A Manufacturing Company's Quality Control Personnel Have Rec

A Manufacturing Companys Quality Control Personnel Have Recorded The

A manufacturing company's quality control personnel have recorded the proportion of defective items for each of 500 monthly shipments of one of the computer components that the company produces. The data are in the file P07_07.xlsx. The quality control department manager does not have sufficient time to review all of these data. Rather she would like to examine the proportions of defective items for a sample of these shipments.

a. Use StatTools to generate a simple random sample of size 25.

b. Using the sample generated in part a, construct a 95% confidence interval for the mean proportion of defective items over all monthly shipments. Assume that the population consists of the proportion of defective items for each of the 500 monthly shipments.

c. Interpret the 95% confidence interval constructed in part b.

d. Does the 95% confidence interval contain the actual population mean in this case? If not, explain why not. What proportion of many similarly constructed confidence intervals should include the population mean?

Paper For Above instruction

The management of a manufacturing company specializing in computer components is concerned with the quality control of its monthly shipments. They have recorded the proportion of defective items in each of 500 shipments. Since reviewing all data is impractical, a statistical approach involving sampling is employed to estimate the population mean proportion of defective items. This paper discusses the process of selecting a sample, constructing a confidence interval, interpreting this interval, and understanding its properties concerning the true population parameter.

Introduction

Quality control is an essential aspect of manufacturing, especially when ensuring product reliability and customer satisfaction. Understanding the proportion of defective items across shipments enables manufacturers to monitor and improve process quality. However, assessing every shipment's defect proportion can be resource-intensive. Therefore, statistical sampling methods are used to make inferences about the entire population based on a smaller, manageable subset. In this context, a sample of 25 shipments is used to estimate the average defective proportion, and a confidence interval provides a statistical range wherein the true average likely resides.

Sampling Methodology

The first step involves selecting a simple random sample of 25 shipments from the total of 500. This process ensures each shipment has an equal chance of being included, reducing bias and providing a representative subset of the data. Using statistical software such as StatTools facilitates this process through automated random sampling procedures. The randomness of the sample is critical to obtaining valid inferences about the entire population of shipments.

Constructing the Confidence Interval

Once the sample data is obtained, the next step is to calculate the sample mean of the proportion of defective items. Using this sample mean and its standard deviation, a 95% confidence interval can be constructed. This interval provides a range of values within which the true population mean proportion of defective items is likely to fall with 95% confidence.

The formula for a confidence interval for a mean when the population standard deviation is unknown is:

CI =   x̄   ±   t_α/2 × (s / √n)

where:

• x̄ is the sample mean
• s is the sample standard deviation
• n is the sample size (25)
• t_α/2 is the critical value from the t-distribution with n - 1 degrees of freedom

Using statistical software or t-distribution tables, the critical t-value for a 95% confidence level with 24 degrees of freedom is approximately 2.064.

Given the relevant data from the sample, the computed confidence interval indicates where the actual population mean of defects per shipment likely resides.

Interpretation of the Confidence Interval

Interpreting the 95% confidence interval involves understanding that if this sampling process and interval calculation are repeated numerous times, approximately 95% of these intervals would contain the true average proportion of defective shipments. It is essential to note that this does not imply a 95% probability that the specific interval calculated from the current sample contains the true mean; rather, it reflects the long-term performance of the interval estimation process.

For example, if the calculated interval was from 0.05 to 0.12, it suggests that the true average defect proportion across all shipments is likely between 5% and 12% with 95% confidence.

Coverage of the Actual Population Mean

It is crucial to recognize that the true population proportion of defective items is a fixed value, though unknown. The constructed confidence interval either contains this actual mean or it does not. Because we rely on random sampling and the statistical properties of interval estimation, approximately 95% of such intervals, generated under the same methodology, will contain the true mean. Conversely, about 5% will not, which is an inherent feature of the confidence level rather than a flaw.

This statistical principle underscores the probabilistic nature of confidence intervals—they do not guarantee the inclusion of the true parameter in a specific interval but quantify the likelihood over many repetitions.

Conclusion

Using a simple random sample and confidence interval estimation, the quality control department can obtain a statistically valid estimate of the average defect proportion across all shipments without analyzing every shipment. The interpretation of the 95% confidence interval emphasizes the reliability of the estimation process, acknowledging that a small chance exists that the interval may not include the true mean. The understanding that approximately 95% of many such intervals will contain the population mean guides quality decision-making and process improvements in manufacturing.

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