Chapter 06, Problem 8
A tire manufacturer has been concerned about the number of defective tires found recently. In order to evaluate the true magnitude of the problem, a production manager selected ten random samples of twenty units each for inspection. The number of defective tires found in each sample are as follows:
Develop a p-chart with a z = 3.
Note. If the lower control limit is negative, round the LCL to zero.
CL = , round your answer to 3 decimal places, the tolerance is +/-0.001
UCL = , round your answer to 3 decimal places, the tolerance is +/-0.002
LCL = , round your answer to 3 decimal places, the tolerance is +/-0.002
Suppose that the next four samples selected had 6, 3, 3, and 4 defects. What conclusion can you make?
The process is out of control.
The process is in control.
The process is out of control
To develop a p-chart, we need to calculate the following values:
1. The proportion of defective tires in each sample:
- For the provided ten random samples of twenty units each, we have the following number of defective tires:
6, 4, 5, 5, 4, 6, 7, 3, 5, 4
- To calculate the proportion of defective tires in each sample, divide the number of defective tires by 20 (the total number of units in each sample):
6/20, 4/20, 5/20, 5/20, 4/20, 6/20, 7/20, 3/20, 5/20, 4/20
2. The overall proportion of defective tires (p̂):
- To get the overall proportion of defective tires, sum up all the defective tires (6+4+5+5+4+6+7+3+5+4) and divide by the total number of units (10*20):
(6+4+5+5+4+6+7+3+5+4) / (10*20)
3. The center line (CL) of the control chart is the overall proportion of defective tires (p̂).
4. The upper control limit (UCL):
- To calculate the UCL, we use the formula: UCL = CL + z * sqrt(CL*(1-CL)/n)
- In this case, z = 3 (given), CL = overall proportion of defective tires (p̂), and n = 20 (the number of units in each sample).
- Calculate the UCL using the formula.
5. The lower control limit (LCL):
- To calculate the LCL, we use the formula: LCL = max(0, CL - z * sqrt(CL*(1-CL)/n))
- In this case, z = 3 (given), CL = overall proportion of defective tires (p̂), and n = 20 (the number of units in each sample).
- Calculate the LCL using the formula.
Now, let's calculate these values step-by-step.
Step 1: Calculate the proportion of defective tires in each sample:
- 6/20, 4/20, 5/20, 5/20, 4/20, 6/20, 7/20, 3/20, 5/20, 4/20
Step 2: Calculate the overall proportion of defective tires (p̂):
- (6+4+5+5+4+6+7+3+5+4) / (10*20) = [sum of all defective tires] / [total number of units]
Step 3: Calculate the center line (CL) of the control chart:
- CL = overall proportion of defective tires (p̂)
Step 4: Calculate the upper control limit (UCL):
- UCL = CL + z * sqrt(CL*(1-CL)/n)
Step 5: Calculate the lower control limit (LCL):
- LCL = max(0, CL - z * sqrt(CL*(1-CL)/n))
After calculating these values, we can determine if the process is in control or out of control.
Now, let's calculate these values:
Step 1: Calculate the proportion of defective tires in each sample:
- 6/20, 4/20, 5/20, 5/20, 4/20, 6/20, 7/20, 3/20, 5/20, 4/20
- 0.3, 0.2, 0.25, 0.25, 0.2, 0.3, 0.35, 0.15, 0.25, 0.2
Step 2: Calculate the overall proportion of defective tires (p̂):
- (6+4+5+5+4+6+7+3+5+4) / (10*20)
- 49 / 200
- 0.245
Step 3: Calculate the center line (CL) of the control chart:
- CL = overall proportion of defective tires (p̂)
- CL = 0.245
Step 4: Calculate the upper control limit (UCL):
- UCL = CL + z * sqrt(CL*(1-CL)/n)
- UCL = 0.245 + 3 * sqrt(0.245*(1-0.245)/20)
- Calculate the UCL using the formula.
Step 5: Calculate the lower control limit (LCL):
- LCL = max(0, CL - z * sqrt(CL*(1-CL)/n))
- LCL = max(0, 0.245 - 3 * sqrt(0.245*(1-0.245)/20))
- Calculate the LCL using the formula.
Now, based on the calculated CL, UCL, and LCL, we can draw the p-chart and interpret the conclusion.
Suppose the next four samples selected had 6, 3, 3, and 4 defects. We can compare these values with the control limits on the p-chart to determine if the process is in control or out of control. If any of these values fall outside the control limits (UCL or LCL), the process is considered out of control. Otherwise, if all values are within the control limits, the process is considered in control.
Based on the given data and calculated control chart values, we can determine if the process is in control or out of control. Since the conclusion is not provided in the question, we cannot determine the correct answer.
To develop a p-chart, we need to calculate the control limits and the center line.
Step 1: Calculate the average proportion of defectives (p-bar):
- Add up the number of defective tires found in each sample and divide by the total number of units inspected.
- In this case, the total number of samples is 10, and the total number of units inspected is 10 * 20 = 200.
- So, p-bar = (sum of defective tires) / (total number of units) = (number of defective tires in all samples) / 200.
Step 2: Calculate the standard deviation of the proportion (sigma-p):
- We use the formula sigma-p = sqrt(p-bar * (1 - p-bar) / (total number of units)).
- Plug in the values to calculate sigma-p.
Step 3: Calculate the control limits:
- The center line (CL) is equal to p-bar.
- The upper control limit (UCL) is equal to CL + z * sigma-p, where z is the number of standard deviations.
- The lower control limit (LCL) is equal to CL - z * sigma-p.
Step 4: Round the control limits to the specified decimal places.
Let's calculate the p-chart for this problem.
Step 1: Calculate p-bar:
- The number of defective tires found in each sample is given.
- Add up the numbers: x1 + x2 + ... + x10.
- Divide by the total number of units inspected: (x1 + x2 + ... + x10) / 200.
- Calculate p-bar.
Step 2: Calculate sigma-p:
- Use the formula: sigma-p = sqrt(p-bar * (1 - p-bar) / 200).
- Plug in the value of p-bar and calculate sigma-p.
Step 3: Calculate the control limits:
- CL = p-bar.
- UCL = CL + z * sigma-p.
- LCL = CL - z * sigma-p.
Step 4: Round the control limits to the specified decimal places.
Now, if the next four samples have 6, 3, 3, and 4 defects, we can compare the number of defects to the control limits to make a conclusion. If any value falls outside the control limits, it indicates that the process is out of control. If all values are within the control limits, the process is considered in control.
Based on the information given in the question, follow the steps outlined above to calculate the p-chart and compare the values to make your conclusion.