Concept explainers
In the SIR model, we assume that everyone in the population is susceptible at time t= 0 except the very small fraction that is already infected. Suppose that some fraction of the population has received a vaccine, so they cannot catch the disease. The vaccine makes the fraction of the population that is susceptible at time t = 0 smaller.
(a) Using HPGSystemSoIver applied to the SIR model with
(b) If
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Differential Equations
- Repeat Example 5 when microphone A receives the sound 4 seconds before microphone B.arrow_forwardThe estimated least-squares regression equation for profit (in $100,000) is: y' = 2.5 + 0.054t Sales for time period 32 (t = 32) are 5% below the average of yearly sales. Forecast sales, including the seasonal component, for t = 32.arrow_forwardThe longitude, rt of an air plane is affected by a random component e, due to the wind effect and its speed t It = -1 + 20t-1 + 6t. The speed of the plane is affected by a constant global linear trend, ß = B₁-1 = 3, and varies randomly due to weather conditions, v₁ = V₁-1 + B₁-1+w₂. In the above equations, is assumed to be white Gaussian noise with zero mean and o = 3. Similarly w is assumed to be white Gaussian noise with zero mean and 2,=0.5. A GPS transmitter mounted on the plane sends a noisy measurement to a receiver about its position, X₁ = 0.5xt + nt, where n, is assumed to be white Gaussian noise with zero mean and o2 = 2. (a) Using the following definition for the state vector, 0t. 0₁ It + ve B₂ write the motion of the plane in state space form. Note: You need to provide the exact form of the h, G and W matrices. (b) Evaluate the initial estimate for the state vector 03-arrow_forward
- Consider a model with an interaction term between being female and being married. The dependent variable is the log of the hourly wage: log(wage) = 0.151 - 0.038 female + 0.1 married - 0.301 female* married + 0.079 educ + 0.027 exper+0.029 tenure (0.072) (0.056) (0.055) (0.007) (0.005) (0.007) n = 536, R2 = 0.461 Numbers in parantheses are standard errors of coefficients. Given the estimation result and the observation number fill in the blanks below which aim at discussing the statistical significance of variables. The test statistic of the interaction term is The critical value at 1% significance level is Then the interaction term statistically significant at 1% significance level. (Hint: to fill the blank make a choice between "is" and "is not".)arrow_forwardConsider an equation to explain salaries of CEOs in terms of annual firm sales, return on equity (roe in percentage form), and return on the firm’s stock (ros, in percentage form): log (salary) = β0+ β1 log(sales) + β2roe + β3ros – u. In terms of the model parameters, state the null hypothesis that, after controlling for sales and roe, ros has no effect on CEO salary. State the alternative that better stock market performance increases a CEO’s salary. Using the data in CEOSAL1, the following equation was obtained by OLS: = 4.32 + .280 log(sales) + .0174 roe + .00024 ros (.32) (.035) (.0041) (.00054) n = 209, R2 = .283 By what percentage is salary predicted to increase if ros increases by 50 points? Does ros have a practically large effect on salary? Test the null hypothesis that ros has no effect on salary against the alternative that ros has a positive effect. Carry out the test at the 10% significance…arrow_forwardConsider an equation to explain salaries of CEOs in terms of annual firm sales, return on equity (roe in percentage form), and return on the firm’s stock (ros, in percentage form): log (salary) = β0+ β1 log(sales) + β2roe + β3ros – u. In terms of the model parameters, state the null hypothesis that, after controlling for sales and roe, ros has no effect on CEO salary. State the alternative that better stock market performance increases a CEO’s salary. Using the data in CEOSAL1, the following equation was obtained by OLS: log(salary) ˆ = 4.32 + .280 log(sales) + .0174 roe + .00024 ros (.32) (.035) (.0041) (.00054) n = 209, R2 = .283 By what percentage is salary predicted to increase if ros increases by 50 points? Does ros have a practically large effect on salary? Test the null hypothesis that ros has no effect on salary against the alternative that ros has a positive effect. Carry out…arrow_forward
