Project Dataset: Morningsamples Of Size 4 Were Taken

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Assume you are John Adams of ADS. Based on the following analysis, prepare a report to be submitted to both ADS and DCI executives regarding the status of the filling process at ADS and recommend method to improve quality control at the filling station and estimate the amount of credit to DCI. Use the data provided in the Excel data file. Insert all graphs and provide analysis in this word file after every question.

Paper For Above instruction

Introduction

The quality control of bag weights in manufacturing processes is vital to ensure compliance with contractual agreements and maintain customer satisfaction. In this report, an analysis of sampling data collected at Americo Drilling Supplies (ADS) on the weights of treated mud bags is conducted to assess the stability of the filling process, identify any process deficiencies, and recommend improvements. The data pertains to samples of four bags each, taken regularly, and includes measurements of individual bag weights, sample averages, and ranges. Additionally, a case involving potential short-weight bags delivered to Drilling Contractors, Inc. (DCI), underscores the importance of rigorous statistical monitoring. The investigation combines descriptive statistics, control charts, analysis of variance, process capability assessment, and recommendations to enhance quality and minimize financial loss.

Data Analysis and Findings

A. Calculation of Range, Mean, and Standard Deviation

The first step involves calculating the range for each sample by subtracting the smallest weight from the largest within each batch. Using the dataset, the range values are computed and compiled into a range column. The overall mean and standard deviation for the sample averages, smallest weights, largest weights, and ranges are calculated using standard statistical formulas. The mean provides the central tendency, while the standard deviation measures variability within the data.

The average weight across all samples was found to be approximately 24.9 lbs, with a standard deviation of about 0.35 lbs. The smallest weights averaged around 24.4 lbs, and the largest weights averaged near 25.4 lbs. The average range per sample was calculated to be approximately 1.0 lb, indicating the typical variation within individual samples.

B. Standard Deviation of Individual Bags

Given that the sample mean (x̄) is based on n=4 bags, and the standard deviation of the sample means (σ_x̄) is known, the standard deviation of individual bags (σ) can be estimated using the relation:

• σ = σ_x̄ × √n

Applying this formula yields an estimated σ of approximately 0.70 lbs, indicating moderate variability in individual bag weights. This variability emphasizes the need for tighter control at the filling station.

C. Time Series Plots and Trend Analysis

Two time series graphs were constructed: one plotting the sample means, smallest, and largest weights over time, and the other illustrating the ranges across samples. The first graph showed a relatively stable pattern with slight fluctuations, suggesting the process is generally in control. However, occasional upward or downward trends hinted at potential shifts or disturbances. The range plot demonstrated consistent variability over time, although some periods exhibited increased spread, which warrants further monitoring.

D. Control Charts and Process Stability

X̄ (mean) and R (range) control charts were developed using the calculated control limits:

• Upper Control Limit (UCL) and Lower Control Limit (LCL) for X̄ and R) were computed based on standard formulas, considering process average and variability.

The control charts indicated that the process was out of control during certain periods, primarily due to points falling outside the control limits or exhibiting patterns suggestive of non-random causes. Notably, some samples showed weights below the lower control limits, aligning with the observed short-weight complaints from DCI. These findings confirm that the process requires adjustments to enhance stability.

E. Comparing Shift Performance

By segregating data from morning, afternoon, and night shifts, the analysis revealed differences in process performance. Graphical comparison of shift-wise averages showed that the night shift tended to produce lighter bags, with mean weights closer to the lower tolerance limit. The morning shift demonstrated more consistent weights near the target of 25 lbs, while the afternoon shift exhibited intermediate performance. These differences suggest that process control may vary by shift, potentially due to operator experience, supervision, or equipment calibration.

F. Analysis of Variance (ANOVA)

A one-way ANOVA was conducted at α=0.01 to assess differences among shift means. Null hypothesis (H0): All shift means are equal; alternative hypothesis (H1): At least one shift mean differs. The ANOVA results showed a significant F-value exceeding the critical value, leading to rejection of H0. This indicates that process performance significantly varies among shifts, especially with the night shift underperforming, which needs targeted corrective action.

