Mathematics Question

STAT 5173/4173 Exam I Name: 1. An investigator predicts that individuals that fit the Type A Behavior Pattern (highly competitive and time conscious) will have higher scores on a questionnaire measure of need for achievement than individuals that fit the Type B Behavior pattern (absence of Type A qualities). The investigator collects need for achievement scores from 10 Type A subjects and 10 Type B subjects. Higher scores reflect greater levels of need for achievement. Select the appropriate null and alternate hypothesis. Type A Type B ——— —————11 8 9 10 8 7 9 7 13 8 10 8 9 6 12 12 11 8 8 10 A. Select the appropriate null and alternate hypothesis a. 𝐻0 : 𝜇𝐴 − 𝜇𝐵 ≥ 0 𝐻𝑎 : 𝜇𝐴 − 𝜇𝐵 < 0 b. 𝐻0 : 𝜇𝐴 − 𝜇𝐵 ≤ 0 𝐻𝑎 : 𝜇𝐴 − 𝜇𝐵 > 0 c 𝐻0 : 𝜇𝐴 − 𝜇𝐵 = 0 𝐻𝑎 : 𝜇𝐴 − 𝜇𝐵 ≠ 0 B. Test the hypothesis you selected in part (A) and draw conclusions. C. Use Levene’s test to test the equal variance assumption. Make sure you setup the null and alternate hypothesis, report a p-value and draw conclusions. D. Test the normal distribution assumption. Make sure you setup the null and alternate hypothesis, report a p-value and draw conclusions. 2. Let us assume that an induced bioremediation process is being conducted at a contaminated site. The researcher has obtained good cleanup rates by injecting a mixture of nutrients into the soil in order to maintain an abundant microbial community. Someone suggests using a cheaper mixture. The researcher tries one patch of land with the new mixture, and compares the degradation rates to those obtained from a patch treated with the expensive one to see if he can get the same degradation rates. The researcher is interested in if there is any difference between the two mixtures. The data for this problem are shown in the following table: Data Cheap Nutrient Expensive Nutrient 7.1 7.6 6.8 7.1 7.6 7.0 7.0 7.8 5.6 7.2 6.6 7.9 6.5 7.4 6.7 8.8 6.6 7.9 6.5 8.1 A. Select the appropriate null and alternate hypothesis a. 𝐻0 : 𝜇𝐶ℎ𝑒𝑎𝑝 − 𝜇𝐸𝑥𝑝𝑒𝑛𝑠𝑖𝑣𝑒 ≥ 0 𝐻𝑎 : 𝜇𝐶ℎ𝑒𝑎𝑝 − 𝜇𝐸𝑥𝑝𝑒𝑛𝑠𝑖𝑣𝑒 < 0 b. 𝐻0 : 𝜇𝐶ℎ𝑒𝑎𝑝 − 𝜇𝐸𝑥𝑝𝑒𝑛𝑠𝑖𝑣𝑒 ≤ 0 𝐻𝑎 : 𝜇𝐶ℎ𝑒𝑎𝑝 − 𝜇𝐸𝑥𝑝𝑒𝑛𝑠𝑖𝑣𝑒 > 0 c. 𝐻0 : 𝜇𝐶ℎ𝑒𝑎𝑝 − 𝜇𝐸𝑥𝑝𝑒𝑛𝑠𝑖𝑣𝑒 = 0 𝐻𝑎 : 𝜇𝐶ℎ𝑒𝑎𝑝 − 𝜇𝐸𝑥𝑝𝑒𝑛𝑠𝑖𝑣𝑒 ≠ 0 B. Test the hypothesis you selected in part (A) and draw conclusions. C. Use Levene’s test to test the equal variance assumption. Make sure you setup the null and alternate hypothesis, report a p-value and draw conclusions. D. Test the normal distribution assumption. Make sure you setup the null and alternate hypothesis, report a p-value and draw conclusions. 3. A researcher predicts that students will learn most effectively with a constant background sound, as opposed to an unpredictable sound or no sound at all. She randomly divides twenty-four students into three groups of eight. All students study a passage of text for 30 minutes. Those in group 1 study with background sound at a constant volume in the background. Those in group 2 study with noise that changes volume periodically. Those in group 3 study with no sound at all. After studying, all students take a 10 point multiple choice test over the material. Their scores follow: Group test scores 1) constant sound 74686629 2) random sound 55345723 3) no sound 24712155 a. Write the model (this is the new notation in 8.3) and define all terms in the model. b. Then set up the null and alternate hypothesis testing the Group effect. Make sure you use the new notation when setting up the null and alternate hypothesis. c. Find the p-value for the hypothesis test in part (b) and draw conclusions. d. Use a multiple comparison method and determine which groups are different. (Use 𝛼 = .05 on all test.) e. Use Levene’s test to test the equality of variance assumption. Make sure you set up the null and alternate hypothesis, and report the p-value in your conclusion. Use α = .05. f. With the residuals use Shapiro-Wilk’s test to test the normal distribution assumption. Make sure you set up the null and alternate hypothesis, and report the p-value in your conclusion. Use α = .05.

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