Suppose that you are using a 3-sigma p chart to monitor the percentage of times that you fail to get phone numbers at bars. Each day for 30 days, you ask 15 different potential "friends" for their phone number. On average, 6 people declined to give you their phone number each day. What would be the upper control limit for the p chart? OD. 2.05% OE. 52.65% OF. 66.83% OG. 100.00% OH. 97.95% 4 CIE
Suppose that you are using a 3-sigma p chart to monitor the percentage of times that you fail to get phone numbers at bars. Each day for 30 days, you ask 15 different potential "friends" for their phone number. On average, 6 people declined to give you their phone number each day. What would be the upper control limit for the p chart? OD. 2.05% OE. 52.65% OF. 66.83% OG. 100.00% OH. 97.95% 4 CIE
Practical Management Science
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ISBN:9781337406659
Author:WINSTON, Wayne L.
Publisher:WINSTON, Wayne L.
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Transcribed Image Text:**Example Problem: Calculating the Upper Control Limit for a 3-Sigma p Chart**
Suppose that you are using a 3-sigma p chart to monitor the percentage of times that you fail to get phone numbers at bars. Each day for 30 days, you ask 15 different potential "friends" for their phone number. On average, 6 people declined to give you their phone number each day. What would be the upper control limit for the p chart?
**Options:**
- O. 2.05%
- E. 52.65%
- F. 66.83%
- G. 100.00%
- H. 97.95%
- I. 50.98%
![## Understanding Control Charts and Upper Control Limits
In this exercise, we explore using a 3-sigma p chart to monitor performance. Specifically, we examine the scenario where you're tracking the percentage of unsuccessful attempts at obtaining phone numbers in a social setting over a 30-day period.
### Problem Context
Every day for 30 days, you approach 15 different individuals to ask for their phone numbers. On average, 6 people decline. The goal is to determine the upper control limit for the p chart.
### Calculating the Upper Control Limit
To find the upper control limit (UCL) for the p chart, we perform the following steps:
1. **Calculate the Proportion of Failures:**
- The proportion of failures (p) is the number of declines divided by the total attempts per day:
\[ p = \frac{6}{15} = 0.4 \]
2. **Determine the Standard Deviation (σ) of the Proportion:**
- The standard deviation formula for a proportion is:
\[ \sigma = \sqrt{\frac{p(1-p)}{n}} \]
- Where \( n = 15 \), the number of trials per day:
\[ \sigma = \sqrt{\frac{0.4(1-0.4)}{15}} \approx 0.126 \]
3. **Calculate the Upper Control Limit:**
- The UCL using a 3-sigma approach is:
\[ UCL = p + 3\sigma \]
\[ UCL = 0.4 + 3(0.126) \approx 0.778 \text{ or } 77.8\% \]
### Options Given:
- **A.** 44.80%
- **B.** 0.00%
- **C.** 77.95%
- **D.** 2.05%
- **E.** 52.65%
- **F.** 66.83%
The closest match to the calculated UCL is Option C: 77.95%.
### Conclusion
This example illustrates how to apply statistical process control techniques to real-world scenarios by determining the UCL on a p chart, helping to monitor and understand variability in processes.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd04645ef-6cf6-4d1b-a92f-7bdee3700327%2F2848b30b-a359-4bfe-adeb-8c0c288f1d40%2Ff4kop4a_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Understanding Control Charts and Upper Control Limits
In this exercise, we explore using a 3-sigma p chart to monitor performance. Specifically, we examine the scenario where you're tracking the percentage of unsuccessful attempts at obtaining phone numbers in a social setting over a 30-day period.
### Problem Context
Every day for 30 days, you approach 15 different individuals to ask for their phone numbers. On average, 6 people decline. The goal is to determine the upper control limit for the p chart.
### Calculating the Upper Control Limit
To find the upper control limit (UCL) for the p chart, we perform the following steps:
1. **Calculate the Proportion of Failures:**
- The proportion of failures (p) is the number of declines divided by the total attempts per day:
\[ p = \frac{6}{15} = 0.4 \]
2. **Determine the Standard Deviation (σ) of the Proportion:**
- The standard deviation formula for a proportion is:
\[ \sigma = \sqrt{\frac{p(1-p)}{n}} \]
- Where \( n = 15 \), the number of trials per day:
\[ \sigma = \sqrt{\frac{0.4(1-0.4)}{15}} \approx 0.126 \]
3. **Calculate the Upper Control Limit:**
- The UCL using a 3-sigma approach is:
\[ UCL = p + 3\sigma \]
\[ UCL = 0.4 + 3(0.126) \approx 0.778 \text{ or } 77.8\% \]
### Options Given:
- **A.** 44.80%
- **B.** 0.00%
- **C.** 77.95%
- **D.** 2.05%
- **E.** 52.65%
- **F.** 66.83%
The closest match to the calculated UCL is Option C: 77.95%.
### Conclusion
This example illustrates how to apply statistical process control techniques to real-world scenarios by determining the UCL on a p chart, helping to monitor and understand variability in processes.
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