- Consider a simple linear regression model Y; = Bo + B1X; + E;, where e; are in- dependent identically distributed (iid) N(0, o²). Suppose the parameter values are Bo = 200, B1 = 5.0, and o = 4. a) Explain the meaning of the parameters Bo Assume that the scope of the model includes X = 0. 200, В1 5.0, and o = 4. b) What is the distribution of Y? What is the mean and variance for this distribution?arrow_forwardThe following model was specified to investigate the relationship between mineral exploitation and energy consumption in a small state over 1970-2006: ln(mining)t= β0 + β1ln(GDP_m)t + β2ln(exports)t + β3ln(energy)t + et where mining = the output of the mining and quarrying industry (constant 2000 US dollars); GDP_m = GDP less mining and quarrying industrial output (constant 2000 US dollars); exports = exports (constant 2000 US dollars); and energy = energy consumption (million kilowatts hours). OLS estimation of the preceding equation yields ln(mining)t=-5.76+0.81ln(GDP_m)t+0.21ln(exports)t-0.09ln(energy)t t-stat = (-3.82) (12.76) (3.53) (-2.49) p-value = (0.001) (0.000) (0.001) (0.017) R2=0.8905 n = 37 Durbin-Watson stat = 0.98 d) Interpret the coefficient on ln(energy). e) Are coefficients’ signs on explanatory variables within your expectation? Answer this question with a…arrow_forwardThe following model was specified to investigate the relationship between mineral exploitation and energy consumption in a small state over 1970-2006: ln(mining)t= β0 + β1ln(GDP_m)t + β2ln(exports)t + β3ln(energy)t + et where mining = the output of the mining and quarrying industry (constant 2000 US dollars); GDP_m = GDP less mining and quarrying industrial output (constant 2000 US dollars); exports = exports (constant 2000 US dollars); and energy = energy consumption (million kilowatts hours). OLS estimation of the preceding equation yields ln(mining)t=-5.76+0.81ln(GDP_m)t+0.21ln(exports)t-0.09ln(energy)t t-stat = (-3.82) (12.76) (3.53) (-2.49) p-value = (0.001) (0.000) (0.001) (0.017) R2=0.8905 n = 37 Durbin-Watson stat = 0.98 a) Use the p value approach to find out whether energy contributes negatively to mineral exploitation at the 5% level. Hint: specify the 5+1 steps. b)…arrow_forward
- Sports scientists want to use nuclear magnetic resonance spectroscopy, NMR, to predict the muscle fibre composition in the thighs of athletes. They obtained the data in the screenshot, which contains three variables: FTF – the percentage of fast twitch fibres in the muscle. T1 – the T1 relaxation time measured in ms. T2 – the T2 relaxation time measured in ms. (a) Perform a linear regression to determine the equation allowing FTF to be predicted by T1, i.e. FTF = b0 + b1T1. i) What is the equation of the best-fit line? ii) What is the F statistic and P value for the regression? Do these indicate that the regression is significant? iii) What is the value of R2? How much variation does the regression take account of?arrow_forwardA sociologist investigating the recent upward shift in homicide trends throughout the country studied the extent to which the homicide rate per 100 000 people (v) is associated with population size in thousands (x ), the rate of unemployment (x,), and the percentage of families with annual incomes less than R24 000 (x3). Data are provided in the table for a sample of 20 cities. A regression analysis was performed on the data and the results indicated that the interaction model E(y) = B,+ Bx + Bx, + Bx; + B,xqx,+ B;xx; + Bzxzx; was a good model to fit to the data. %3D However, the residual plot for this model indicates that heteroscedasticity may exists for the percentage of families with annual incomes less than R24 000. How do I Conduct a test for heteroscedasticity by dividing the data into two subsamples, x,21 and x, > 21 using MSE for the first subsample = 582.3, MSE for the second subsample = 364.7 and a= 0.05. And how do I explain my conclusion.arrow_forwardFind the least linear regression of (1, 0), (3, 3), and (5, 6).arrow_forward
- Calculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,Trigonometry (MindTap Course List)TrigonometryISBN:9781337278461Author:Ron LarsonPublisher:Cengage Learning