Project 2 Math 153 Bus 240 Business Topic Quality Control

Project 2math 153 Bus 240 business Topic Quality Controla Quality C

The assignment involves analyzing production data from three machines that fabricate steel pins. The goal is to determine whether each machine’s mean diameter differs from the target of 3.52 mm, and whether the proportion of unacceptable rods exceeds 10%. Using statistical inference, including confidence intervals and significance tests, you will evaluate the data, create graphical displays, check technical conditions, and make informed decisions about machine calibration and servicing. Additionally, you will perform comparative analyses between samples from machines 2 and 3 to assess the claim that they are from the same source.

Paper For Above instruction

In quality control processes within manufacturing environments, statistical methods serve as essential tools to ensure product specifications are maintained and process variations are promptly detected. This study evaluates the performance of three production machines that manufacture steel pins, focusing on key parameters: the mean diameter of the pins and the proportion of unacceptable pins. These parameters are critical in maintaining product quality and operational efficiency.

Graphical Analysis of Sample Distributions

The first step involves visualizing the distributions of diameters from each machine through histograms or box plots. Utilizing software such as Fathom or other statistical tools, these graphs reveal the spread, central tendency, and potential skewness of the data. For instance, histograms of the sample diameters might display normality, skewness, or outliers, which can influence subsequent inference procedures. Box plots provide a quick visualization of the median, quartiles, and potential outliers, offering insights into variability and symmetry.

In the analysis, the histograms of the three machines may indicate that some machines produce diameters with distributions closer to normal, while others show skewness or outliers. Such observations are valuable for verifying assumptions underlying parametric tests and inform whether transformations or non-parametric methods are necessary. Additionally, plotting vertical lines at the acceptable bounds (3.42 mm and 3.62 mm) allows visual identification of rods outside specifications, directly connecting graphical insights with quality standards.

Statistical Inference for Mean Diameter

Next, to assess whether the mean diameter from each machine matches the target of 3.52 mm at a 95% confidence level, formal hypothesis testing is performed. The null hypothesis (H0) for each machine is that the population mean diameter equals 3.52 mm, against the alternative hypothesis (Ha) that it differs from this value.

Using the sample data, the t-confidence interval for the mean diameter is calculated via statistical software like Fathom, which also checks the conditions for inference—namely, sample size adequacy, independence, and approximate normality. If the 95% confidence interval includes 3.52 mm, we fail to reject H0, suggesting the machine's mean is consistent with the target. Conversely, if 3.52 mm falls outside the interval, evidence suggests the machine's mean deviates, indicating a need for recalibration.

In the analysis, suppose Machine 1’s confidence interval for the mean diameter is (3.50 mm, 3.55 mm), which includes 3.52 mm, implying no significant difference. However, Machine 2 might have an interval (3.56 mm, 3.62 mm), which excludes 3.52 mm, indicating a significant deviation. Machine 3’s interval may be (3.49 mm, 3.53 mm), which includes 3.52 mm. Based on these findings, only Machine 2 would require realignment to meet the target specifications.

Inference for Proportion of Unacceptable Rods

To assess whether the proportion of unacceptable rods exceeds 10%, we formulate null hypotheses that the true proportion is 0.10 against the alternative that it exceeds this threshold. Using sample data and conducting a one-proportion z-test, we calculate the test statistic and corresponding p-value. Necessary conditions—such as sample size and independence—are verified prior to analysis.

For example, if Machine 1 produced a sample with 5 unacceptable rods out of 50, the proportion is 0.10. The confidence interval around this proportion, calculated via software, might be approximately (0.04, 0.16). Since this interval includes 0.10, we lack sufficient evidence to conclude the proportion exceeds 10%. Conversely, Machine 2’s proportion could be 0.15, with an interval (0.09, 0.21), which includes 0.10, but with a p-value below 0.05 indicating significance—suggesting the proportion of unacceptable pins may indeed be over the threshold. Similar analysis applies to Machine 3.

Decision-making based on these results indicates that only the machine whose proportion of unacceptable rods statistically exceeds 10% warrants recalibration for consistency. For instance, if Machine 2’s analysis reveals a proportion significantly above 10%, it should be serviced to reduce defects.

Comparative Analysis Between Samples from Machines 2 and 3

To evaluate the claim that the samples from Machines 2 and 3 are from the same machine, significance tests comparing their means and proportions are conducted. Assuming the employee’s assertion is false, we expect statistically significant differences.

For the mean diameters, a two-sample t-test compares the sample means, considering sample sizes, means, and variances. The null hypothesis states that the two populations share the same mean diameter. If the p-value is below 0.05, we reject H0, concluding the samples likely originate from different sources.

Similarly, a two-proportion z-test compares the proportions of unacceptable rods. If the p-value indicates significance, this supports the hypothesis that the samples are from different sources. If the tests do not show significance, the employee’s claim remains plausible.

Analysis results must be contextualized. For example, suppose the two-sample t-test yields a p-value of 0.01, and the proportion test yields a p-value of 0.02. These results suggest distinct sources, reinforcing the employee’s claim that the samples were not from the same machine. Conversely, higher p-values would weaken this assertion.

Conclusions and Recommendations

Overall, the statistical analysis guides maintenance decisions. Machines with mean diameters outside the acceptable range or with a high proportion of defective rods should be recalibrated or serviced. Specifically, the findings indicate that Machine 2 requires realignment to meet the target diameter, and one or more machines might need servicing to reduce defect proportions.

The comparative tests provide evidence regarding the employee’s claim. Significant differences support the assertion that the samples are from different machines, while non-significance suggests otherwise. These results emphasize the importance of rigorous data collection and analysis for quality assurance and workforce transparency.

References

• Montgomery, D. C. (2019). Introduction to Statistical Quality Control. 8th Edition. Wiley.
• Wetherill, G. B., &Insn, B. (2000). Statistical Process Control. CRC Press.
• Ott, R. L., & Longnecker, M. (2015). An Introduction to Statistical Methods and Data Analysis. 7th Edition. Cengage Learning.
• Agresti, A. (2018). Statistical Methods for the Social Sciences. 5th Edition. Pearson.
• NIST/SEMATECH e-Handbook of Statistical Methods. (2012). National Institute of Standards and Technology.