MATH 1320 FINAL PROJECT To practice statistical analysis on real data 1. To practice using R /R-studio 2. To practice interpreting R output and drawing conclusions 3. To practice writing report Description: This project provides two data sets for you to analyze. Please first read the background information of the data sets and find out the questions that researchers wanted to answer in these two cases. Then carry out a thorough statistical analysis on these data sets and write a report to answer these questions. The report is expected to have the following components: Exploratory data analysis Clear statement of statistical questions you are going to answer Complete R codes for the analysis work Interpreting R output Summary of statistical findingsCASE I: Bumpus's Data on Natural Selection In an 1898 biology lecture at Woods Hole, Massachusetts, Hermon Bumpus reminded his audience that the process of natural selection for evolutionary change was an unproved theory: "Even if the theory of natural selection were as firmly established as Newton's theory of attraction of gravity, scientific method would still require frequent examination of its claims." As evidence in support of natural selection, he presented measurements on house sparrows brought to the Anatomical Laboratory of Brown University after an uncommonly severe winter storm. Some of these birds had survived and some had perished. Bumpus asked whether that those perished did so because they lacked physical characteristics enabling them to withstand the intensity of that particular instance of selective elimination. Table 1 exhibits the humerus (arm bone) lengths for the 24 adult male sparrows that are perished and for the 35 adult males that survived. Do humerus lengths tend to be different for survivors than for those that perished? If so, how large is the difference? Perished Survived 9 65 66 67 9 68 7 69 932 70 39 3 71 5 96600 72 13368889 988761 73 0033569 543 74 111139 422 75 12256 5 76 679 77 0 78 0 Lenengd: 69 7 represents 0.697 inch Table 1: Humerus Lengths of adult male house sparrows, 24 perished and 35 that survived in a winter stormCase II: The Big Bang - An Observational Study Edwin Hubble used the power of the Mount Wilson Observatory telescopes to measure features of nebulae outside the Milky Way. He was surprised to find a relationship between a nebula's distance from earth and the velocity with which it was going away from the earth. Hubble's initial data on 24 nebulae are shown in Table 2. (Data from Hubble, "A Relation Between Distance and Radial Velocity Among Extra-galactic Nebular," Proceedings of the National Academy of Science 15 (1929): 168-73.) The recession velocity is measured in kilometers per second, which was determined with considerable accuracy by the red shift in the spectrum of light from a nebula. The distances -from the earth, in megaparsecs: 1 megaparsec is 1 million parsecs, and 1 parsec is about 30.9 trillion kilometers- were measured by comparing mean luminosities of the nebulae to those of certain star types, a method that is not particularly accurate. The apparent statistical relationship between distance and velocity led scientists to consider how much a relationship could arise. It was proposed that the universe came into being with a Big Bang, a long time ago. The material in the universe traveled out from the point of the Big Bang, and scattered around the surface of an expanding sphere. If the material were traveling at a constant velocity (V from the point of the bang, then the earth and any nebulae would appear as in Figure 1. Big Bang theory model for distance-velocity relationship of nebulae 3 Half the distance 5 Half the total recession between the nebula velocity divided by V is th, divided by the sine of the same D. is the sine of half half-angle. the angle between the trajectories of the two bodies. A Nebula Earth 4 The velocities V of the nebula and of the earth are Bang! Traveling at velocity V. the earth and the nebula each have covered the ponents receding distance D = VTsince from each other the bung occurred. 1 T years ago, the universe came into being with a bang Figure 1: Big Bang theory model for distance-velocity relationship of nebulae MacBook ProThe distance ()) between them and the velocity (X) at which they appear to be going away from each other satisfy the relationship Y ENIX VT - sin(A) , where A is the half angle between them In that case, Y = TX is a straight line relationship between distance and velocity. The points in Figure 1 do not fall exactly on a straight line. It might be, however, that the mean of the distance measurements is TX. The slope parameter T in the equation Mean {Y}=TX is the time elapsed since the Big Bang (that is the age of universe). Several questions arise. Is the relationship between distance and velocity indeed a straight line? Is the y-intercept in the straight line equation zero, as the Big Bang theory predicts? Nebula Velocity Xi Distance Yi S. Mag. 170 0.032 L. Mag. 290 0.034 NGC 6822 130 0.214 NGC 598 -70 0.263 NGC 221 -185 0.275 NGC 224 -220 0.275 NGC 5457 200 0.450 NGC 4736 290 0.500 NGC 5194 270 0.500 NGC 4449 200 0.630 11 NGC 4214 300 0.800 12 NGC 3031 -30 0.900 13 NGC 3627 650 0.900 14 NGC 4626 150 0.900 15 NGC 5236 500 0.900 16 NGC 1068 920 1.000 450 1.100 17 NGC 50551 do not fall exactly on a straight line. It might be, however, that the mean of the distance measurements is TX. The slope parameter T in the equation Mean {Y}=TX is the time elapsed since the Big Bang (that is the age of universe). Several questions arise. Is the relationship between distance and velocity indeed a straight line? Is the y-intercept in the straight line equation zero, as the Big Bang theory predicts? Nebula Velocity X, Distance Y; S. Mag. 170 0.032 L. Mag. 290 0.034 NGC 6822 -130 0.214 NGC 598 -70 0.263 NGC 221 -185 0.275 NGC 224 -220 0.275 NGC 5457 200 0.450 NGC 4736 290 0.500 NGC 5194 270 0.500 10 NGC 4449 200 0.630 11 NGC 4214 300 0.800 12 NGC 3031 30 0.900 13 NGC 3627 650 0.900 14 NGC 4626 150 0.900 NGC 5236 500 0.900 15 16 NGC 1068 920 1.000 17 NGC 5055 450 1.100 18 NGC 7331 500 1.100 1.400 19 NGC 4258 500 20 NGC 4151 960 1.700 21 NGC 4382 500 2.000 22 NGC 4472 850 2.000 23 NGC 4486 800 2.000 24 NGC 4649 1090 2.000 Table 2: Big Band Study Data