Answer :
### Step-by-Step Solution
#### Given Data
You have 13 samples of MRI retests, each consisting of 200 observations. The number of retests in each sample is provided as follows:
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 \\ \hline \text{No. of retests} & 1 & 2 & 3 & 1 & 0 & 1 & 2 & 3 & 2 & 6 & 2 & 5 & 3 \\ \hline \end{array} \][/tex]
#### Calculate the Proportion of Retests for Each Sample
The proportion [tex]\( p_i \)[/tex] of retests for each sample is calculated as:
[tex]\[ p_i = \frac{\text{Number of retests in sample } i}{n} \][/tex]
Here, [tex]\( n = 200 \)[/tex].
[tex]\[ \begin{align*} p_1 &= \frac{1}{200} = 0.005 \\ p_2 &= \frac{2}{200} = 0.01 \\ p_3 &= \frac{3}{200} = 0.015 \\ p_4 &= \frac{1}{200} = 0.005 \\ p_5 &= \frac{0}{200} = 0.0 \\ p_6 &= \frac{1}{200} = 0.005 \\ p_7 &= \frac{2}{200} = 0.01 \\ p_8 &= \frac{3}{200} = 0.015 \\ p_9 &= \frac{2}{200} = 0.01 \\ p_{10} &= \frac{6}{200} = 0.03 \\ p_{11} &= \frac{2}{200} = 0.01 \\ p_{12} &= \frac{5}{200} = 0.025 \\ p_{13} &= \frac{3}{200} = 0.015 \\ \end{align*} \][/tex]
#### Calculate the Overall Sample Proportion
The overall sample proportion [tex]\( \bar{p} \)[/tex] is given by:
[tex]\[ \bar{p} = \frac{\sum_{i=1}^{13} p_i}{13} \][/tex]
[tex]\[ \bar{p} = \frac{0.005 + 0.01 + 0.015 + 0.005 + 0.0 + 0.005 + 0.01 + 0.015 + 0.01 + 0.03 + 0.01 + 0.025 + 0.015}{13} \approx 0.011923076923076923 \][/tex]
#### Calculate the Standard Deviation of the Sample Proportion
The standard deviation [tex]\( \sigma_p \)[/tex] of the sample proportion is calculated using the formula:
[tex]\[ \sigma_p = \sqrt{\frac{\bar{p} (1 - \bar{p})}{n}} \][/tex]
[tex]\[ \sigma_p = \sqrt{\frac{0.011923076923076923 \times (1 - 0.011923076923076923)}{200}} \approx 0.007674932299298579 \][/tex]
#### Calculate the Control Limits (Using 2-Sigma Limits)
The lower control limit (LCL) and upper control limit (UCL) are given by:
[tex]\[ \text{LCL} = \bar{p} - 2\sigma_p \][/tex]
[tex]\[ \text{UCL} = \bar{p} + 2\sigma_p \][/tex]
[tex]\[ \text{LCL} = 0.011923076923076923 - 2 \times 0.007674932299298579 \approx -0.0034267876755202346 \approx -0.0034 \quad \text{(rounded to 4 decimal places)} \][/tex]
[tex]\[ \text{UCL} = 0.011923076923076923 + 2 \times 0.007674932299298579 \approx 0.02727294152167408 \approx 0.0273 \quad \text{(rounded to 4 decimal places)} \][/tex]
#### Control Limits
The control limits are thus:
[tex]\[ \text{LCL} = -0.0034 \][/tex]
[tex]\[ \text{UCL} = 0.0273 \][/tex]
#### Check If the Process is in Control
To determine if the process is in control, we check if all the proportions [tex]\( p_i \)[/tex] fall within the control limits.
Here, all the given proportions [tex]\( p_i \)[/tex] lie between -0.0034 and 0.0273. Specifically:
[tex]\[ \begin{align*} 0.005 &\text{ (within limits)} \\ 0.01 &\text{ (within limits)} \\ 0.015 &\text{ (within limits)} \\ 0.005 &\text{ (within limits)} \\ 0.0 &\text{ (within limits)} \\ 0.005 &\text{ (within limits)} \\ 0.01 &\text{ (within limits)} \\ 0.015 &\text{ (within limits)} \\ 0.01 &\text{ (within limits)} \\ 0.03 &\text{ (not within limits)} \\ 0.01 &\text{ (within limits)} \\ 0.025 &\text{ (within limits)} \\ 0.015 &\text{ (within limits)} \end{align*} \][/tex]
Since [tex]\( p_{10} = 0.03 \)[/tex] is outside the control limits, the process is not in control.
### Answers
a. Control Limits:
[tex]\[ \begin{align*} \text{LCL} &= -0.0034 \\ \text{UCL} &= 0.0273 \end{align*} \][/tex]
b. Is the process in control?
