✅ Assignments

Spread, standard error and variation

1. P1 A coffee roaster weighs single bags, and also reports the average weight of each pallet of 10 bags. Which of those numbers bounces around less from pallet to pallet, and what do we call the amount the pallet average bounces? How could you estimate that bounce in R by simulation, with no formula at all?
   Follow-up: If you wanted to cut that bounce in half, how many bags per pallet would you need, and why?
2. P2 Two labs report their results: one measures battery life in hours, the other measures cable length in metres. Your manager asks two things: which process is more variable, and how much to trust each lab's reported average. Which statistic answers each question, how is each one computed, and why can one statistic not answer both?
   Follow-up: Lab 1 doubles its sample size. Which of your two numbers changes, and which stays about the same?
3. P3 A call centre's average answer time wobbles a little every day, and on one Tuesday it tripled when the phone system crashed. Which of those is common-cause variation and which is special-cause? Who is responsible for reducing each kind, and how would a control chart help you tell them apart?
   Follow-up: A supervisor wants to hold a team meeting every time the daily average goes up a bit. What would you advise, and why?
4. P4 A lab weighs 5 tablets from every hour of a shift. Describe two ways to summarise the spread of those weights: one that looks only at variation inside each hour, and one that uses every tablet at once. When would these two numbers disagree, and which one belongs in the control limits?
   Follow-up: A customer cares about every tablet they receive over a month. Would you report Cpk or Ppk to them, and why?

Stability and capability

1. P5 Explain to a new engineer the difference between a stable process and a capable process. What tool tells you about each, and whose 'voice' sets each set of limits? Describe a real process that is stable but not capable, and one that is capable but not stable.
   Follow-up: Why should you check stability before you report a capability index?
2. P6 A resistor maker reports Cp = 1.8 and Cpk = 0.7 for resistance, and Pp and Ppk are close to those. Sketch what the histogram must look like against the spec limits. Is the process capable, and what single change would help most?
   Follow-up: What would have to happen for Cpk to equal Cp?
3. P7 A soda filler reports Cp = Cpk = 0.9 and Pp = Ppk = 0.6, and the process is well centred. What do the numbers tell you, and why are the P indices lower than the C indices?
   Follow-up: Which would you work on first: tightening each short run, or stopping the drift between runs? Why?
4. P8 Write Cpk as a formula and explain each piece in plain words: which parts come from the customer and which come from the process? Why does it look at both spec limits and then keep only one?
   Follow-up: Sketch a process where Cp is high but Cpk is low, and one where the two are equal.

Designing a study and choosing a chart

1. P9 Diners at a campus dining hall say the lunch line has gotten slower this month. How would you check whether the line has really changed? What would you record, how would you group the readings, and which charts would you make?
   Follow-up: The dining hall promises 'served within 10 minutes'. Where does that number belong in your analysis, and where does it not?
2. P10 Baristas say a coffee shop's espresso shots 'taste weak lately'; the machine is supposed to pull 18 g of coffee into a 36 g shot. How would you find out whether the machine has drifted? What would you weigh, how often, grouped how, and plotted on what?
   Follow-up: The recipe says 36 g plus or minus 2 g. At what point in your analysis would you bring that in?
3. P11 A bakery checks 200 cookies every day and throws out any that are broken. You have a month of daily reject counts. Which control chart fits these data, where do its limits come from, and what would count as a signal?
   Follow-up: A colleague wants the mean and standard deviation of each cookie's 'broken or not'. Why does an averages chart not fit here?
4. P12 A city inspects some of its streetlights every month and counts every fault it finds (dead bulbs, cracked covers, bad wiring). Some months it inspects 50 lights, some months 300. Which chart would you use, and how do its limits behave from month to month?
   Follow-up: A council member points to the month with the most faults. Why might that not be the worst month?
5. P13 From memory, write down the upper and lower control limits for an averages chart, then explain what each symbol stands for. Why do we multiply by 3, and why divide by the square root of the subgroup size?
   Follow-up: If you estimated sigma from all the data pooled together instead of from within subgroups, would your limits get wider or narrower, and what would the chart then miss?

