Mathematics Question

Final Exam STAT 4173/5173 Spring 2024 1. Name: The table below gives the yield (in bushels) for 14 equal-sized plots planted in tomatoes for different levels of fertilization. Yield (y) (in bushels) 24 18 31 33 26 30 20 25 25 27 21 29 29 26 Amount of Fertilizer, (x) (in pounds per plot) 12 5 15 17 20 14 6 23 11 13 8 18 22 25 a. Write the simple linear regression model with y (yield) and x (amount of fertilizer). b. Predict yield when the amount of fertilizer is 20. c. ̂1 in context of the problem. Interpret the value of 𝐵 d. Identify 𝑆𝜀 , the sample standard deviation of the regression line. e. Test the hypothesis that there is a linear relationship between yield and fertilizer. Make sure you setup the null and alternate hypothesis, find the p-value and draw conclusions. f. Find a 95% confidence interval for 𝐵1. g. Test the following hypothesis: Ho: 𝐵1 = .5 𝑣𝑒𝑟𝑠𝑢𝑠 Ha: 𝐵1 ≠ .5 h. ̂1? What is the standard error of 𝐵 i. Find the 95% confidence interval for the average yield when fertilizer is 20. Interpret the confidence interval. j. Find the 95% prediction limits on yield when fertilizer is 20. k. Test the assumption that the residual can be modeled by the normal distribution. Make sure you setup the null and alternate hypothesis, report a p-value, and draw conclusions. 2. The data below is from a report on the relationship between aspirin use and heart attacks by the Physicians’ Heath Study Research Group at Harvard Medical School. The Physicians’ Heath Study is a randomized clinical trial testing whether aspirin taken regularly reduces mortality from cardiovascular disease. Every other day, physicians participating in the study took either one aspirin tablet or a placebo. The study was blind – those in the study did not know which they were taking. Myocardial Infraction Placebo Fatal Attack 18 Non-Fatal Attack 171 No Attack 10,845 Aspirin 7 99 10,933 Since we are interested in whether the aspirin reduces mortality, we are going to combine the non-fatal and no heart attack groups into one group. Myocardial Infraction Placebo Fatal Attack 18 Aspirin 7 Non-Fatal and No Attack 11,016 11,032 a. Compute the odds of a fatal attack, given they took the placebo. b. Compute the odds of a fatal attack, given they took an aspirin. c. Calculate the odds ratio. Interpret its meaning. d. Find a 95% confidence interval for the true odds ratio. Interpret the confidence interval. e. Using the confidence interval for OR…Are the results statistically significant? 3. The paper “Serum beta-2-microglobulin (SB2M) in patients with multiple myeloma treated with alpha interferon” (Journal of Medicine (1997) 28:311-318) reports on the influence of alpha interferon administration in the treatment of patients with multiple myeloma (MM). Twenty newly diagnosed patients with MM were entered into the study. The researchers randomly assigned the 20 patients into the two groups. Ten patients were treated with both intermittent melphalan and sumiferon (treatment group), whereas the remaining 10 patients were treated with only the intermittent melphalan (control group). The SB2M levels were measured before and at days 3, 8, 15, and months 1, 3, and 6 from the start of therapy. The measurement of SB2M was performed using a radioimmunoassay method. The measurements before treatment are given here: Treatment Group 2.9, 2.7, 3.9, 2.7, 2.1, 2.6, 2.2, 4.2, 5.0, 0.7 Control Group 3.5, 2.5, 3.8, 8.1, 3.6, 2.2, 5.0, 2.9, 2.3, 2.9 Since this data was taken before the start of therapy, the researches would like there to be no statistical difference between the two groups. If the two groups are statistically different before the start of the therapy then it would be difficult (maybe impossible) to compare the two groups after the therapy. Therefore, we want to test to see if there is any difference between the two groups. a. Select the appropriate null and alternate hypothesis 𝐻𝑜: 𝜇 𝑇𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡 − 𝜇𝐶𝑜𝑛𝑡𝑟𝑜𝑙 ≥ 0 𝐻𝑎: 𝜇 𝑇𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡 − 𝜇𝐶𝑜𝑛𝑡𝑟𝑜𝑙 < 0 𝐻𝑜: 𝜇 𝑇𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡 − 𝜇𝐶𝑜𝑛𝑡𝑟𝑜𝑙 ≤ 0 𝐻𝑎: 𝜇 𝑇𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡 − 𝜇𝐶𝑜𝑛𝑡𝑟𝑜𝑙 > 0 𝐻𝑜: 𝜇 𝑇𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡 − 𝜇𝐶𝑜𝑛𝑡𝑟𝑜𝑙 = 0 𝐻𝑎: 𝜇 𝑇𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡 − 𝜇𝐶𝑜𝑛𝑡𝑟𝑜𝑙 ≠ 0 b. What is the observed t-value for the test? c. What is the p-value for the test (assuming equal variances)? d. What are your conclusions? (Use α = .05) e. Find the 95% confidence interval for the true difference between the treatment group and the control group. Interpret the confidence interval. 4. A researchers record the yields of corn, in bushels per plot, for 4 different varieties of corn, A, B, C, and D. In a controlled greenhouse experiment the researchers randomly assign each variety to 5 of 25 plots available for the study. The researchers want to determine if there is any difference between the 4 varieties of corn. The yields are listed here: A 2.5, 3.6, 2.8, 2.7, 3.1 B 3.6, 3.9, 4.1, 4.3, 2.9 C 4.3, 4.4, 4.5, 4.1, 3.5 D 2.8, 2.9, 3.1, 2.4, 3.2 a. Write the model (use the notation in 8.3) and define all terms in the model. b. When you set this problem up in SPSS you need to define a variable, let’s call it Variety, which contains the 4 varieties of corn. Set up the null and alternate hypothesis testing the Variety effect. c. Find the p-value and draw conclusions. d. Use a multiple comparison method and determine which varieties of corn are different. (Use 𝛼 = .05 on all test.) e. Test the equal variance assumption. Make sure you set up the null and alternate hypothesis, report the p-value and draw conclusions. f. Test the normal distribution assumption. Make sure you setup the null and alternate hypothesis, report the p-value and draw conclusions.