This exam is open book, open notes; you may use a calculator. Show your work to receive full credit. Use the back if more space is needed. Each question (or part) is worth 10 points.
1. Describe an appropriate experimental design for each of the following situations. Be sure to tell me about blocks, factors, nesting, crossing, fixed, random, etc.
a) A crayon company wants to study ways to decrease breakage of their product. They have two ``recipes'' for the crayon mixture, two different molds, and two different cooling regimes. They can use four batches of each recipe. The molds and cooling regimes are easy to change, but changing the recipes requires cleaning the apparatus.
b) A political scientist wishes to study how polling methods affect results. Two candidates (A and B) are seeking endorsement at their party convention. A random samples of 3600 voters has been taken and divided at random into 9 sets of 400. All voters will be asked if they support candidate A. However, before the question is asked, they will either be told a) that the poll is funded by candidate A, b) that the poll if funded by candidate B, or c) nothing. Due to logistical constraints, all voters in a given set (of 400) must be given the same information; the response for a set of 400 is the number supporting candidate A.
c) A garden club wishes to compare two nontraditional fungicides to a traditional fungicide and to ``no treatment''. Forty-eight rose bushes will be planted in a 12 by 4 arrangement. Four volunteers will plant and care for the bushes (each being responsible for a column of 12 specific bushes). There is also a soil moisture gradient, with the low numbered rows (1-4) being wetter than than high numbered rows (9-12), and we expect more fungus in the moister soils.
d) Open top chambers are used to provide growth environments for plants somewhat intermediate between real field conditions and greenhouse conditions. A series of air pumps, filters, sprayers, etc. is used to modify the environment. We wish to study the effects of ozone (2 levels of added ozone), sulfur dioxide (2 levels of added sulfur dioxide), amount of additional ``artificial rain'' (two levels), and pH of additional ``artificial rain'' on the productivity of soybeans. However, growth chambers are expensive, and we can only use eight of them.
e) We wish to study four foot powders for their ability to prevent athlete's foot. Sixty junior high boys will be the subjects. We expect huge subject to subject variation in the incidence of athlete's foot.
f) Scientists wish to understand how several factors influence (refrigerated) shelf life of yogurt. The factors are amount of sugar (two levels), culture type (two levels), type of fruit (blueberry or strawberry), and pH (two levels). They may produce 32 batches of yogurt, but the cooler only has enough room to hold 8 bathes at a time.
2. Describe the experimental design used in each of the following situations and give a skeleton ANOVA (sources and df only). Be sure to tell me about blocks, factors, nesting, crossing, fixed, random, etc.
a) We wish to test the efficacy of dental sealants for reducing tooth decay on molars in children. There are 5 treatments (sealants A or B applied at either 6 or 8 years of age, and a control of no sealant). We have 40 children and the 5 treatments are assigned at random to the 40 children. As a response, we measure the number of cavities on the molars by age 10. In addition, we measure the number of cavities on the nonmolar teeth (this may be a general measure of quality of brushing or resistance to decay).
b) We wish to study the acidity of orange juice available at our grocery store. We choose two national brands. We then choose 3 days at random (from the next month) for each brand; cartons of brand A will be purchased only on the days for brand A, and similarly for brand B. On a purchase day for brand A, we choose 5 cartons of brand A orange juice at random from the shelf, and similarly for brand B. Each carton is sampled twice and the samples are measured for acidity.
c) A national travel agency is considering new computer hardware and software. There are two hardware setups and three competing software setups. All three software setups will run on both hardware setups, but the different setups have different strengths and weaknesses. Twenty branches of the agency are chosen to take part in an experiment. Ten are high sales volume, 10 are low sales volume. Five of the high sales branches are chosen at random for hardware A, the other 5 get hardware B. The same is done in the low sales branches. All three software setups are tried at each branch. One of the three software systems is randomly assigned to each of the first three weeks of May (this is done separately at each branch). The measured response for each hardware/software combination is a rating score based on the satisfaction of the sales personnel.
d) Wafer board is a manufactured wood product made from wood chips. One potential problem is warping. Consider an experiment where we compare 3 kinds of glue and two curing methods. All six combinations are used 4 times, once for each of 4 different batches of wood chips. The response is the amount of warping.
e) Plant breeders wish to study six varieties of corn. They have 24 plots available, 4 in each of 6 locations. The varieties are assigned to location as follows (there is random assignment of varieties to plot within location)
f) We wish to determine the number of warblers that will respond to three recorded calls. We will get 18 counts, 9 from each of two forest clearings. We expect variation in the counts from early to mid to late morning, and we expect variation in the counts from early to mid to late in the breeding season. Each recorded call is used three times at each clearing, arranged in such a way that each call is used once in each phase of the breeding season and once in each morning hour.
3. A 
fractional factorial was run with aliasing generated by
I = -ACDE = ABCF.
a) Find the factor/level combinations used in this fraction.
b) Find the alias structure for this design.
4. City hall wishes to learn about the rate of parking meter use. They choose 8 downtown blocks at random (these are city blocks, not statistical blocks!), and on each block they choose 5 meters at random. Six weeks are chosen randomly from the year, and the usage (money collected) on each meter is measured every day (Monday through Sunday) for all the meters on those weeks.
a) Write out an appropriate linear model, telling me what is random, fixed, crossed, nested, etc.
b) Give a skeleton ANOVA (sources and df only) for this design.
5. Consider a completely randomized design with 5 treatments, 4 units per treatment, and treatment means
The MSE is 4.012. a) Construct an ANOVA table for this experiment and test the null hypothesis that all treatments have the same mean.
b) Test the null hypothesis that the average response in treatments
1 and 2 is the same as the average response in treatments 3, 4, and 5.