Ratio, Rate and Proportion

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Exercise 29 Calculating Simple Linear Regression Simple linear regression is a procedure that provides an estimate of the value of a dependent variable (outcome) based on the value of an independent variable (predictor). Knowing that estimate with some degree of accuracy, we can use regression analysis to predict the value of one variable if we know the value of the other variable ( Cohen & Cohen, 1983 ). The regression equation is a mathematical expression of the influence that a predictor has on a dependent variable, based on some theoretical framework. For example, in Exercise 14 , Figure 14-1 illustrates the linear relationship between gestational age and birth weight. As shown in the scatterplot, there is a strong positive relationship between the two variables. Advanced gestational ages predict higher birth weights. A regression equation can be generated with a data set containing subjects' x and y values. Once this equation is generated, it can be used to predict future subjects' y values, given only their x values. In simple or bivariate regression, predictions are made in cases with two variables. The score on variable y (dependent variable, or outcome) is predicted from the same subject's known score on variable x (independent variable, or predictor). Research Designs Appropriate for Simple Linear Regression Research designs that may utilize simple linear regression include any associational design ( Gliner et al., 2009 ). The variables involved in the design are attributional, meaning the variables are characteristics of the participant, such as health status, blood pressure, gender, diagnosis, or ethnicity. Regardless of the nature of variables, the dependent variable submitted to simple linear regression must be measured as continuous, at the interval or ratio level. Statistical Formula and Assumptions Use of simple linear regression involves the following assumptions ( Zar, 2010 ): 1. Normal distribution of the dependent ( y ) variable 2. Linear relationship between x and y 3. Independent observations 4. No (or little) multicollinearity 5. Homoscedasticity 320 Data that are homoscedastic are evenly dispersed both above and below the regression line, which indicates a linear relationship on a scatterplot. Homoscedasticity reflects equal variance of both variables. In other words, for every value of x , the distribution of y values should have equal variability. If the data for the predictor and dependent variable are not homoscedastic, inferences made during significance testing could be invalid ( Cohen & Cohen, 1983 ; Zar, 2010 ). Visual examples of homoscedasticity and heteroscedasticity are presented in Exercise 30 . In simple linear regression, the dependent variable is continuous, and the predictor can be any scale of measurement; however, if the predictor is nominal, it must be correctly coded. Once the data are ready, the parameters a and b are computed to obtain a regression equatio.

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FAST School of Computing Project Differential Equations (MT-224) Due Date: 14th, June 2021. Max Marks: 70 A Brief Literature Review: We have studied the population growth model i.e., if P represents population. Since the population varies over time, it is understood to be a function of time. Therefore we use the notation P (t) for the population as a function of time. If P (t) is a differentiable function, then the first derivative dP dt represents the instantaneous rate of change of the population as a function of time, which is proportional to present population in case of the exponential growth and decay of populations and radioactive substances. Mathematically dP dt ∝ P. We can verify that the function P (t) = P0e rt satisfies the initial-value problem dP dt = rP, P (0) = P0. This differential equation has an interesting interpretation. The left-hand side represents the rate at which the population increases (or decreases). The right-hand side is equal to a positive constant multiplied by the current population. Therefore the differential equation states that the rate at which the population increases is proportional to the population at that point in time. Furthermore, it states that the constant of proportionality never changes. One problem with this function is its prediction that as time goes on, the population grows without bound. This is unrealistic in a real-world setting. Various factors limit the rate of growth of a particular population, including birth rate, death rate, food supply, predators, diseases and so on. The growth constant r usually takes into consideration the birth and death rates but none of the other factors, and it can be interpreted as a net (birth minus death) percent growth rate per unit time. A natural question to ask is whether the population growth rate stays constant, or whether it changes over time. Biologists have found that in many biological systems, the population grows until a certain steady-state population is reached. This possibility is not taken into account with exponential growth. However, the concept of carrying capacity allows for the possibility that in a given area, only a certain number of a given organism or animal can thrive without running into resource issues. • The carrying capacity of an organism in a given environment is defined to be the maxi- mum population of that organism that the environment can sustain indefinitely. • We use the variable K to denote the carrying capacity. The growth rate is represented by the variable r. Using these variables, we can define the logistic differential equation. dP dt = rP ( 1 − P K ) . 1 • An improvement to the logistic model includes a threshold population. The threshold population is defined to be the minimum population that is necessary for the species to survive. We use the variable T to represent the threshold population. A differential equation that incorporates both the threshold population T and carrying capacit ...

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