Question

A tire manufacturing company receives an order for a large quantity of tires with mean breaking energy \(1000 \mathrm{~J}\) and standard deviation of breaking energy \(10 \mathrm{~J}\). The company wants to establish a control chart for mean breaking energy of tire based on the aforesaid parameters and samples of size 9 such that the probability of rejecting good tires produced by a statistically controlled process is \(0.01\). Under this situation can you find out what are the upper and lower control limits for mean breaking energy of tire?

Short answer

The lower control limit for the mean breaking energy of the tires is 992.3 J and the upper control limit is 1007.7 J.

Step-by-step solution

Identifying the parameters of the problem

First, identify the key parameters of the problem. The mean (\(\mu\)) is 1000J, the standard deviation (\(\sigma\)) is 10J, the desired probability of rejecting a good tire is 0.01, and the sample size (n) is 9.

Compute population mean and standard deviation

The control chart will be centered around the population mean. We also need to compute the standard deviation of the sample mean, which is the standard deviation of the population (\(\sigma\)) divided by the square root of the sample size (n). In this case, \(\sigma/\sqrt{n} = 10/\sqrt{9} = 10/3 \approx 3.33 J\).

Determine the z-score for the desired probability

Next, we want to determine the z-score that corresponds to the desired probability of 0.01. From z-tables or using a z-score calculator, we find that the z-score corresponding to a tail probability of 0.01 is -2.33 for the lower limit and 2.33 for the upper limit.

Calculate the control limits

Now we can calculate the control limits. These are given by the formula \(\mu \pm Z\sigma/\sqrt{n}\), where Z is the z-score, \(\sigma\) is the standard deviation of the population, and n is the sample size. For the lower limit, this is \(1000 - 2.33*3.33 = 992.3 J\). For the upper limit, this is \(1000 + 2.33*3.33 = 1007.7 J\).

Key concepts

Statistical Process Control

Statistical Process Control (SPC) is a method used in manufacturing to monitor, control, and ensure the quality of processes. By collecting and analyzing data, particularly from manufacturing processes, it enables workers and operations managers to detect and correct variations in the process before they result in defective products.

SPC involves using control charts, like the one in our example with the tire manufacturing company, to track the performance of key process parameters—such as the mean breaking energy of tires. Control charts typically feature a central line for the average, an upper control limit (UCL), and a lower control limit (LCL), which indicate the threshold at which the process variation is acceptable. If a measured sample falls outside of these control limits, it suggests that the process may be out of control and requires corrective action.

Quality Control in Manufacturing

Quality control in manufacturing is a systematic process designed to ensure that products meet specified quality standards and satisfy customer requirements. This entails having a series of actions or techniques in place that detect, reduce, and eliminate non-conformities or defects in manufactured products.

Applying SPC and control charts aids quality control by providing a visual representation of the process variability. Manufacturers use these charts to ensure that the process stays within predefined control limits, which helps maintain product quality and consistency. Continuous monitoring of quality parameters, like mean breaking energy of tires, is crucial for maintaining the trust of customers and minimizing the cost of quality, which includes reworks, scrap, and warranty claims.

Standard Deviation

The standard deviation is a statistical measurement that sheds light on the amount of variation or dispersion within a set of values. In manufacturing, knowing the standard deviation of a quality parameter, such as breaking energy of tires, is essential for creating control charts and ensuring process stability.

A high standard deviation signifies that the values are spread out over a wider range, implying greater process variability, while a low standard deviation indicates that the values are closer to the mean, resulting in a more consistent process. When combined with the sample size, the standard deviation gives insights into the accuracy and reliability of the quality measurements being performed.

Sample Size

Sample size plays a critical role in statistical process control. It refers to the number of observations or measurements taken from a process to assess its quality. The larger the sample size, the more reliable the estimates of process parameters, such as the mean and standard deviation, will be. However, as seen in the exercise with the tire manufacturing company, larger sample sizes can also mean more resources spent on testing.

When calculating control limits, using the appropriate sample size is crucial. The central limit theorem tells us that as sample size increases, the distribution of the sample means will tend to be normally distributed regardless of the shape of the process distribution. This justifies the use of the z-score in our control chart calculations for mean breaking energy, as we approximate the sampling distribution of the mean to be normal, even for small samples, such as n=9.