Reading control charts and process overviews

1. P14 Sketch an averages chart with a centre line and control limits where every point is inside the limits, but the last eight points all sit above the centre line. Is that process in control? Explain why that pattern is unlikely if only common-cause noise is at work.
   Follow-up: What would you go and look for at the subgroup where that run began?
2. P15 Sketch three averages charts: one with a single point below the lower limit, one with seven points in a row steadily climbing, and one that just wiggles inside the limits. Which would you flag, and what would you tell the process owner about each?
   Follow-up: Why is a lot of wiggling inside the limits not, on its own, a reason to act?
3. P16 Picture an averages chart where the points zigzag up, down, up, down for fourteen subgroups in a row, all inside the limits. Another chart has two of three points in a row sitting close to the upper limit. Are these signals? Why would a pure-noise process rarely do either?
   Follow-up: What kinds of real causes produce a strict zigzag?
4. P17 Picture a process overview of the length of cut lumber boards (target 2.40 m): boxplots of 20 subgroups over time, plus an overall histogram. The boxes stay centred on the target but get taller and taller from left to right. What would you say about centre, shape and change over time, and which chart would you draw next to test it?
   Follow-up: What would the overall histogram, on its own, have hidden from you?
5. P18 Picture a process overview of delivery times for a food-delivery app, in minutes. The boxes are steady over time, but each has a long upper whisker, and the histogram has a long right tail. Describe the centre and the shape, and say what you would look into next.
   Follow-up: Would you report the mean or the median delivery time to customers, and why?
6. P19 Picture a process overview of paint thickness on car doors (target 120 microns). The first ten subgroups sit around 120 and the last ten all sit a bit higher. What does the overall histogram probably look like, and what would you do about it?
   Follow-up: Why can a histogram of all the data make a process look more variable than it is from moment to moment?

Lifetimes and failure functions

1. P20 A component has a constant failure rate lambda. Write the probability that it has failed by time t, and the probability that it is still working at time t. Sketch both curves on one set of axes and say what the mean time to failure is.
   Follow-up: Why would this model be a poor fit for a car tyre that wears out?
2. P21 Your supplier gives you R(t), the reliability function of a fan motor. How would you get the chance that it has failed by time t, and the density of failures at time t? Sketch all three and say what each one means in plain words.
   Follow-up: Can f(t) be bigger than 1? What does the area under it mean?
3. P22 Explain the hazard function to someone who only knows f(t) and R(t). How is it computed, and why do we call it a conditional rate?
   Follow-up: Sketch the bathtub curve and say which part of a light bulb's life each section describes.
4. P23 Start from R(t) = e^(-lambda t) for an exponential lifetime. Derive F(t), f(t) and the hazard h(t). What do you notice about h(t), and what does that mean for a part that has already run for a year?
   Follow-up: Name one kind of product whose failures fit this model and one that does not.

System reliability

1. P24 Read this case. A food truck's espresso service needs power from either its generator or a shore-power plug, then one grinder, then either of two espresso machines. Describe the series and parallel parts, and explain how you would work out the reliability of 'coffee gets served' without doing any arithmetic.
   Follow-up: Both espresso machines draw water from the same tank. What does that do to your independence assumption?
2. P25 Read this case. An airliner's landing gear can be lowered by the main hydraulic system or by a backup electric motor, or as a last resort by gravity free-fall; the gear then has to lock. Describe the structure, and explain how you would compute the reliability of 'the gear is down and locked', without numbers.
   Follow-up: Where is the weak point, and where would one more backup help least?
3. P26 Read this case. A website runs on two load-balanced servers; each server needs its own disk, and both sit behind one network switch and one internet connection. Describe the series, parallel and nested blocks, and explain how you would compute the reliability of 'the website is up'.
   Follow-up: Both servers sit in one rack on one power strip. How does that change your calculation?
4. P27 Read this case. A home aquarium keeps its fish alive overnight if the water is filtered by either of two filters, heated by one heater, and aerated by an air pump that runs off the wall or a battery backup. Describe the structure, and explain how you would calculate the reliability of 'the fish are fine overnight'.
   Follow-up: Which single component would you duplicate first, and why?

How to practise them with a teammate, plus a tip per topic: the full Practice questions page.