A process filling small bottles with baby formula has a target of 3 ounces ± 0.150 ounce. Two hundred bottles from the process were sampled. The results showed the average amount of formula placed in the bottles to be 3.042 ounces. The standard deviation of the amounts was 0.034 ounce. Determine the value of Cpk. Roughly what proportion of bottles meet the specifications? The process capability index (Cpk) is (round your response to three decimal places). Slightly more than % of the bottles meet the specifications. 80 90 95.45 99.99 99.73
A process filling small bottles with baby formula has a target of 3 ounces ± 0.150 ounce. Two hundred bottles from the process were sampled. The results showed the average amount of formula placed in the bottles to be 3.042 ounces. The standard deviation of the amounts was 0.034 ounce. Determine the value of Cpk. Roughly what proportion of bottles meet the specifications? The process capability index (Cpk) is (round your response to three decimal places). Slightly more than % of the bottles meet the specifications. 80 90 95.45 99.99 99.73
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter10: Statistics
Section: Chapter Questions
Problem 22SGR
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![### Process Capability and Quality Control in Baby Formula Bottling
#### Overview:
A process at a manufacturing facility is designed to fill small bottles with baby formula with a targeted amount of 3 ounces. The tolerance range for this process is ± 0.150 ounce. To assess the process capability, a sample of 200 bottles was collected and analyzed. The findings indicated an average fill of 3.042 ounces per bottle, with a standard deviation of 0.034 ounce.
#### Objective:
Determine the value of the process capability index (Cpk) and estimate the proportion of bottles that meet the specified tolerance limits.
#### Process Capability Index Calculation:
The process capability index (Cpk) is a measure of how well a process can produce output within specified limits. It is calculated using the following formula:
\[ C_{pk} = \min \left ( \frac{USL - \mu}{3\sigma}, \frac{\mu - LSL}{3\sigma} \right ) \]
Where:
- \( \mu \) = Process mean (average)
- \( \sigma \) = Standard deviation
- USL = Upper Specification Limit (Target + 0.150)
- LSL = Lower Specification Limit (Target - 0.150)
Given:
- \( \mu \) = 3.042 ounces
- \( \sigma \) = 0.034 ounce
- USL = 3.150 ounces
- LSL = 2.850 ounces
The Cpk computation will provide insight into how well the process adheres to specifications.
#### Proportion of Bottles Meeting Specifications:
Upon calculating the Cpk value, the next step involves determining the percentage of bottles that meet the specification limits. This is typically derived from standard normal distribution tables or relevant statistical software.
#### Dropdown Menu for User Input:
1. The dropdown menu offers options for the percentage of bottles that meet specifications, including:
- 80%
- 90%
- 95.45%
- 99.99%
- 99.73%
Users can select the correct response based on the Cpk value calculated.
#### Interactive Section:
- **Text Box for Cpk Value:** An input field where students can enter the calculated Cpk value, rounded to three decimal places.
- **Dropdown for Proportion:** A dropdown selection to input the percentage of bottles that meet the specifications.
This educational activity teaches students how to apply statistical methods](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5705588d-52dd-4872-ae93-60cb2dff5836%2F5be3e42b-8b02-4452-9c33-77729b6fc83d%2F7nub7zf_processed.png&w=3840&q=75)
Transcribed Image Text:### Process Capability and Quality Control in Baby Formula Bottling
#### Overview:
A process at a manufacturing facility is designed to fill small bottles with baby formula with a targeted amount of 3 ounces. The tolerance range for this process is ± 0.150 ounce. To assess the process capability, a sample of 200 bottles was collected and analyzed. The findings indicated an average fill of 3.042 ounces per bottle, with a standard deviation of 0.034 ounce.
#### Objective:
Determine the value of the process capability index (Cpk) and estimate the proportion of bottles that meet the specified tolerance limits.
#### Process Capability Index Calculation:
The process capability index (Cpk) is a measure of how well a process can produce output within specified limits. It is calculated using the following formula:
\[ C_{pk} = \min \left ( \frac{USL - \mu}{3\sigma}, \frac{\mu - LSL}{3\sigma} \right ) \]
Where:
- \( \mu \) = Process mean (average)
- \( \sigma \) = Standard deviation
- USL = Upper Specification Limit (Target + 0.150)
- LSL = Lower Specification Limit (Target - 0.150)
Given:
- \( \mu \) = 3.042 ounces
- \( \sigma \) = 0.034 ounce
- USL = 3.150 ounces
- LSL = 2.850 ounces
The Cpk computation will provide insight into how well the process adheres to specifications.
#### Proportion of Bottles Meeting Specifications:
Upon calculating the Cpk value, the next step involves determining the percentage of bottles that meet the specification limits. This is typically derived from standard normal distribution tables or relevant statistical software.
#### Dropdown Menu for User Input:
1. The dropdown menu offers options for the percentage of bottles that meet specifications, including:
- 80%
- 90%
- 95.45%
- 99.99%
- 99.73%
Users can select the correct response based on the Cpk value calculated.
#### Interactive Section:
- **Text Box for Cpk Value:** An input field where students can enter the calculated Cpk value, rounded to three decimal places.
- **Dropdown for Proportion:** A dropdown selection to input the percentage of bottles that meet the specifications.
This educational activity teaches students how to apply statistical methods
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