Fall 2021 MEES:698B Take HomeMidterm 1 (53 points) Due 11:59 pm Octo- ber 15th, 2021 Please, submit your solutions in a PDF format. Late submission penalty is 25% of the total grade per each 12-hour delay (thus, a submission with a delay of 12-24 h will receive only a half of earned points). This is a take home exam. You can not communicate among yourselves (or others) like you could with your home works - we do not tolerate cheating (we know none of you will, but we have to say it anyway). For all analyses show your R code (so that you can get partial credit if you get the answer wrong), your results, and a brief interpretation of what it means. We are NOT asking you to articulate the null and alternative hypotheses unless we specifically ask. When asked for, we ARE looking for a thorough analysis using the techniques and analyses of assumptions (unless otherwise stated) that you learned in class to address such questions. As many of you are probably now aware, there are often multiple ways to get at the answer in R - there may not be a singular, correct answer, but make sure that your analysis has a firm, logical basis to adequately address the question. 1. The dataset1 cuttgrowth_3.csv contains information on the condition (a measure of an individual’s physical condition) of cutthroat trout in a stream in southeast Idaho. Within the dataset are 3 groups: GenPop95=a ran- dom sampling of the population in 1995; GenPop96=a random sample of the population in 1996; and mobile95=fish caught moving through traps in 1995 from a random sample. Perform a thorough analysis by answering the questions below concerning conditions among the 3 groups. (22 points total) (a) If we are asked to compare the mean conditions of each group against some standard value (0.2), and we know a priori that the condition of these groups might be better or larger than 0.2 (8 points total). i. What are the null and alternative hypotheses? (2 points) ii. To compare the mean condition of the 1995 sample with 0.2 (6 points total), A. Address test assumptions for the 1995 sample. (2 points) B. State the most appropriate test for the 1995 sample. (2 point) C. Show test results and interpret findings for the 1995 sample. (2 point) iii. To compare the mean condition of the fish caught moving through traps in 1995 with 0.2 (6 points total), A. Address test assumptions for the fish caught moving through traps in 1995. (2 point) B. State the most appropriate test for the fish caught moving through traps in 1995. (2 point) C. Show test results and interpret findings for the fish caught moving through traps in 1995. (2 point) (b) Is there a difference in mean conditions between the 1995 (GenPop95) and 1996 samples? (8 points total) i. State the null and alternative hypotheses. (1 point) ii. Address test assumptions. (2 points) iii. What form of analysis is the most appropriate ? (2 points) iv. Show test results and interpret findings in light of hypotheses (3 points) 2. Regression analysis ( 15 points total) Use the same data set cuttgrowth_3.csv to run a regression analysis for GenPop96. (Subset the entire data set to include only observations for GenPop96) (a) Use pairwise scatter plots and correlations to decide whether to include length and/or weight in the re- gression model. (2 points) (b) Estimate a regression model relating condition to the variable(s) you chose. Provide residuals diagnostics for the model: check the two assumptions about residuals’ stable (homogeneous) variance, and normality. Suggest some approaches for how might you deal with violations, if any. (5 points) (c) Apply F-test for the global utility of the model. State the hypotheses and interpret the results. (2 points) (d) Calculate 95% confidence intervals for the mean condition for (2 intervals total; 4 points) i) a fish of average length and average weight of the GenPop96 population (although such a combination might not be possible or typical in nature); ii) a fish of length 100 and weight 100. (e) Write down the equation (formula) of the estimated model as you would present it in a publication; use the actual estimates (i.e., numbers) in place of the regression coefficients. (2 points) 3. Count the objects (16 points total) In the following figure, the locations of objects (e.g., tree, mines, dwellings) in a study region are given by the solid dots. The goal is to estimate the number of objects in the study region. 1Data set re-used with permission from Dr. Robert Hilderbrand from AL at UMCES 1 (a) A simple random sample of size 20 units has been selected from the N= 400 units in the population. Units selected are indicated by shading in the following figure. The sample data of counts per unit is listed below. 0,1,0,1,0,0,2,0,0,1,0,0,2,1,1,0,1,0,1,0 Use the sample to estimate the number of objects in the figure. Provide a 95% confidence interval using the formula of plus and minus twice the standard errors. (4 point) (b) Use the sampled data from the question 3a, but estimate the total area that was occupied by the objects. Assume that each unit has an area of 1. Provide a 95% confidence interval using the formula of plus and minus twice the standard errors. (4 point) (c) A stratified random sample without replacement of n = 20 units has been selected from the N = 400 units in the population. n1 = 10 units selected from the stratum with N1 = 200 units are indicated by red shading; n2 = 10 units selected from the stratum with N2 = 200 units are indicated by green shading.The sampled data are listed below. stratum 1 1,0,1,1,0,0,1,1,0,0 stratum 2 0,0,1,0,0,0,1,1,1,0 Use the stratified sample data to estimate the number of objects in the figure. Provide a 95% confidence interval using the formula of plus and minus twice the standard errors. (4 point) (d) You are interested in identifying how many units need to be sampled (using a simple random sample) in order to estimate the total of units (τ) with margin of error δ = 50 and 95% confidence. That is≈ τˆ± 2× δ. How large a sample size (n) should you take to estimate τ the total? Use data provided herein to derive a prior guess of σ2 the variance of object counts among pixels. (4 point) 2