1) In March 2022, the Institut national de sante publique du Quebec (INSPQ) released the following standardized mortality rates per 100,000 for COVID-19 related deaths by sex in the province of Quebec during the first five waves of the pandemic. Waves Gender 1 2 3 4 5 Men 69 64 8 5 36 Women 55 5 3 A chi-square test of independence was performed on these data to determine whether men and women were hit in the same proportions across all waves. (a) How many degrees of freedom does the chi-square statistic have? A) 1 B) 4 C) 5 D) 9 E) 10 (b) What is the conclusion of the chi-square test at the 10% level? A) Ho is rejected. B) Ho is not rejected. (c) What is the p-value of the test? 3 MARKS 2) In a paper published in 1905 in The Journal of Physiology (vol. 32, pp. 501-510), the famous Scottish pharmacologist and physiologist Arthur Cushny (1866-1926) and an instructor at the University of Michigan, Alvin Peebles (1884-1917), compared various tropane alkaloids as hypnotics. Ten patients with chronic sleep loss were each tested on four treatments, namely a control and three drugs: duboisine and a left or right isomer of hyoscine (also known as Devil's Breath). The table below gives the average hours of sleep for each patient with no drug, and their amount of sleep averaged over the three drugs. Patients Condition 1 2 3 4 5 6 7 8 9 10 Drug 1.97 3.20 5.00 4.90 6.30 7.50 7.47 4.10 4.20 5.87 Control 0.60 1.10 2.50 2.80 2.90 3.00 3.20 4.70 5.50 6Nonparametric tests were conducted in order to determine whether the patients are sleeping longer on average under the "drug" condition than under the "no drug" condition. (a) Does the sign test reject the null hypothesis at the 1% level? TRUE O FALSE O (b) What is the expected value of Wilcoxon's signed rank statistic under Ho? (c) What is the p-value of the signed rank test in this case? (d) Which of the sign test and the signed rank test is more likely to detect a difference between the two groups, given that a real difference exists? Justify your answer. 4 MARKS 3) Gunning's Fog Index is a readability test for English writing. The index estimates the years of formal education a person needs to understand the text on the first reading. For instance, a fog index of 12 requires the reading level of a United States high school senior (around 18 years old). The test was developed in 1952 by Robert Gunning, an American businessman who had been involved in newspaper and textbook publishing. In 1981, the Journal of Advertising Research (vol. 21, no. 5, pp. 45-51) published a study of the readability of magazine advertisements in which Kelly Shuptrine and Daniel McVicker computed Gunning's Fog Index for a random sample of six advertisements from Scientific American and Newsweek. Their results are summarized in the table below. Scientific American 15.75 11.55 8.20 9.23 11.16 9.92 Newsweek 9.66 7.67 10.21 5.12 3.12 4.88 Conduct a suitable nonparametric test to see if there is any difference in Gunning's Fog Index between Scientific American and Newsweek, and answer the following questions.(a) Which test is most appropriate in this case? (b) What is the value of the test statistic? (c) Should the test be one-sided or two-sided? Justify your answer. (d) What is the p-value of the test? (e) What conclusion can be drawn? 5 MARKS 4) Newton's universal law of gravitation states that F = Gmim2/r, where F is the gravitational force between points with masses m, and my separated by distance r, and G is the gravitational constant. The experimental data validation of this theory has a long history. In the 1920s, the famous American physicist and inventor Paul R. Heyl (1872- 1961) contributed to its solution through time-of-swing" measurements in which source masses modulate the oscillation frequency of a torsion pendulum. His experiments were conducted with small masses of glass, gold, and platinum. The table below reports the values of r for Heyl's actual measurements of G = (6600 + x) x 10-14 m kg-1sec-2. Substance Measurements Glass 78 71 75 72 74 Gold 83 81 76 78 79 Platinum 61 61 67 67 64 (a) Which of the following tests is most suitable to determine whether these data contain any evidence of a significant difference between the substances? Friedman test O Kruskal-Wallis test (b) What is the p-value of the test? (c) What conclusion can be drawn? 5 MARKS6) The November 2007 issue of the American Journal of Critical Care (vol. 16, pp. 575- 585) carried a paper by Kathie Cole, Anna Gawlinski, Neil Steers, and Jenny Kotlerman reporting on a clinical trial to determine whether a 12-minute hospital visit with a therapy dog could improve hemodynamic measures, lower neurohormone levels, and decrease state anxiety in patients with advanced heart failure. To this end, 76 heart failure patients were randomly assigned to one of three groups. Patients in Group T were visited by a volunteer with a trained dog, those in Group V were visited by a volunteer only, and those in Group C were not visited at all. The anxiety level of each patient was measured (in points) both before and after the visits. The accompanying table gives summary statistics for the drop in anxiety level for patients in the three groups. Sample Size Mean Drop Standard Deviation Group T: Volunteer + Trained Dog 26 10.5 7.6 Group V: Volunteer Only 25 3.9 7.5 Group C: Control Group (No Visit) 25 1.4 7.5 An analysis of variance (ANOVA) of the data was used to compare the mean drops in anxiety level corresponding to these three treatments. In what follows, SST is an acronym for the "sum of squares for treatments" and SSE refers to the "sum of squares for error." (a) What is the value of SST for the ANOVA? (b) What is the value of SSE for the ANOVA? (c) What is the value of the F ratio? (d) What is the p-value of the F test? (e) Is there sufficient evidence to conclude, at the 1% level, that there are differences among the mean drops in anxiety levels for the patients in the three groups? YES O NO O (f) List the ANOVA assumptions and comment on their validity. How might this affect the results of the study? 8 MARKS