1. The Lorenz Curve: In this problem, we'll work step-by-step through constructing values of a Lorenz function from a set of raw data. This will help you to understand the meaning of the Lorenz curve, which you'll use in your pod project. (a) A small village has a population of 10 people. The yearly incomes, in thousands of CAD per year, for each of the 10 residents is listed below. 95, 46, 74, 150, 60, 110, 46, 34, 80, 105 Rearrange these incomes in increasing order, from smallest to largest. (b) What is the total sum of the income made by all people in this town each year? Answer: thousand CAD. (c) What fraction of the total income is made by top 20% of earners? Please express your answer as a decimal value between 0 and 1. Answer: (d) What fraction of the total income is made by bottom 20% of earners? Please express your answer as a decimal value between 0 and 1. (Note: Even if there is a tie for the second-lowest earner, you should only count two of the 10 people as the lowest-earning 20%.) Answer: (e) Please continue by filling in the chart below. All numbers in the second row should be in units of thousands of CAD, while all numbers in the third row should be decimal values between 0 and 1. Please round numbers in the third row to the nearest 0.01. Lowest-earning fraction x | 0.0 | 0.2 | 0.4 | 0.6 | 0.8 | 1.0 Total income (thousand CAD) made by lowest-earning fraction x Fraction of income made by lowest-earning fraction x
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(a) To rearrange the incomes in increasing order, we start with the given list: \[ 95,46,74,150,60,110,46,34,80,105 \] Rearranging this list in increasing order, we get: \[ 34, 46, 46, 60, 74, 80, 95, 105, 110, 150 \] (b) Show more…
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Let x be the per capita income in thousands of dollars. Let y be the number of medical doctors per 10,000 residents. Six small cities in Oregon gave the following information about x and y. x 8.4 9.6 10.4 8.0 8.3 8.7 y 9.8 19.0 21.0 10.2 11.4 13.1 Complete parts (a) through (e), given Σx = 53.4, Σy = 84.5, Σx^2 = 479.46, Σy^2 = 1303.65, Σxy = 773.31, and r ≈ 0.9733. (a) Draw a scatter diagram displaying the data. (b) Verify the given sums Σx, Σy, Σx^2, Σy^2, Σxy, and the value of the sample correlation coefficient r. (Round your value for r to four decimal places.) Σx = 53.4 Σy = 84.5 Σx^2 = 479.46 Σy^2 = 1303.65 Σxy = 773.31 r = 0.9733 (c) Find x̄ and ȳ. Then find the equation of the least-squares line ŷ = a + bx. (Round your answer to four decimal places.) x̄ = ȳ = ŷ = + x (d) Graph the least-squares line. Be sure to plot the point (x, y) as a point on the line. (e) Find the value of the coefficient of determination r^2. What percentage of the variation in y can be explained by the corresponding variation in x and the least-squares line? What percentage is unexplained? (Round your answer for r^2 to four decimal places. Round your answers for the percentages to two decimal places.) r^2 = explained % unexplained % (f) Suppose a small city in Oregon has a per capita income of 8.4 thousand dollars. What is the predicted number of M.D.s per 10,000 residents? (Round your answer to two decimal places.) M.D.s per 10,000 residents
Shyam P.
Exercises $49-51 :$ The Lorenz curve $y=F(r)$ is used by economists to study income distribution in a given country (see Figure 14$) .$ By definition, $F(r)$ is the fraction of the total income that goes to the bottom rth part of the population, where $0 \leq r \leq 1 .$ For example, if $F(0.4)=0.245,$ then the bottom 40$\%$ of households receive 24.5$\%$ of the total income. Note that $F(0)=0$ and $F(1)=1$ The following table provides values of $F(r)$ for Sweden in 2004 Assume that the national average income was $A=30,000$ euros. (a) What was the average income in the lowest 40$\%$ of households? (b) Show that the average income of the households belonging to the interval $| 0.4,0.61$ was $26,700$ euros. (c) Estimate $F^{\prime}(0.5) .$ Estimate the income of households in the 50 th percentile? Was it greater or less than the national average?
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For the scatterplot on the right, determine which, if any, of these functions might be used as a model for the data. Quadratic: f(x) = ax^2 + bx + c Polynomial, not quadratic Exponential: f(x) = ae^kx, k > 0 Exponential: f(x) = ae^-kx, k > 0 Logarithmic: f(x) = a + b ln x Logistic: f(x) = a / (1 + be^-kx) Which of the functions above might be used as a model for the data? A. Logistic: f(x) = a / (1 + be^-kx) B. Quadratic: f(x) = ax^2 + bx + c C. Logarithmic: f(x) = a + b ln x D. Exponential: f(x) = ae^kx, k > 0 E. Polynomial, not quadratic F. Exponential: f(x) = ae^-kx, k > 0 A ship carrying 1000 passengers is wrecked on a small island from which the passengers are never rescued. The natural resources of the island restrict the population to a limiting value of 5730, to which the population gets closer and closer but which it never reaches. The population of the island after time t, in years, is approximated by the logistic equation given below. Complete parts (a) through (c). P(t) = 5730 / (1 + 4.73e^-0.6t) a) Find the population after 19 years. (Round to the nearest integer as needed.) b) Find the rate of change, P'(t). A. (1 + 4.73e^-0.6t)^2 / (16,261.74e^-0.6t) B. (27,102.90e^-0.6t) / (1 + 5730e^-0.6t)^2 C. (27,102.90e^-0.6t) / (1 + 4.73e^-0.6t)^2 D. (16,261.74e^-0.6t) / (1 + 4.73e^-0.6t)^2 c) Sketch a graph of the function. Choose the correct graph below.
Andrew N.
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