Feb 21, 2006Statistics 512 Midterm 1 Spring 2006Statistics 512Midterm 1Feb 21, 2006The 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. You may not use any device, such as a cell-phone or PDA that allows you to access the internet or other outside information.5. Failure to comply with items 3 or 4 could lead to reduction in your grade, or disciplinary action.I have read the rules above and agree to comply with them.Signature ________________________________________________Name (printed) ___________________________________________problem your score total points1 23Total1Statistics 512 Midterm 1 Spring 20061. The Enzyme-Linked Immunosorbent Assay (ELISA for short) is a biochemical technique used to detect the quantity of a protein in a sample. The measurement is the optical density of the assay.In planning an experiment to determine the sensitivity of ELISA for detecting the protein DNase, samples were prepared with known quantities of DNase.a) From previous experience, the standard deviation of optical density in this type of experiment is known to be about 0.03. The investigators would like to be able to detect a difference in means of 0.04 with 80% power when testing at the 10% level. Assuming that the data are normally distributed and that they will use a 2-sample t-test, what sample size will be required?b) Under the assumptions of part a and with the sample size you estimated in part a, what will be the approximate width of a 90% confidence interval for the difference in means?2Statistics 512 Midterm 1 Spring 2006The investigators decided to use 5 different known concentrations of DNase. The plan is to do n replicates using one batch of reagent for each where n was determined from the previous sample size computation. For each replicate, 1 sample will be prepared independently at each of the 5 concentrations. Each sample will be measured twice, so that 10n measurements will be taken in all.c) Identify the factors, factor levels, blocks, experimental units and sampling units in this study.Factors:Factor Levels:Blocks (if any)Experimental UnitsSampling Unitsd) Is this an experimental, observational or mixed study? Briefly explain your answer.3Statistics 512 Midterm 1 Spring 2006e) What 3 assumptions about the response are required for analyzing this experiment using the one-way ANOVA model. f) Do you think that it would be appropriate to analyze this experiment using one-way ANOVA with 5 treatments and 10n observations? Briefly justify your answer.4Statistics 512 Midterm 1 Spring 20062. In planning an experiment to determine the sensitivity of ELISA for detecting the protein DNase, the experimenters decided to prepare samples with 8 known concentrations of DNase, and 22 replicates at each concentration. Each sample was prepared and measured independently (with reagents prepared as needed for each sample). After speaking with the Statistical Consulting Center, the investigators felt that the one-way ANOVA model would be appropriate for the analysis of their data.a) Write the one-way Normal theory ANOVA model for these data using the factor effects model. Be sure to define all of your notation. Include the ranges of the subscripts, the constraints and any distribution assumptions.5Statistics 512 Midterm 1 Spring 2006b) Below are a boxplot and a scatterplot of optical density versus concentration.0 . 0 4 8 8 2 8 1 2 0 . 1 9 5 3 1 2 5 0 . 3 9 0 6 2 5 0 . 7 8 1 2 5 1 . 5 6 2 5 3 . 1 2 5 6 . 2 5 1 2 . 500 . 51 . 01 . 52 . 02 . 5optdensc o n cr u n 1 2 3 4 5 67 8 9 1 0 1 1o p t d e n s0 . 00 . 10 . 20 . 30 . 40 . 50 . 60 . 70 . 80 . 91 . 01 . 11 . 21 . 31 . 41 . 51 . 61 . 71 . 81 . 92 . 02 . 1c o n c0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3In both plots, the relationship between concentration and optical density appears to be curved, but the curvature is different in the 2 plots. Why is this?c) What ANOVA assumptions appear to be violated by the data? Briefly explain your answer.6Statistics 512 Midterm 1 Spring 2006Level of concentration -------optdens------- N Mean0.04882812 22 0.053318180.1953125 22 0.150954550.390625 22 0.239727270.78125 22 0.406772731.5625 22 0.666318183.125 22 1.037727276.25 22 1.4285909112.5 22 1.76986364d) Above is a table of means of optical density for each level of concentration.Suppose that the mean square error of the ANOVA is used to estimate 95% prediction intervals for the predicted observed optical density at concentrations 1.5625 and 12.5. Which interval would you expectto be more accurate in predicting the range of 95% of the observations, or would they be equally good?Briefly explain your answer.e) After fitting the ANOVA model, what is your estimate of the overall mean optical density?7Statistics 512 Midterm 1 Spring 2006f) Much of the ANOVA table is below. Fill in the missing parts of the table.Sum ofSource DF Squares MeanSquareFValue Pr>FModel <.0001Error 0.00213957XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXCorrectedTotalXXXXXXXXXXXXXXXXXXXXg) The statistical consultant noted that the F-test is robust to the violations of the ANOVA assumptions observed here (although the prediction intervals are not). Assuming this is true, test the hypothesis thatall means are equal, filling in the blanks below. H0:_______________________________ HA:_______________________d.f. for test ______________________________p value ________________________________Conclusion: ______________________________________________8Statistics 512 Midterm 1 Spring 2006h) The investigators expected the relationship between mean optical density and concentration to be quadratic. They fitted a quadratic regression in concentration. The plot of studentized residuals versusconcentration is below. Do you think a quadratic regression is a good fit to these data? Briefly justify your answer.r- 3- 2- 101234c o n c0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3i) The quadratic regression is highly significant (p<.001), R2 = 0.72. Is this evidence that a quadratic regression is a good fit to these data? Briefly justify your answer.9Statistics 512 Midterm 1 Spring 2006j) Do a test of the fit (lack of fit) of the quadratic regression to the data. (You may assume that this F-test is also robust to the type of violation of assumptions in this example.)(Hint: You DO have enough information to compute the test statistic. If you are stuck, consider the