Please check my work (statistics assignment - ANOVA)

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I have attached my previous assignment answers and it's questions and grading criteria (all combined, my answers start on pg. 7), please check my work (combinepdf 2) to that criteria.

Please pay attention to my rounding and explanations (particularly the last page), as I am unsure about them.

Questions are in one document (Problems_2) and my answers are in the last one separate (combinepdf 2).

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GLENDON COLLEGE - DEPARTMENT OF PSYCHOLOGY Introduction to Statistics 2 – GL/PSYC 2531.03 E – Fall 2017 Problem set #1 - distributed Oct 2, 2017 Value: 25% of final grade due: Oct 16, 2017 Answer all questions below. You should do all calculations by hand, because the point is to make sure that you understand the operations and the steps involved. You can use a standard calculator to help with the mechanics of calculating, but not computers. You should calculate so that your answer is correct to 2 decimal places: that is, calculate all intermediate calculations to 3 correct decimal places (4 is even better), then round the final value that you report to 2 places. Grades focus on getting the correct answers, but you must show your calculations. If your answer is wrong, calculations may show how errors occurred and get you partial marks. 1. A nutritionist studying weight gain in college freshmen obtains a sample of n=20 first-year students at the state college. Each student is weighed on the first day of school and again on the last day of the semester. The scores below measure the change in weight, in pounds, for each student. Positive scores indicate weight gain (# pounds) during the semester and negative scores indicate weight loss. 4 pt +5, +6, +3, +1, +8, +5, +4, +4, +3, –1, +2, +7, +1, +5, +8, 0, +4, +6, +5, +3 a. Sketch a histogram showing the distribution of weight-change scores. b. Calculate mean, median, and mode weight-change scores for this sample. c. Does there appear to be a consistent trend in weight change during the semester? Explain their answer d. What shape is the distribution and what does that tell you about weight change? ANS: a. histogram (1 pt) : 1 pt for correct axes, frequencies, layout, labeling : 0.5 pt if H has errors but is “more or less” correct (your judgment) : 0 pt if H has errors so serious that the picture is incorrect . 5 │ │ 4 │ ┌───┐ │ │ │ 3 │ ┌───┬───┤ │ │ │ │ │ │ 2 │ ┌───┐ │ │ │ ├───┐ ┌───┐ f │ │ │ │ │ │ │ │ │ │ 1 │ ┌───┬───┤ ├───┤ │ │ │ ├───┤ │ │ │ │ │ │ │ │ │ │ │ │ │ └─┴───┴───┴───┴───┴───┴───┴───┴───┴───┴───┴─ X 1 0 +1 +2 +3 +4 +5 +6 +7 +8 Weight change (pounds) during the first semester, freshman year 1 1 pt 1.5 pt 0.5 pt 1 pt b. mean - median - mode of weight change (1.5 pt = @ 0.5 pt) (grading: 0.5 pt if both M and calculations correct (calcs must be shown) [0.5 pt OK if minor calculation errors, use your judgment] Mean: ∑x/n = 79/20 = 3.950 pounds (calc) = 3.95 pounds (final ans). In words: on average, these students gained 3.95 pounds during the semester. NB: make sure to check calculated to 3+ decimal places. Here not so critical, other places important. Median: Mdn = 4 (score in the middle = 50:50 above:below) Mode: Md = 5 (most commom score, 5 occurs 4 times; visible in histogram) c. consistent trend in weight change (0.5 pt)? Yes, nearly all students gained weight. d. shape & wt change pattern (1.0 pt) : skewed negatively (0.5 pt) : i.e., longer ‘tail’ on the left = low side wt change pattern (0.5 pt) : small wt change for most, large for few 2. For each of the following, find the exam score X that should give the better grade. In each case, explain your answer. a. X=56 on an exam with μ=50 and σ=4, or X=60 on an exam with μ=50 and σ=20. b. X=40 on an exam with μ=45 and σ=2, or X=60 on an exam with μ=70 and σ=20. c. X=62 on an exam with μ=50 and σ=8, or X=23 on an exam with μ=20 and σ=2. 