**Problem 13-7 (Algo)**Ten samples of 15 parts each were taken from an ongoing process to establish a p-chart for control. The samples and the number of defectives in each are shown in the following table.| SAMPLE | n | NUMBER OF DEFECTIVE ITEMS IN THE SAMPLE ||--------|----|----------------------------------------|| 1 | 15 | 1 || 2 | 15 | 3 || 3 | 15 | 2 || 4 | 15 | 2 || 5 | 15 | 0 || 6 | 15 | 2 || 7 | 15 | 1 || 8 | 15 | 3 || 9 | 15 | 5 || 10 | 15 | 2 |a. Determine the \( \bar{p} \), \( S_p \), UCL, and LCL for a p-chart of 95 percent confidence (1.96 standard deviations). *(Leave no cells blank. Round up any negative LCL value to "0". Round your answers to 3 decimal places.)*| \( \bar{p} \) | \( S_p \) | UCL | LCL ||---------------|-----------|-----|-----|| | | | |### Explanation**Table Overview:**The table lists 10 samples, each consisting of 15 parts from a production process, along with the number of defective items found in each sample. This data is intended for establishing a control chart, known as a p-chart, which is used to monitor the proportion of defective items in a process.**Objective:**- Calculate the average proportion of defects (\( \bar{p} \)).- Compute the standard deviation of the sample proportion (\( S_p \)).- Determine the Upper Control Limit (UCL) and Lower Control Limit (LCL) for a p-chart with a 95% confidence level.**Additional Details:**- Use a confidence level corresponding to 1.96 standard deviations for calculations.- In case the LCL is negative, round it up to zero.- Results should be rounded to three decimal places to ensure precision in control chart analysis.This problem provides a practical application of statistical quality control techniques, illustrating how data ### Statistical Process Control with a p-Chart#### Task Overviewa. **Objective**: Calculate the parameters for a p-chart at a 95% confidence level (1.96 standard deviations). Ensure no cells remain empty and adjust any negative Lower Control Limit (LCL) to zero. Answers should be rounded to three decimal places.#### Table for Calculation- **p**: - **Sp**: - **UCL** (Upper Control Limit): - **LCL** (Lower Control Limit): #### Key Considerations- Compute using the confidence interval to determine the control limits.- In cases where the LCL calculation results in a negative value, round it up to zero to maintain process validity.#### Analysis Questionb. **Evaluate the Process**:- Choose whether the process is: - ☐ Out of statistical control - ☐ In statistical controlThrough these calculations, assess if the process remains stable over time or if significant variations are present, indicating a process that may require adjustments or further scrutiny.

Firstly calculating: Sample n No.of defectives Proportion of defects 1 15 1 0.067 2 15 3 0.200 3 15 2 0.133 4 15 2 0.133 5 15 2 0.133 6 15 0 0.000 7 15 2 0.133 8 15 1 0.067 9 15 3 0.200 10 15 2 0.133 Proportion of defects Sample = No. of Defects / n Above given that n = 15, also given that Z = 1.96 Proportion of defects Sample 1 = 1/15 = 0.067 Sample 2 = 3/15 = 0.2, Similarly all other samples can be calculated. Total No. of Defects = 1+3+2+2+2+0+2+1+3+2 = 18 P bar = No. of Defects / Total of NSp = SQRT((P bar*(1-P bar))/n)UCL = P bar + Z value * spLCL = P bar - Z value * sp (If negative, It should be 0)Calculating: P bar = 18/150 = 0.12sp = ((0.12*(1-0.12))/15) = 0.084UCL = 0.120 + 1.96*0.084 = 0.284LCL = 0.120 - 1.96*0.084 = -0.044 = 0 Go to Excel, Select Proportion of defects, UCL and LCL, CLICK on Insert on 2-D line Chart: FINAL ANSWERS: (A) (B) Above chart clearly shows "Process is in Control"