General recommendations:
QUALITY DATA ANALYSIS
• Write the solutions in CLEAR and READABLE way on paper and show (qualitatively) all the relevant plots.
• Avoid (if not required) theoretical introductions or explanations covered during the course.
• Always state the assumptions and report all relevant steps/discussion/formulas/expression to present and
motivate your solution.
• When using hypothesis tests provide the numerical value of the test statistic and the test conclusion in terms
of p-value.
• Exam duration: 2h
• For multichance students only: you can skip Exercise 2, point 4), Exercise 3, question 2).
Exercise 1 (14 points)
The measured diameters of the shafts produced over two shifts are reported in `diameter_phase1.csv`. The columns of the table report a sequential index (‘idx’ column), the measurements in mm (‘diam’ column) and the shift (‘shift’ column) at which the data were collected.
1) Find an adequate model to fit the data.
2) Design the appropriate control charts to monitor the diameter of the shafts (use K = 3). Note: in case of
violations of control limits, assume no assignable cause was found.
3) Using the control chart(s) designed in point 1 (phase 1), check if the data collected for the following 20 shafts produced during shift 2 (stored in `diameter_phase2.csv`) are in control. Report the index of the OOC points, if any.
Exercise 2 (15 points)
A German company operating in the aerospace sector needs to monitor the manufacturing process for a new type of titanium bracket. The quality characteristic of interest is the Brinell hardness. Four hardness measurements are performed in four pre-defined locations of the component, and parts are randomly picked up from the shop floor and inspected every two hours. Data to be used for control chart design are reported in the file `AERO_phase1.csv`. Each column refers to one location where the hardness measurement is performed. Assume the measurements within each sample were performed following the same order shown in the provided table.
1) Check the assumptions and discuss the result.
2) Design a statistical test to check if the hardness in location 1 is statistically higher than the hardness in
location 2. Discuss the result.
3) Design a suitable univariate control charting method for these data, using K = 3. In case of violation of
control limits, assume no assignable cause is known.
4) Using the control chart designed in point 3) determine if the new Brinell hardness measurements in `AERO_phase2.csv` are in control or not. Discuss the result.
09/02/2024
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Exercise 3 (4 points)
Question 1)
The Shewhart control chart for the mean of a process, where the data are known to be normally distributed, sets the control limits (i.e., LCL and UCL) to be 𝐾 standard deviations away from the center line, for some
constant, 𝐾 > 0. If a user will decide to replace 𝐾 by 𝐾
valid for the control chart performance:
> 𝐾, then which of the following statements will be
On a data set, we run the linear model: 𝑌 = 𝛽0 + 𝛽1𝑋1 + 𝛽2𝑋2 + ⋯ + 𝛽𝑘𝑋𝑘 + 𝜀 and we derive the ANOVA table for the fitted model. If the overall model’s F-ratio has a p-value that is smaller than the predetermined level of significance (alpha) for this problem, then which of the following statements will be valid?
a) All model coefficients (𝛽1, 𝛽2, … , 𝛽𝑘) are statistically significantly different from zero.
b) All model coefficients (𝛽1, 𝛽2, … , 𝛽𝑘) are not statistically significantly different from zero.
c) At least one of the model coefficients (𝛽1, 𝛽2, … , 𝛽𝑘) is statistically significantly different from zero.
d) At most one of the model coefficients (𝛽1, 𝛽2, … , 𝛽𝑘) is statistically significantly different from zero.
a) The false alarms will increase.
b) The false alarms will decrease.
c) The out-of-control detection power will increase.
d) We cannot tell from the above information only.
Question 2)
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