Feb. 20, 2003Statistics 512 Midterm 1 Spring 2003Statistics 512Midterm 1Feb. 20, 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 points12Total 401Statistics 512 Midterm 1 Spring 20031. In a study of the effects of ozone exposure on wheat, plants were grown in climate chambers. Wheat sprouts were placed in pots the chambers and then ozone was added to the air in the chamber. After 3 weeks, the plants were removed and the dry weight of healthy above-ground tissue (DW) was recorded.4 levels of ozone were used: 20 ppm, 50 ppb., 80 ppb. 100 ppb and 120 ppb.The investigators plan to do a balanced experiment.a. Current environmental regulations pay special attention to 80 ppm and 120 ppm for regulatory purposes. So the investigators are particularly interested in estimating the contrastL1 =DW80 – DW120Previous experience with this type of wheat at 50 ppm ozone suggests that the variance ofDW should be about 25 gm2. What sample size (for each treatment) will be required so that the width of a 95% confidence interval for this contrast will be less than 4 gm.? You may assume that the data are normally distributed with constant variance.2Statistics 512 Midterm 1 Spring 2003b. Due to limitations on space in the climate chambers, the investigators were able to run only 4 samples at each level of ozone. The observed mean squared error for the experiment was 30 gm2. As well, one of the samples at 120 ppm ozone was infested witha fungal disease, as a result of which the investigators decided discard the data (i.e. the sample size for 120 ppm was 3). Fill in the missing entries in the ANOVA Table below:Source degrees of freedomozone level ____________error ____________total ____________3Statistics 512 Midterm 1 Spring 2003c. The observed summaries are listed below. The MSE from the ANOVA table is 30.80 ppm 120 ppmsample size 4 3sample mean 84 gm 74 gmsample s.d. 7 gm 5 gmTest whether there is a statistically significant difference between DW at 80 ppm and 120 ppm. Null and Alternative HypothesesTest Statistic FormulaComputed value of Test Statisticp-valueConclusion (as a statement)4Statistics 512 Midterm 1 Spring 2003d. What is the smallest difference in DW80 and DW120 that the investigators can detect at =0.05 given the observed sample sizes and variance estimates? (i.e. What is the smallest estimate of the difference in means that will be declared statistically significant?)e. The investigators were also interested in the contrastL2 = DW50 – DW2020 ppm 50 ppmsample size 4 4sample mean 117 gm 112 gmsample s.d. 3.5 gm 5 gmIs L2 orthogonal to L1? Your answer should consider orthogonality of both the contrasts and their estimates. Justify for your response.5Statistics 512 Midterm 1 Spring 20032. In a (rather old) study of fuel efficiency, the miles per gallon (MPG) of makes of cars was considered as a function of the number of cylinders in the vehicle engine. Cars may have 4, 6, 8 or occasionally 5 cylinders, so it is convenient to treat “number of cylinders” as a factor in the experiment.One of the questions of interest is whether MPG decreases linearly with the number of cylinders.a. Write a factor effects model for MPG for this study. Define all of the terms.b. Write a cubic polynomial regression model for MPG with predictor CY (cylinders) for this study. Define all terms.c. Consider E(MPGij) = i. Express i in terms of the parameters of the two models in a. and b.6Statistics 512 Midterm 1 Spring 2003d. Below is a boxplot of the MPG as a function of number of cylinders. Point out 2 features of the plot that are of interest for the statistical analysis, such as the pattern of themeans, the variances, outliers, etcetera.4 5 6 81 52 02 53 03 54 0mpgc y l i n d e r7Statistics 512 Midterm 1 Spring 2003The last 2 pages of the exam include computer output for the following problems. You may assume that any relevant residual and normal probability plots appear to be fine.d. Is there a statistically significant effect of number of cylinders on MPG? Justify your answer with an appropriate statistical test.Null and Alternative Hypothesis:Formula for Test StatisticValue of Test Statistic (if this is on the computer output, you do not need to recompute it)P-value of Test StatisticConclusions written in a sentence.8Statistics 512 Midterm 1 Spring 2003e. Compute a 95% confidence interval for the mean MPG of 8 cylinder vehicles.FormulaComputed Interval9Statistics 512 Midterm 1 Spring 2003f. The investigator thinks the regression of MPG on the number of cylinders should be linear. Is there any evidence of lack of fit of this model? Justify your answer with one ormore statistical tests.Null and alternative hypothesesFormula for Test Statistic(s) Value of Test Statistic(s) (if this is on the computer output, you do not need to recompute it)P-value(s) Conclusions written in a sentence.10Statistics 512 Midterm 1 Spring 2003car data The GLM Procedure Class Level InformationClass Levels Valuescylinder 4 4 5 6 8Number of observations 40Dependent Variable: mpg Sum ofSource DF Squares Mean Square F Value Pr > FModel 3 1114.764504 371.588168 27.69 <.0001Error 36 483.129246 13.420257Corrected Total 39 1597.893750R-Square Coeff Var Root MSE mpg Mean0.697646 14.85400 3.663367 24.66250Source DF Type I SS Mean Square F Value Pr > Fcylinder 3 1114.764504 371.588168 27.69 <.0001Source DF Type III SS Mean Square F Value Pr > Fcylinder 3 1114.764504 371.588168 27.69 <.0001Level of -------------mpg-------------cylinder N Mean Std Dev4 19 30.0210526 4.182447315 3 21.9666667 2.081666006 10 21.0800000 4.077526488