No
#### Given Data
You have 13 samples of MRI retests, each consisting of 200 observations. The number of retests in each sample is provided as follows:
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 \\ \hline \text{No. of retests} & 1 & 2 & 3 & 1 & 0 & 1 & 2 & 3 & 2 & 6 & 2 & 5 & 3 \\ \hline \end{array} \][/tex]
#### Calculate the Proportion of Retests for Each Sample
The proportion [tex]\( p_i \)[/tex] of retests for each sample is calculated as:
[tex]\[ p_i = \frac{\text{Number of retests in sample } i}{n} \][/tex]
Here, [tex]\( n = 200 \)[/tex].
[tex]\[ \begin{align*} p_1 &= \frac{1}{200} = 0.005 \\ p_2 &= \frac{2}{200} = 0.01 \\ p_3 &= \frac{3}{200} = 0.015 \\ p_4 &= \frac{1}{200} = 0.005 \\ p_5 &= \frac{0}{200} = 0.0 \\ p_6 &= \frac{1}{200} = 0.005 \\ p_7 &= \frac{2}{200} = 0.01 \\ p_8 &= \frac{3}{200} = 0.015 \\ p_9 &= \frac{2}{200} = 0.01 \\ p_{10} &= \frac{6}{200} = 0.03 \\ p_{11} &= \frac{2}{200} = 0.01 \\ p_{12} &= \frac{5}{200} = 0.025 \\ p_{13} &= \frac{3}{200} = 0.015 \\ \end{align*} \][/tex]
#### Calculate the Overall Sample Proportion
The overall sample proportion [tex]\( \bar{p} \)[/tex] is given by:
[tex]\[ \bar{p} = \frac{\sum_{i=1}^{13} p_i}{13} \][/tex]
[tex]\[ \bar{p} = \frac{0.005 + 0.01 + 0.015 + 0.005 + 0.0 + 0.005 + 0.01 + 0.015 + 0.01 + 0.03 + 0.01 + 0.025 + 0.015}{13} \approx 0.011923076923076923 \][/tex]
#### Calculate the Standard Deviation of the Sample Proportion
The standard deviation [tex]\( \sigma_p \)[/tex] of the sample proportion is calculated using the formula:
[tex]\[ \sigma_p = \sqrt{\frac{\bar{p} (1 - \bar{p})}{n}} \][/tex]
[tex]\[ \sigma_p = \sqrt{\frac{0.011923076923076923 \times (1 - 0.011923076923076923)}{200}} \approx 0.007674932299298579 \][/tex]
#### Calculate the Control Limits (Using 2-Sigma Limits)
The lower control limit (LCL) and upper control limit (UCL) are given by:
[tex]\[ \text{LCL} = \bar{p} - 2\sigma_p \][/tex]
[tex]\[ \text{UCL} = \bar{p} + 2\sigma_p \][/tex]
[tex]\[ \text{LCL} = 0.011923076923076923 - 2 \times 0.007674932299298579 \approx -0.0034267876755202346 \approx -0.0034 \quad \text{(rounded to 4 decimal places)} \][/tex]
[tex]\[ \text{UCL} = 0.011923076923076923 + 2 \times 0.007674932299298579 \approx 0.02727294152167408 \approx 0.0273 \quad \text{(rounded to 4 decimal places)} \][/tex]
#### Control Limits
The control limits are thus:
[tex]\[ \text{LCL} = -0.0034 \][/tex]
[tex]\[ \text{UCL} = 0.0273 \][/tex]
#### Check If the Process is in Control
To determine if the process is in control, we check if all the proportions [tex]\( p_i \)[/tex] fall within the control limits.
Here, all the given proportions [tex]\( p_i \)[/tex] lie between -0.0034 and 0.0273. Specifically:
[tex]\[ \begin{align*} 0.005 &\text{ (within limits)} \\ 0.01 &\text{ (within limits)} \\ 0.015 &\text{ (within limits)} \\ 0.005 &\text{ (within limits)} \\ 0.0 &\text{ (within limits)} \\ 0.005 &\text{ (within limits)} \\ 0.01 &\text{ (within limits)} \\ 0.015 &\text{ (within limits)} \\ 0.01 &\text{ (within limits)} \\ 0.03 &\text{ (not within limits)} \\ 0.01 &\text{ (within limits)} \\ 0.025 &\text{ (within limits)} \\ 0.015 &\text{ (within limits)} \end{align*} \][/tex]
Since [tex]\( p_{10} = 0.03 \)[/tex] is outside the control limits, the process is not in control.
### Answers
a. Control Limits:
[tex]\[ \begin{align*} \text{LCL} &= -0.0034 \\ \text{UCL} &= 0.0273 \end{align*} \][/tex]
b. Is the process in control?
No
