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IND E 521 Statistical Quality Engineering

Midterm Exam

Fall 2022

Problem 1 (16—2 pts each)

True/False. Each of the following statements can be TRUE or FALSE. Mark “T” if true or “F” if false in the parenthesis for each statement. There is no need to show the justification.

(1) (       ) When the observations  are independently distributed with non-identical distributions with a large n, the sample mean  approximately follows a normal distribution.

(2) (       ) The average run length for an  control chart increases when the associated specification limits are widened while keeping other things the same.

(3) (       ) The Shewhart control chart assumes that the statistic it is monitoring follows a normal distribution.

(4) (       ) The overall type II error rate for a control chart increases if more sensitizing rules, which are independent and have non-zero detection powers, are used to signal out-of-control.

(5) (      ) In estimating process capability, it is assumed that the data follows a normal distribution.

(6) (       ) Control limits are based on both types of variation inherent in a process.

(7) (      ) A process that exhibits random variability would be judged to be out of control.

(8) (      ) The out-of-control ARL, ARL₁, is 1/(Type II Error Rate).

Problem 2 (12—2 points each)

Multiple-choice Questions. Please circle the MOST appropriate answer (only ONE). There is no need to show the justification.

(1) Given desired type I and II error rates, which of the following parameters is NOT important to determine the sample size to use for an X-bar control chart?

 (2) Which of the following statements is TRUE if we increase  for L-sigma limit control charts while keeping other things the same?

Problem 3 (6)

A bottling company claims that the mean fill volume of their bottling process is equal to 120 ml. It is known that the standard deviation of fill volume of the bottling process is 6 ml. To test the validity of the company’s claim on the mean fill volume, you randomly collected 15 bottles of the company and measured their fill volumes (). Based on the following summary statistic from your sample, please answer the questions below.

(a) State the hypotheses for testing (1 point).

(b) Test your hypotheses in part (a) using α = 0.05 and draw conclusion (3 points).

(c) Construct the 95% two-sided confidence interval for the mean fill volume (2 point).

Problem 4 (10 pts)

The following table summarizes the Phase I data that was collected to establish control charts to monitor a quality characteristic. The sample size is 5.

(a) Construct the initial  and S control charts with 3-sigma limits (4 points).

(b) Construct 3-sigma  and S control charts for Phase II monitoring, assuming that all out-of-control points, if any, in Phase I are due to assignable causes (4 points).

(c) Suppose a 3-sigma R chart will be substituted for the S chart in part (b). What would be the appropriate parameters of the R chart, assuming the process variance remains the same (2 points)?

Problem 5 (4 pts)

The  and R control charts with 3-sigma limits (n = 4) are given as follows:

Assume the process is in control and LSL = 790, USL=810.

a) Estimate the process standard deviation. (2)

b) Estimate the process capability ratio. (2)

Problem 6 (6 pts)

To monitor a quality characteristic, X and MR control charts are constructed. Based on 50

observations (), we obtain  and , where . Assume all data are in-control and specification limits are 19  4.

a) Estimate the process mean and standard deviation. (2 point)

b) Estimate the number of parts per million (ppm) that will be outside of the specifications, assuming the normality of the quality characteristic. (4 points)

Problem 7 (10 pts)

Twenty samples of size n = 4 are taken from an in-control process to measure the thickness of piston rings in microns. The sum of the sample means is  and the sum of the sample standard deviations is .

a) Construct the  and  control charts with 3-sigma control limits. (4pts)

Assume a shift with the magnitude of 8 microns occurs in the process mean (with no change in the process variance).

b) Estimate the probability of detecting the shift by the first plotted point on the 3-sigma

control chart after the shift. (3)

c) Estimate the probability of detecting the change using the following rule only.

Rule: Signal an out-of-control if two consecutive points are plotted beyond the 3-sigma limits on the same side of the  control chart. (3)

Problem 8 (16 pts)

A small public utility in the San Francisco area monitored daily usage of water (1 unit=748,000) gallons) for a period of ten weeks (Mon-Fri) during September, October, and November.  The data, found in the spreadsheet, presents the daily water usage for 50 weekdays.  Records show that on day 5, a water main broke and a major leak occurred.  

a) Is it reasonable to assume that usage is normally distributed? Why? (3)

b) If not, transform the data to achieve a normal distribution by using an inverse transformation. In other words, set Y=1/x and then use the Y data for the rest of this question. (This is also known as a Box-Cox transformation with l=-1.0.). Use a histogram to show if this transformed data, Y, is normally distributed. (3)

c) Using the appropriate data from a) or b) what are the parameters of your control charts (e.g. UCL, etc)? (4)

d) Is the process in control? Explain. (3)

e) What is your next step for long-term monitoring of daily water usage? (3)

Problem 9 (24 pts)

Krypton Ltd. Manufactures silicon wafers for the semiconductor industry.  The business is highly competitive with pressure from customers to design wafers with more elements per surface area.  There are two ways to do this:  larger diameter wafers and thinner lines on the wafer surface.  The industry standard has been 200mm diameter wafers.  A 300mm wafer is an option but requires major capital and tooling.

Thinner lines is appealing because the chips (from the wafer) can be smaller, operate at faster speeds, consume less energy, and generate less heat.  The attribute, line width, is expressed in microns (mM).  The goal is to produce a wafer with a line width of 0.11mM.  The production steps create many sources of variation.  On the wafer itself, the line width measure depends on where the measurement is taken.  Wafers are produced in cassettes (25 in a batch) and the location within the cassette can influence the measure.  In addition, the cassettes may be different.  

The data for this problem may be found in the spreadsheet file.  The data are the measurements that were taken from three wafers selected at random from a cassette.  10 cassettes were also selected at random; this gives a total of 30 wafers. Each wafer was measured at 5 locations:  top, left, center, right, and bottom.  The original micron measurements have been multiplied by 10 for ease of analysis.

a) and R control charts with a sample size of n=5 were constructed. The control charts exhibited a definite lack of control with many OOC points on the chart.  Why are there so many OOC points on the chart? (4)

b) Use descriptive statistics or a histogram or a box plot to determine if there are differences between the five measurement locations? (4)

Design a control chart to monitor the following. You have the data, but do not have to build a control chart. Describe the sampling mechanism, number of plot points, control chart, etc.

c) The average of the overall process and its variability. (3)

d) The proportion of variability that is due to location differences within the wafer. (3)

e) The proportion of variability that is due to location differences within the cassette. (3)

f) The proportion of variability is due to different cassettes. (3)