April 3, 2003Statistics 512 Midterm 2 Spring 2003Statistics 512Midterm 2April 3, 2003The following rules apply.1. You may bring with 3 sheets of paper, double-sided with any information you need.2. You may use a calculator.3. You may not collaborate or copy.4. Failure to comply with item 3 could lead to reduction in your grade, ordisciplinary action.I have read the rules above and agree to comply with them.Signature ________________________________________________Name (printed) ___________________________________________problem your score total points1 23 2 18Total 40 (1)1Statistics 512 Midterm 2 Spring 20031. An investigator is interested in determining the effects of %protein and %calcium in cow food on milk production (measured in kg/day). 60 cows were used - 10 assigned at random to each of 6 “treatments”. The treatments are all combinations of 3 levels of protein (low, medium, high) treatment P and two levels of calcium (medium, high) treatment C.a. Write an ANOVA model for this experiment that includes interaction. Be sure to define all of your terms, including any constraints or distribution assumptions.2Statistics 512 Midterm 2 Spring 2003b. The 2 “interaction plots” for this experiment are below and on p. 4.. Do you expect statistically significant interaction between %protein and %calcium?Answer: Plot of yield*calcium. Symbol is value of protein. yield ‚ 20 ˆ 3 ‚ ‚ ‚ ‚ ‚ ‚ 18 ˆ ‚ ‚ ‚ ‚ ‚ ‚ 16 ˆ ‚ ‚ ‚ ‚ 1 ‚ ‚ 2 14 ˆ ‚ ‚ ‚ ‚ ‚ 3 ‚ 12 ˆ ‚ ‚ ‚ ‚ ‚ ‚ 2 10 ˆ ‚ ‚ 1 ‚ Šƒƒˆƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒˆƒƒ 1 2 calcium3Statistics 512 Midterm 2 Spring 2003 Plot of yield*protein. Symbol is value of calcium.yield ‚ 20 ˆ 2 ‚ ‚ ‚ ‚ ‚ ‚ 18 ˆ ‚ ‚ ‚ ‚ ‚ ‚ 16 ˆ ‚ ‚ ‚ ‚2 ‚ ‚ 1 14 ˆ ‚ ‚ ‚ ‚ ‚ 1 ‚ 12 ˆ ‚ ‚ ‚ ‚ ‚ ‚ 2 10 ˆ ‚ ‚1 ‚ ‚ ‚ ‚ 8 ˆ Šˆƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒˆƒ 1 2 3 protein4Statistics 512 Midterm 2 Spring 2003c. How would you test whether there is significant interaction between %protein and%calcium? Using the model that you set up in part (a) state your null and alternative hypotheses, the test statistic and the Null distribution for the test statistic, including the degrees of freedom for the test.Hypotheses:Test Statistic:Null distribution:Conclusion if you reject:5Statistics 512 Midterm 2 Spring 2003d. Assuming that the interaction between %protein and %calcium is statistically significant, explain how you would test whether or not the effect of %protein is linear.(You may use SAS MODEL statements or statistical models if that helps your explanation.) Your explanation should include a test of lack of fit and a test of linearity. As well, you should include the degrees of freedom for the tests.6Statistics 512 Midterm 2 Spring 2003e. The brand of cow food usually used for “in-barn” feeding would be considered “medium calcium, medium protein”, whereas allowing the cows to graze in the field would be considered “medium calcium, low protein”. Using the notation from the model set up in part (a), set up a contrast to determine if there is a significant effect of “in-barn” versus “in field” feeding. Your contrast should be expressed in terms of the model parameters, not the data. You do not need to show how to compute standard errors.7Statistics 512 Midterm 2 Spring 2003f. The hormone bovine somatotropin (bST) is sometimes injected into the cows to increase milk production. However, there is some evidence that only cows on high protein food increase production when bST is used. The investigator decides to add a3rd treatment bST (none, injected) to the experiment, and so randomly assigned 5 cows to each level of protein (P), calcium (C) and bST (B). There are two hypotheses:(i) The effect of bST on yield is not affected by the level of calcium.(ii) The effect of bST on yield is affected by the level of protein.Remembering that the Null hypothesis is always “no effect” set up the null and alternative hypotheses for the investigator and state which effect in the model should be tested (i.e. which of the 7 effects P, C, B, PC, PB, CB, PCB). You may state your hypotheses in terms of these effects instead of the parameters (which would be very tedious.)Null and Alternative Hypotheses for : effect of bST on yield is not affected by the level of calcium.Null and Alternative Hypothesis for : effect of bST on yield is affected by the level ofprotein.8Statistics 512 Midterm 2 Spring 20032. The cost of trimming trees growing close to above-ground electrical wires can be large.An investigator is interested in trying out 3 growth inhibitors, which appear to reduce the amount of growth of maple trees. In an experiment, 20 trees of about the same size were selected. In March, 5 trees at random were injected with each of the growth inhibitors, and the remaining 5 trees were used as a control. Although the trees were about the