3 pt ANS: 1 pt @ question a-b-c : 0.5pt (z-calculations) + 0.5 (decision) a. X=56 corresponds to z = 1.50, X=60 corresponds to z = 0.50. SO X=56 better grade b. X=60 corresponds to z = –0.50, X=40 corresponds to z = –2.50. SO X =60 better grade c. X=62 corresponds to z = 1.50, and X=23 also corresponds to z = 1.50. SO the two scores have the same relative position & shld receive the same grade (neither better). 3. A distribution with mean μ = 56 and standard deviation of σ = 20 is transformed into a standardized distribution with μ. = 50 and σ = 10. Find the new, standardized score for each of the following values from the original population. a. X = 46 b. X = 40 c. X = 80 3 pt ANS: 1 pt @ a, b, c. Calculations : z = (X-56)/20¸then standardized X = Xs = z*10+50 a. Xs = 45 (z = –0.50) z = (46-56)/20 = -10/20 = -0.5 Xs = -0.5*10+50 = -5+50 = 45 Xs = -0.8*10+50 = -8+50 = 42 b. Xs = 42 (z = –0.80) z = (40-56)/20 = -16/20 = -0.7 c. Xs = 62 (z = 1.20) z = (80-56)/20 = 24/20 = 1.2 Xs = 1.2*10+50 = 12+50 = 62 4. Over the past 10 years, the local school district has measured physical fitness for all high school freshmen. During that time, the average score on a treadmill endurance task has been μ=19.8 minutes with a standard deviation of σ=7.2 minutes. Assuming 2 3 pt that the distribution is approximately normal, find each of the following probabilities. a. What is the probability of randomly selecting a student with a treadmill time greater than 25 minutes? In symbols, p(X > 25) = ? b. What is the probability of randomly selecting a student with a time greater than 30 minutes? In symbols, p(X > 30) = ? c. If the school required a minimum time of 10 minutes for students to pass the physical education course, what proportion of the freshmen would fail? ANS: @ Q a,b,c = 1 pt : z value for target X-value = 0.5 pt : p (> or < z) + answer the Q = 0.5 pt a. for X=25, z = 0.72, p(z > 0.72) = 0.2358 b. for X=30, z = 1.42, p(z > 1.42) = 0.0778 c. for X=10, z = -1.36, p(z < –1.36) = 0.0869 5. Government critics were concerned that the quality of the public schools in their 12 pt community was poor, so they ran a study to compare school satisfaction of teachers working private versus public schools in their community. The teachers’ school satisfaction ratings they obtained are below (higher rating = greater satisfaction). Teacher A B C D E Public School 2 4 6 8 10 Private School 7 8 10 8 12 a. Assume these ratings represent samples of 5 randomly selected teachers from 5 pt each type of school (i.e., every score is from a different teacher). Use an appropriate statistical test to assess whether public vs. private school satisfaction ratings differ (use α = 0.05). i. State the test you will use and why State your hypotheses (H1 & H0) and critical region ii. iii. Calculate the test statistic iv. Interpret your findings b. Now assume these ratings represent a sample of 5 randomly selected teachers 5 pt and each teacher is tested twice - once before and once after their school was converted from public to private (i.e., pre-post). Use an appropriate statistical test to assess whether public vs. private school satisfaction ratings differ (α = 0.05). i. State the test you will use and why State your hypotheses (H1 & H0) and critical region ii. iii. Calculate the test statistic iv. Interpret your findings c. Compare your findings from these two tests (i.e., what do they allow you to 2 pt conclude. Do they differ or not, and why? Assess whether one test is better than the other and explain your answer. 