Question

Leah's Toys makes rubber balls. The current process is capable of producing balls that weigh, on average, 2.5 ounces, with a standard deviation of 0.34 ounces. a. The upper and lower tolerance limits are 2.9 ounces and 2.1 ounces respectively. The process capability ratio is 392 . (Enter your response rounded to three decimal places.) Leah's capable of meeting the tolerance limits 99.7% of the time. b. In order to exactly meet the tolerance limits 99.7% of the time, Leah's Toys would need to reduce the standard deviation to olices. (Enter your response rounded to two decimal places.) c. Suppose Leah's ioys invests in process improvements that lower the standard deviation to just 0.13 ounces. This en gh for Leah's to achieve Six Sigma quality levels with regard to the weight of the balls, because a new process capat y ratio is (Enter your response rounded to three decimal places.)

Answer 1

Let's take this step by step.

a. The process capability ratio is given as Cp = 392. However, I believe there may be a misunderstanding or typo in the original question. Process capability ratio typically falls within the range 0-3. The formula for the process capability ratio (Cp) is:

Cp = (USL - LSL) / (6σ)

Where:

- USL is the upper specification limit (in this case, 2.9 ounces)

- LSL is the lower specification limit (in this case, 2.1 ounces)

- σ is the standard deviation (in this case, 0.34 ounces)

Let's recalculate the Cp given these inputs:

Cp = (2.9 - 2.1) / (6 * 0.34) = 0.800

This indicates that Leah's Toys' process is currently capable of producing balls within the tolerance limits about 80% of the time, assuming a normal distribution of weights. There may be a misunderstanding with the provided Cp of 392.

b. If Leah's Toys wants to meet the tolerance limits 99.7% of the time, then they would need to reduce the standard deviation such that the output falls within +/- 3σ (3 standard deviations from the mean). This is also known as achieving a "Six Sigma" level of quality.

We can rearrange the Cp equation to solve for σ:

σ = (USL - LSL) / (6 * Cp)

Assuming a Cp of 1.0 (which represents a process that meets tolerance limits 99.73% of the time under a normal distribution), we find:

σ = (2.9 - 2.1) / (6 * 1.0) = 0.13 ounces

This is the standard deviation Leah's Toys would need to achieve to meet the tolerance limits 99.7% of the time.

c. If Leah's Toys invests in process improvements and lowers the standard deviation to 0.13 ounces, then the new process capability ratio (Cp) would be:

Cp = (2.9 - 2.1) / (6 * 0.13) = 1.026

This means Leah's Toys could achieve Six Sigma quality levels (99.7% of products within specification limits) with this new standard deviation. Six Sigma is often represented by a Cp or Cpk (which takes into account mean shift) of 1.5 or more, but in a perfect process centered between the limits, a Cp of 1.0 represents 99.73% within limits, which aligns with your 99.7% target.