Question

Parts manufactured by an injection-molding process are subjected to a compressive strength test. We wish to monitor the compressive strength of the parts manufactured by this process, using an x̄ and an s control chart.
Samples of size 9 are to be taken at regular intervals, and their mean compressive strength (in psi = pounds per square inch) and standard deviation are plotted on the charts in time order.
The target values for the compressive strengths are a mean of μ = 90 psi and a standard deviation of σ = 9 psi.

Sample size n c4 2.606 276 2.088 1.964 9515 0.029 1.874 0.9594 0.113 1.806 0.965 0.179 1.751 9693 0.232 1.707 97270.276 1.669 0.7979 8862 0.9213 .94 10

Q1: The center line for the three sigma x̄ control chart would be

• 9
• 81
• 90
• 99

Q2: The upper control limit for the x̄ control chart would be

• 117
• 99
• 94.5
• 90

Q3: The lower control limit for the x̄ control chart would be

• 81
• 85.5
• 90
• 63

Q4: Suppose at the time of sample 10, we observe a mean of 97.5 psi. We should

• continue sampling, but increase the sample size to 16. The process is barely in control.
• continue sampling. The process is still in control.
• declare the process out of control.
• continue sampling, but reduce the sample size to 4. The process is well in control.

Q5: Suppose at the time of sample 15, we observe a mean of 77.4 psi. We should

• continue sampling. The process is still in control.
• continue sampling, but reduce the sample size to 4. The process is well in control.
• continue sampling, but increase the sample size to 16. The process is barely in control.
• declare the process out of control.

Q6: Suppose now that we observe 10 sample means in a row that are all below 90. We should

• continue sampling, but increase the sample size to 16. The process is barely in control.
• continue sampling. The process is still in control.
• declare the process out of control.
• continue sampling, but reduce the sample size to 4. The process is well in control.

Q7: Suppose now that we observe 2 sample means in a row that are all above 90. We should

• continue sampling. The process is still in control.
• continue sampling, but reduce the sample size to 4. The process is well in control.
• continue sampling, but increase the sample size to 16. The process is barely in control.
• declare the process out of control.

Q8: The center line for the three sigma s chart would be

• 4.5
• 18
• 8.7237
• 9

Q9: The upper control limit for the three sigma s chart would be

• 100
• 4.5
• 15.363
• 9

Q10: The lower control limit for the three sigma s chart would be

• 2.088
• 18
• 0
• 9

I need help figuring out how to answer these questions. I don't just want the answers, I would like to know how to solve for them, as I am feeling completely lost. Please!!

Answer 1

We have ,

$\dpi{100} \fn_jvn \mu = 90$ , $\dpi{100} \fn_jvn \sigma = 9$ and   $\dpi{100} \fn_jvn n = 9$

The 3 sigma control limits for X bar chart are given by :

$\dpi{100} \fn_jvn E(\bar{X}) \pm 3S.E.(\bar{X})$

$\dpi{100} \fn_jvn \Rightarrow \mu \pm 3\sigma /\sqrt{n}$

Center line : $\dpi{100} \fn_jvn CL_{\bar{X}} = \mu$

Lower Control Limit : $\dpi{100} \fn_jvn LCL_{\bar{X}} = \mu - 3\sigma /\sqrt{n}$

Upper control Limit :  $\dpi{100} \fn_jvn UCL_{\bar{X}} = \mu + 3\sigma /\sqrt{n}$

1. Center line = $\dpi{100} \fn_jvn \mu = 90$

2. Upper control Limit = $\dpi{100} \fn_jvn \mu + 3\sigma /\sqrt{n}$

= $\dpi{100} \fn_jvn 90 + 3*9 /\sqrt{9}$

= $\dpi{100} \fn_jvn 90 + 9$

= $\dpi{100} \fn_jvn 99$

3. Lower Control Limit   $\dpi{100} \fn_jvn = \mu - 3\sigma /\sqrt{n}$

  = $\dpi{100} \fn_jvn 90 - 3*9 /\sqrt{9}$

= $\dpi{100} \fn_jvn 90 - 9$

= $\dpi{100} \fn_jvn 81$

4. At the time of sample 10, we observe a mean of 97.5 psi. We should "continue sampling, the process is still in control" as it is within control limits.

5. At the time of sample 15, we observe a mean of 77.4 psi. We should "declare the process out of control"

as it is outside the control limits.

6. We observe 10 sample means in a row that are all below 90. We should "continue sampling, but increase the sample size to 16. The process is barely in control." as there is a pattern of 7 or more points below the center line which indicates presence of assignable causes.

7. We observe 2 sample means in a row that are all above 90. We should "continue sampling, the process is still in control." as it is within control limits.

The 3 sigma control limits for three sigma s chart are given by:

$\dpi{100} \fn_jvn E(s) \pm 3S.E.(s)$

$\dpi{100} \fn_jvn \Rightarrow C_{4} .\sigma \pm 3.C_{5} .\sigma$

Center line : $\dpi{100} \fn_jvn CL_{s} = C_{4} .\sigma$

Lower Control Limit : $\dpi{100} \fn_jvn LCL_{s} = C_{4} .\sigma - 3.C_{5} .\sigma = B_{5}.\sigma$

Upper control Limit :  $\dpi{100} \fn_jvn UCL_{s} = C_{4} .\sigma + 3.C_{5} .\sigma = B_{6}.\sigma$

8. Center line for the three sigma s chart = $\dpi{100} \fn_jvn CL_{s} = C_{4} .\sigma$

= 0.9693* 9

= 8.7237

9. The upper control limit for the three sigma s chart = $\dpi{100} \fn_jvn UCL_{s} = C_{4} .\sigma + 3.C_{5} .\sigma = B_{6}.\sigma$

= 1.707 * 9

= 15.363

10. The lower control limit for the three sigma s chart = $\dpi{100} \fn_jvn LCL_{s} = C_{4} .\sigma - 3.C_{5} .\sigma = B_{5}.\sigma$

= 0.232 * 9

= 2.088