3 ANSWERS a. Ratings represent samples of 5 randomly selected teachers from each type of 5 pt school (i.e., @ score - different teacher). Use an appropriate statistical test to assess whether public vs. private school satisfaction ratings differ (use α = 0.05). i. State the test you will use and why: Independent samples t-test 0.5 pt Why: each score is from an independent ‘source’ (teacher) 0.5 pt ii. iii. State your hypotheses (H1 & H0) and critical region H1: pub-priv school satisfaction ratings differ (i.e., 2-tailed test) & H0: pub-priv school satisfaction ratings do not differ Critical region: tcrit (0.05, 2-tailed, df=8) = 2.306 (t-table G&W Appendix B2) Calculate the test statistic 0.5 pt 0.5 pt 2.0 pt Calculations below; highlighted key values students should have obtained Teacher A B C D E n sum‐Y sum‐Y2 M SS df Y1 2 4 6 8 10 Public School Y1^2 (Y1‐M1)^2 4 16.000 16 4.000 36 0.000 64 4.000 100 16.000 Private School TOTALS Y2^2 (Y2‐M2)^2 49 4.000 64 1.000 100 1.000 64 1.000 144 9.000 5 10 45 75 421 641 9.000 7.5 16.000 4 8 Y2 7 8 10 8 12 5 30 220 6.000 40.000 4 pooled s2 2 s t 7.000 1.400 ‐1.793 1.400 Grading: means (0.5), both SS (0.5), both s2 (0.5), t value (0.5) iv. Interpret your findings 1.0 pt Cannot reject H0 (stats conclusion: 0.5 pt) can’t confirm pub-priv school satisfaction differs (interpretation: 0.5 pt) b. Now assume these ratings represent a sample of 5 randomly selected teachers 5 pt and each teacher is tested twice - once before and once after their school was converted from public to private (i.e., pre-post). Use an appropriate statistical test to assess whether public vs. private school satisfaction ratings differ (α = 0.05). i. State the test you will use and why: 4 Repeated measures t-test Why: each S rates both public & private school satisfaction so each subjects is tested repeatedly (twice) ii. State your hypotheses (H1 & H0) and critical region H1: pub-priv school satisfaction ratings differ (i.e., 2-tailed test) H0: pub-priv school satisfaction ratings do not differ Critical region: tcrit (0.05, 2-tailed, df=n-1=4) = 2.776 (t-table G&W App’x B2) iii. Calculate the test statistic 0.5 pt 0.5 pt 0.5 pt 0.5 pt 2.0 pt Calculations below; highlighted key values students should have obtained Teacher A B C D E n sum‐Y M SS df Public School Y1 2 4 6 8 10 5 30 6.000 Private School Y2 7 8 10 8 12 5 45 9.000 Diff Formulae D 5 4 4 0 2 (D‐M)^2 4 1 1 9 1 5 5 15 3.000 ** OK +3 or ‐3 16 4 s2 4.00 SS/df s2M 0.80 0.89 3.354 s2 / N sM t sqrt (above) (M‐0)/sM Grading: D-mean (0.5), SS (0.5), sM (0.5), t value (0.5) iv. Interpret your findings Can reject H0 (statistical conclusion: 0.5 pts) Can conclude that pub-private school satisfaction differs (0.5 pt) c. Compare your findings from these two tests (i.e., what do they allow you to conclude). Do they differ or not, and why? Assess whether one test is better than the other and explain your answer. DIFFER: Y/N : Yes, the two tests (IM-RM) differ. RM test : teacher satisfaction clearly differs public-private schools IM test : teacher satisfaction does not clearly differ …. 5 1.0 pt 2 pt 0.5 pt : variability differs (RM lower than IM)  RM higher t  significant 0.5 pt IS ONE TEST BETTER? (several answers OK with suitable reasoning) YES RM better because it is more sensitive Why: able to detect a difference if it is possibly there (impt where consequences are serious if you miss a difference) IM better because it is more cautious/conservative Why: does not identify a difference unless evidence is very strong (impt where consequences are serious if you claim a ‘fake’ diff) : OR not confounded by order effects whereas RM can be NO One isn’t better than the other, because they serve different purposes (per the “why” above) 1.0 pt Why ~ THE END ~ 6 GLENDON COLLEGE - PSYCHOLOGY Introduction to Statistics – GL/PSYC 2531, Fall 2017 Problem set #2 Value: 30% due: Nov. 20, 2017 Answer all questions below, doing all calculations by hand. You can use a standard calculator to help with calculation mechanics but not computers. You must show all your calculations so that it is clear how you got your