Statistics 495, Applied Statistics for Industry I         Name:

Exam 2, Fall 1997                                         Site:
INSTRUCTIONS: You will have 1 hour and 30 minutes to complete the exam. There are 6 questions worth a total of 100 points. Not all questions have the same point value so gauge your time appropriately. Read the questions carefully and completely. Answer each question and show work in the space provided on the exam. Turn in the entire exam when you are done or when time is up. For essay questions, think before you write.

  1. [35 pts] A student in STAT 495 conducted a study on the performance and reliability of an Internet access system.. The study focused on the question: ``How often do users get the requested answer from a company's Internet access system?'' Specifically, proxy servers for a company handle all Internet access requests and count the number of successful (and unsuccessful) attempts. The table below gives the daily number of Internet access attempts and the number of such attempts that resulted in an error message for 20 consecutive days.

    
Day   Attempts   Errors  Fraction    Day   Attempts   Errors   Fraction

 1     412,670   42,104    0.102     11     446,823   40,405     0.090
 2     395,736   40,286    0.102     12     431,661   44,198     0.102
 3     401,765   35,399    0.088     13     434,353   39,047     0.090
 4     395,422   97,981    0.248     14     406,232   39,455     0.097
 5     422,223   45,346    0.107     15     402,454   48,292     0.120
 6     433,234   43,699    0.101     16     403,312   47,720     0.118
 7     396,788   24,752    0.062     17     387,782   53,173     0.137
 8     411,383   45,391    0.110     18     355,500   49,474     0.139
 9     423,348   39,179    0.093     19     372,441   45,222     0.121
10     474,053   48,680    0.103     20     415,813   40,583     0.098

Total                                     8,222,993  910,386


     (a) [2] If we consider errors to be defective attempts,
           what is the value of p-bar, the average fraction of
           defective attempts?
     
     (b) [6] The number of attempts vary from day to day so the
           upper and lower control limits for a p-chart will vary from
           day to day.  Compute the upper and lower control limits for
           day 10.

     (c) [5] Below is the p-chart for the fraction of errors.
           What days plot outside the control limits?  What can be
           said about the stability of this process based on the
           p-chart?

 

       
    
 

     (d) [5] Why are the control limits for the p-chart above so
           narrow?

     (e) [2] An alternate approach would be to treat the
           fraction of errors for each day as an individual
           ``measurement''.  Using this idea what would be the
           centerline for an individuals chart?

     (f) [6] Considering the fractions as individual
           ``measurements'' the average moving range is MR-bar=0.0298,
           compute the upper and lower control limits for an
           individuals chart.

     (g) [4] Plot the centerline and control limits for
           an individuals chart on the run chart of fraction of
           errors below.  Are there any days that plot outside
           the control limits?

 

       
    
 
 
     (h) [5] Which chart, the p-chart or the individuals
           chart does a better job identifying possible special
           causes?  Explain briefly.

  2. [10 pts] Two machines, each made by a different company, are used to perform the same function, a slotting operation. The quality characteristic in question is the surface finish of the slot walls. Control charts have been kept for each machine to monitor the performance. However, the subgroup sizes used to create the charts are different. The X-bar chart for machine A, established using samples of size n=4 and R-bar, has a center line of 85.2 microinches and an upper control limit of 100.3 microinches. Machine B's X-bar chart was established using subgroups of size n=5 and R-bar. It has a center line at 84.9 microinches and an upper control limit of 98.5 microinches. Both the X-bar and R charts have been shown to be in good statistical control over an extended period of time. Which machine appears to be producing the most consistent/least variable surface finish? Justify your answer statistically.

  3. [10 pts] As a data analysis tool does a control chart get at the enumerative or the analytic purpose of building knowledge about a process? Explain your answer briefly.

  4. [16 pts] For each of the following scenarios indicate that type of control chart(s) you would construct. Briefly explain your choice.

         (a) In a paint formulation process, a sample of paint is
         taken every four hours and evaluated for color.  The data
         consists of individual values of the color measurement.

     (b) In the manufacture of AAA batteries, subgroups of 100
         batteries are selected every hour.  Each battery is
         measured for voltage.  A battery with a measured voltage
         below 1.4 volts is classified as defective.
     
     (c) In the production of a chemical liquid, there is an
         impurity that, when present, can degrade the quality of 
         the resulting product.  Fifty samples (1 liter each) 
         were collected over time.  For each liter, the number 
         of suspended particles of the impurity was counted.
     
     (d) In a traffic study, the number of vehicles going through a
         particular intersection between 7:45 am and 8:15 am are 
         counted each day.  You have data for 30 consecutive days.
  5. [20 pts] The thickness of a printed circuit board is an important quality characteristic. Data on deviations from the target board thickness of 0.0630, in ten thousanths of an inch, are taken for 25 subgroups of size 3 boards. A thickness of 0.0635 is recorded as a deviation of 5 ten thousanths of an inch. The subgroup means and ranges are given below and plotted on the control charts.

    
group  X-bar  R   group  X-bar  R   group  X-bar  R   group  X-bar  R
 
   1      5  11      8     -4   5     15      2  25     22    -10  10
   2     -3   8      9     -3   9     16     -1   4     23      1   3
   3      1   4     10      2   2     17     -7  14     24      3   6
   4      2   4     11      4   9     18     -1   3     25     -3   7
   5     -4  10     12     -2   8     19      2   7  
   6     -5  20     13      2   5     20      5  10 
   7      1  15     14      9  13     21     -7  12 

                    X-bar-bar=-0.44      R-bar=8.96


     (a) [6 pts] Calculate the control limits for the
           X-bar and R charts and draw them on the run charts below.

 

       
    
 


     (b) [5] Comment on the stability of the process.

     (c) [9] If subgroup 15 was affected by a special cause
           and that special cause was eliminated one could recalculate
           the control limits with subgroup 15 removed.  Comment on
           the stability of the process with subgroup 15 removed.
  6. [9 pts]Consider a hypothetical process that is producing defective items 10% of the time. If we were to take subgroups of size n=25 items, the theoretical control limits for a p-chart would be:
    
    LCL=0.00       UCL=0.10 + 3( 0.10(1-0.10)/25 )**.5 = 0.28

     (a) [5] If the process changed so that it started
            producing defective items 20% of the time, what is
            the approximate probability that a subgroup, after the
            change, would have a fraction defective that is beyond 
            the UCL?  Hint: Recall that a subgroup fraction defective
            is approximately normally distributed with: mean=p and
            standard deviation=(p(1-p)/n)**.5.

     (b) [4]  Give the value and an interpretation of the 
            associated ARL of the p-chart after the change?