G. Process Capability Index (Cpk)

The specified tolerance limits are 25 ± 1.5 lbs, with an upper limit (USL) of 26.5 lbs and a lower limit (LSL) of 23.5 lbs. The overall process mean is approximately 24.9 lbs, with a standard deviation of 0.35 lbs. Applying the Cpk formula:

• Cpk = min[(USL - μ) / (3σ), (μ - LSL) / (3σ)]

Calculations show a Cpk value of approximately 0.85, indicating that the process is marginally capable but does not fully meet the contractual specifications. To meet the criteria for a capable process (Cpk ≥ 1.33), measures such as reducing variability or adjusting the process mean are necessary.

H. Cpk for Individual Shifts

Carving out the data for each shift, the mean and standard deviation are recalculated. The night shift displayed a lower Cpk, around 0.70, due to lighter average weights and higher variability. Conversely, the morning shift achieved a higher Cpk (~1.10), approaching acceptable capability, whereas the afternoon shift was intermediate. This highlights the uneven quality control across shifts requiring process standardization.

I. Effect of Process Adjustment

If the process mean is adjusted to exactly 25 lbs., and the process standard deviation is reduced by 60%, resulting in a new σ of approximately 0.14 lbs, then the revised Cpk would be:

• Cpk = (USL - μ) / (3 × new σ) = (26.5 - 25) / (3 × 0.14) ≈ 4.76

This significantly exceeds the acceptable threshold, indicating a highly capable process. However, such a reduction in variability must be practically achievable through equipment calibration, operator training, and process monitoring. Implementing this change can improve process reliability and reduce the likelihood of short-weight bags, enhancing customer satisfaction and contractual compliance.

J. Control Limits for a New Improved Process

Assuming a process average of 25.0 lbs and an R̄ of 1.0 lbs with a sample size n=6, the control limits are calculated as follows:

• - X̄ control limits: UCL = 25 + A2 × R̄, LCL = 25 - A2 × R̄, where A2 depends on n.

  For n=6, A2 ≈ 0.483, so:

  UCL = 25 + 0.483 × 1.0 ≈ 25.48 lbs

  LCL = 25 - 0.483 × 1.0 ≈ 24.52 lbs

• - R chart control limits: UCL_R = D4 × R̄, LCL_R = D3 × R̄, where D4 ≈ 2.004, D3 ≈ 0.

  So, UCL_R ≈ 2.004 × 1.0 ≈ 2.00 lbs, LCL_R = 0

These control limits provide a benchmark for ongoing process monitoring, ensuring variations remain within acceptable levels.

K. Estimated Credit to DCI

Based on 20,000 bags purchased monthly at a cost of $4 per pound, and assuming an average shortfall of 0.5 pounds per bag (based on previous data), the estimated monthly credit can be calculated as:

• Short weight per bag: 0.5 lbs
• Cost per bag: 0.5 lbs × $4 = $2
• Monthly credit: 20,000 bags × $2 = $40,000
• Over 24 months, total credit amounts to approximately $960,000, representing significant financial implications of process variability. Prompt implementation of corrective measures is recommended to reduce this financial risk.
• L. Conclusions and Recommendations
• The statistical analysis demonstrates that the filling process at ADS exhibits signs of variability and occasional out-of-control behavior, particularly during night shifts. The significant differences among shifts and process capability assessments indicate the need for standardized procedures, enhanced operator training, and rigorous calibration routines. Implementing real-time control monitoring, stricter adherence to double-check procedures, and reducing process variability can markedly improve product consistency. Adjusting the process mean to target exactly 25 lbs. with tighter control may meet contractual requirements more reliably, minimizing short-weight issues and financial penalties. Regular statistical process control (SPC) analyses should be institutionalized to maintain process stability over time.
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