answers. Calculate so your final answers are correct to 2 decimal places: Numbers used in calculating should be correct to 3-4 decimal places (e.g., OK to write 1.2 if the value is 1.2000; OK to use 0.1667 as 0.167 but not 0.17). Then round the final value to 2 decimal places when you report it, nowhere before that, and not if you are using it in further calculations. Penalty 0.5 pt for rounding when you shouldn’t. SECTION A. For the following questions and data, determine the appropriate statistical test, calculate it, and use your findings to answer the researchers’ questions. (40 pts) 1. Professors tested their students’ claim that drinking coffee while studying improves their exam performance. They randomly selected 36 students from a statistics course, divided them randomly into 3 groups, and gave them coffee while they studied statistics. One group got high-caffeine coffee, one got moderate caffeine, and one got decaf. The professors then tested them on the material studied. Grades (%) on this exam are in the table below. Answer the questions below to test if this study confirms students’ claims. (15 pts) a. b. c. d. e. f. Subject Decaf 1 2 3 4 5 6 68 74 59 61 65 72 ΣX = ΣX2 = SSWG = 815 56009 656.92 Moderate High Subject Caffeine caffeine 68 72 7 80 65 8 64 68 9 65 83 10 69 79 11 79 92 12 865 62891 538.92 922 71622 781.67 Mean s2 s Decaf 80 58 65 60 78 75 67.92 59.72 7.73 Moderate High caffeine caffeine 80 69 63 74 69 78 70 83 83 88 75 71 72.08 48.99 7 76.83 71.06 8.43 What statistical test will you need to conduct? Explain why. (1 pt) State the null and alternative hypotheses. (1 pt) Using α=0.05, what is the tabled critical value to use? Explain how you determined it. (1 pt) Graph the results for the three groups, give an eyeball assessment of findings (2 pt) Calculate the appropriate statistical test. (8 pt) Evaluate the null and alternative hypotheses based on your statistical findings (2 pt) 1 2. Re-analyze Question 1 data, now assuming the professors used a random sample of only 12 students and manipulated each student three times. That is, they gave each student coffee while they studied plus an exam on the material studied three times: once with high-caffeine coffee, once with moderate caffeine coffee, and once with decaf. (The professors counterbalanced the order in which they gave subjects each type of coffee). Answer the questions below to test if this study confirms students’ claims. a. b. c. d. e. f. What statistical test will you need to conduct? Explain why. State the null and alternative hypotheses. Using α=0.05, what is the tabled critical value to use? Explain how you determined it Calculate the appropriate statistical test. Evaluate the null and alternative hypotheses based on your statistical findings Compare your results with those from Q #1, i.e., are they the same or not, and why? (1 pt) (1 pt) (1 pt) (8 pt) (2 pt) (2 pt) 3. Use a two-factor independent measures ANOVA (α=.05) to assess the significance of the effects of drugs, hospitals, and their interaction in the following study. Researchers tested patients suffering from depression in three different hospitals with a three different drugs that may help control depressive symptoms. Patients in each hospital (H1-H3) were divided into three equal groups - one group got Drug 1, one group got Drug 2, one group got Drug 3. Drug 1 Drug 2 Drug 3 H1 H2 H3 H1 H2 H3 H1 H2 H3 10 7 11 1 6 4 3 2 5 8 4 9 2 7 3 2 1 6 7 3 10 1 6 6 3 2 4 9 2 9 4 5 4 3 3 5 1 2 3 4 6 5 These data generated the following summary table Source df SS Hospital 2 145.33 Drug 2 45.24 Interaction 4 93.33 Error Total ?? ?? 