soln%204 - X X X X X X X X X X X X X X X X X AP Statistics...

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X X X X X X X X X X X X X X X X X AP Statistics Solutions to Packet 4 X More on Two-Variable Data Transforming Relationships Cautions about Correlation and Regression Relations in Categorical Data X X X X X X X X X X X X X
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2 HW #23 6, 10, 11, 14, 15, 18, 19 4.6 GYPSY MOTHS Biological populations can grow exponentially if not restrained by predators or lack of food. The gypsy moth outbreaks that occasionally devastate the forests of the Northeast illustrate approximate exponential growth. It is easier to count the number of acres defoliated by the moths than to count the moths themselves. Here are data on an outbreak in Massachusetts: Let L1 = year, L2 = acres Look at scatterplot: year vs acres, scatterplot looks ___ exponential __ Verify that y is being multiplied by about 4 each year by calculating the ratio of acres defoliated each year to the previous year. (start with 1979 to 1978, when the ratio is 226,260/63,042 = 3.59) The ratios are 226,260/63,042 = 3.59, 907,075/226,260 = 4.01, and 2,826,095/907,075 = 3.12. Let L3 = log (acres) = log (L2) log y yields 4.7996, 5.3546, 5.9576, and 6.4512. Look at scatterplot, year vs log (acres), L1 vs L3 Year Acres 1978 63,042 1979 226,260 1980 907,075 1981 2,826,095
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3 Calculate LSRL for this scatterplot. ˆ lo g 1094. 5 0.5558 yx = - + r = 0.999 Look at residual plot. How does it look? residual plot shows random scatter. Perform the inverse transformation to express y ˆ as an exponential equation. eqt: ˆ 1094. 5 .555 8) =10∧( - + 0 Sketch a scatterplot of the original data with the exponential curve model superimposed. Is your exponential function a satisfactory model for the data? YES! Use your model to predict the # of acres defoliated in 1982. 10,719,965 (Postscript: A viral disease reduced the gypsy moth population between the readings in 1981 and 1982. The actual count of defoliated acres in 1982 was 1,383,265.) 4.10 GUN VIOLENCE (Exact exponential growth) A paper in a scholarly journal once claimed (I am not making this up), “Every year since 1950, the number of American children gunned down has doubled.” To see that this is silly, suppose that in 1950 just 1 child was “gunned down” and suppose that the paper’s claim is right. (a) Fill in table with # of children killed in each of the next 10 years. (b) Plot the number of deaths against the year and connect the points with a smooth curve. This is an exponential curve. year # children killed 1950 1 1951 2 1952 4 1953 8 1954 16 1955 32 1956 64 1957 128 1958 256 1959 512 1960 1024
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4 (c) The paper appeared in 1995, 45 years after 1950. How many children were killed in 1995, according to the paper?_ If x = number of years after 1950, then y = the number of children killed x years after 1950 = 2 x . At x = 45, 45 2 3.5 2 , 35,200,000,000,000(3 5 !) y o r trillion 13 = = ×1 0 (d) Let L3 = log (# children) = log (L2). Look at scatterplot of year vs log (# of children) (L1 vs L3) This should appear linear. (e) Find LSRL of scatterplot above. ˆ lo g 587.008 5 .30103 yx = - + r = 1 Make sure to look at residual plot. (f) Perform the inverse transformation to express y ˆ as an exponential equation.
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soln%204 - X X X X X X X X X X X X X X X X X AP Statistics...

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