# Check R-chart for out-of-control points
out_of_control_R = df[(df['R'] > UCL_R) | (df['R'] < LCL_R)]
print("\n--- R-chart Analysis ---")
if not out_of_control_R.empty:
    print("Out-of-control points found in R-chart:")
    print(out_of_control_R[['Hour', 'R']])
else:
    print("No out-of-control points found in R-chart.")

# Check X-bar chart for out-of-control points
out_of_control_Xbar = df[(df['X_bar'] > UCL_Xbar) | (df['X_bar'] < LCL_Xbar)]
print("\n--- X-bar chart Analysis ---")
if not out_of_control_Xbar.empty:
    print("Out-of-control points found in X-bar chart:")
    print(out_of_control_Xbar[['Hour', 'X_bar']])
else:
    print("No out-of-control points found in X-bar chart.")

# Overall conclusion
print("\n--- Overall Process Control Status ---")
if out_of_control_R.empty and out_of_control_Xbar.empty:
    print("Based on the X-bar and R charts, the process appears to be in statistical control.")
else:
    print("Based on the X-bar and/or R charts, the process is NOT in statistical control due to points outside control limits.")
```

```text?code_stdout&code_event_index=2
Subgroup Means and Ranges:
   Hour  X_bar    R
0     1  1.750  1.0
1     2  1.325  1.3
2     3  1.175  0.4
3     4  0.175  1.3
4     5  2.300  1.4
5     6  1.250  1.9
6     7  1.525  1.0
7     8  1.775  1.3
8     9  1.900  2.4
9    10  1.725  2.0

Grand Mean (X_double_bar): 1.490
Average Range (R_bar): 1.400

Control Limits for R-chart:
  UCL_R: 3.195
  LCL_R: 0.000

Control Limits for X-bar chart:
  UCL_Xbar: 2.511
  LCL_Xbar: 0.469

--- R-chart Analysis ---
No out-of-control points found in R-chart.

--- X-bar chart Analysis ---
Out-of-control points found in X-bar chart:
   Hour  X_bar
3     4  0.175

--- Overall Process Control Status ---
Based on the X-bar and/or R charts, the process is NOT in statistical control due to points outside control limits.

```

To determine if the process is in statistical control, we will construct an X-bar chart and an R chart.

**Given Data:**

| Hour | Sample Values |
| :--- | :------------ |
| 1    | 1.7, 2.2, 1.9, 1.2 |
| 2    | 0.8, 1.5, 2.1, 0.9 |
| 3    | 1.0, 1.4, 1.0, 1.3 |
| 4    | 0.4, -0.6, 0.7, 0.2 |
| 5    | 1.4, 2.3, 2.8, 2.7 |
| 6    | 1.8, 2.0, 1.1, 0.1 |
| 7    | 1.6, 1.0, 1.5, 2.0 |
| 8    | 2.5, 1.6, 1.8, 1.2 |
| 9    | 2.9, 2.0, 0.5, 2.2 |
| 10   | 1.1, 1.1, 3.1, 1.6 |

The subgroup size ($n$) is 4.

**1. Calculate Subgroup Means ($\\bar{X}$) and Ranges (R):**

| Hour | Sample Values      | $\\bar{X}$ (Mean) | R (Range) |
| :--- | :----------------- | :--------------- | :-------- |
| 1    | 1.7, 2.2, 1.9, 1.2 | 1.750            | 1.0       |
| 2    | 0.8, 1.5, 2.1, 0.9 | 1.325            | 1.3       |
| 3    | 1.0, 1.4, 1.0, 1.3 | 1.175            | 0.4       |
| 4    | 0.4, -0.6, 0.7, 0.2 | 0.175            | 1.3       |
| 5    | 1.4, 2.3, 2.8, 2.7 | 2.300            | 1.4       |
| 6    | 1.8, 2.0, 1.1, 0.1 | 1.250            | 1.9       |
| 7    | 1.6, 1.0, 1.5, 2.0 | 1.525            | 1.0       |
| 8    | 2.5, 1.6, 1.8, 1.2 | 1.775            | 1.3       |
| 9    | 2.9, 2.0, 0.5, 2.2 | 1.900            | 2.4       |
| 10   | 1.1, 1.1, 3.1, 1.6 | 1.725            | 2.0       |

**2. Calculate the Grand Mean ($\\bar{\\bar{X}}$) and Average Range ($\\bar{R}$):**

  * **Grand Mean ($\\bar{\\bar{X}}$):** Average of all subgroup means
    $\\bar{\\bar{X}} = (1.750 + 1.325 + 1.175 + 0.175 + 2.300 + 1.250 + 1.525 + 1.775 + 1.900 + 1.725) / 10 = 14.900 / 10 = 1.490$
  * **Average Range ($\\bar{R}$):** Average of all subgroup ranges
    $\\bar{R} = (1.0 + 1.3 + 0.4 + 1.3 + 1.4 + 1.9 + 1.0 + 1.3 + 2.4 + 2.0) / 10 = 14.0 / 10 = 1.400$

**3. Determine Control Chart Constants for n=4:**
From control chart tables for subgroup size $n=4$:

  * $A\_2 = 0.729$
  * $D\_3 = 0$
  * $D\_4 = 2.282$

**4. Calculate Control Limits for the R-chart:**

  * **Center Line (CL\_R):** $\\bar{R} = 1.400$
  * **Upper Control Limit (UCL\_R):** $D\_4 \\times \\bar{R} = 2.282 \\times 1.400 = 3.1948 \\approx 3.195$
  * **Lower Control Limit (LCL\_R):** $D\_3 \\times \\bar{R} = 0 \\times 1.400 = 0.000$

**5. Calculate Control Limits for the X-bar chart:**

  * **Center Line (CL\_Xbar):** $\\bar{\\bar{X}} = 1.490$
  * **Upper Control Limit (UCL\_Xbar):** $\\bar{\\bar{X}} + A\_2 \\times \\bar{R} = 1.490 + 0.729 \\times 1.400 = 1.490 + 1.0206 = 2.5106 \\approx 2.511$
  * **Lower Control Limit (LCL\_Xbar):** $\\bar{\\bar{X}} - A\_2 \\times \\bar{R} = 1.490 - 0.729 \\times 1.400 = 1.490 - 1.0206 = 0.4694 \\approx 0.469$

**Summary of Control Limits:**

| Chart  | Center Line | UCL   | LCL   |
| :----- | :---------- | :---- | :---- |
| R-chart | 1.400       | 3.195 | 0.000 |
| X-bar  | 1.490       | 2.511 | 0.469 |

**6. Determine if the process is in statistical control:**

  * **R-chart Analysis:**

      * Examine the 'R' column in the table above. All R values (from 0.4 to 2.4) are within the R-chart control limits (0.000 to 3.195).
      * **Conclusion for R-chart: The process range (variability) appears to be in statistical control.**

  * **X-bar chart Analysis:**

      * Examine the '$\\bar{X}$' column in the table above.

      * We compare each subgroup mean to the X-bar chart control limits (0.469 to 2.511).

      * For Hour 4, the mean is $0.175$. This value is below the LCL\_Xbar of $0.469$.

      * **Conclusion for X-bar chart: The process average (mean) is NOT in statistical control because the mean for Hour 4 ($0.175$) falls below the Lower Control Limit.**

**Overall Conclusion:**

Based on the analysis of both the X-bar and R charts, the process for the bearing diameters is **NOT in statistical control**. The presence of an out-of-control point in the X-bar chart (Hour 4) indicates that there is an assignable cause of variation that needs to be investigated and eliminated.