1.45 41 331.90 2 a. b. c. d. What test should you use to test the three hypotheses? Explain why (1 pt) State the hypothesis for each of the 3 statistical tests (1.5 pt) Find decision criteria for each test using a = 0.05 (first calculate degrees of freedom) (2 pt) Calculate the statistical tests by completing the summary table (fill empty cells and (3.5 pt) ?? columns where suitable) e State your conclusion for each tests (in statistical and real terms) (3 pt) f. Draw a graph to show scores for both factors then, by eye, interpret the interaction (4 pt) SECTION B. For each question below, decide what statistical procedure is the most appropriate for answering the researchers’ question(s) and explain your choice. You do not have to carry out the statistical test—just figure out which test you should use. (@ 2 pts = 10 pts) 1. Students are greatly concerned about the consistency of marking between lecturers. Lecturers quite often gain reputations as ‘hard’ or ‘light’ markers, but there is often little to substantiate these reputations. So, a group of students tested the consistency of marking by submitting the same essay to four different lecturers. The mark given by each lecturer was recorded for each of the 8 essays (table below shows each lecturer’s marks for each essay). What statistical procedure should students use to test whether lecturers grade differently (hard - light)? Essay 1 2 3 4 5 6 7 8 mean Lecturer Dr. Field 62 63 65 68 69 71 78 75 68.875 Dr. Smith 58 60 61 64 65 67 66 73 64.25 Dr. Scrote 63 68 72 58 54 65 67 75 65.25 Dr. Death 64 65 65 61 59 50 50 45 57.375 Mean Variance 61.75 64.00 65.75 62.75 61.75 63.25 65.25 67.00 6.92 11.33 20.92 18.25 43.58 84.25 132.92 216.00 2. A professor conducted a study to test if where students sat in his classroom affected their grades. For one class of 200 students, he divided the class into four equal-sized groups according to where they normally sat in the classroom (front and center, far left side, far right side, back) and recorded their grades on the final exam. He used the appropriate statistical procedure to test his hypothesis and obtained a significant result: where students sit did affect their grades. He then wanted to estimate how large a grade difference seating location makes (i.e., effect size). What statistical tests should he have used (hypothesis, effect size)? 3. The US Census Bureau publishes information on the ages of married people. Suppose we want to decide whether the mean age of married men differs from the mean age of married women in the USA. Ten (10) married couples in the USA were selected at random; their ages (in years) are shown in the table below. What test could you use to make this decision? 3 Couple 1 2 3 4 5 6 7 8 9 10 Husband 59 21 33 78 70 33 68 32 54 52 Wife 53 22 36 74 64 35 67 28 41 44 Difference (D) 6 -1 -3 4 6 -2 1 4 13 8 4. Red palm oil, due to its high content of vitamin A, is thought to reduce the occurrence and severity of malaria for young children. To assess if this is true, researchers developed a food supplement containing a high dose, a low dose, or no (placebo) red palm oil. They tested the supplement on boys and girls (75 each), in case they differ in their exposure to malaria and response to the red palm oil supplement, and randomly assigned 25 children of each sex to each of the supplement levels. The study design is shown below. What statistical test should they use to test their hypotheses about red palm oil supplements, sex, and malaria? Red palm oil Placebo Low Dose High Dose Total Girls 25 25 25 75 Boys 25 25 25 75 Total 50 50 50 150 5. Neurosurgeons wanted to determine whether a dynamic system (Z-plate) reduced neurosurgery operative time relative to a static system (ALPS plate). They obtained the data shown below on operative times (in minutes), for the two systems. What statistical procedure and test criterion should they use to assess whether the data provide sufficient evidence to conclude that the mean operative time is less with the dynamic than the static system? THE END 4
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