Multivariable functions #4 - Questions and Answers

Question 1 In a regression analysis, the variable that is being predicted is the "dependent variable." a. Intervening variable b. Dependent variable c. None d. Independent variable Answer 1 Step 1 Introduction Regression: Regression analysis is used to estimate or predict the dependent variable based on independent variable by using the following Regression equation SSRy = a + b * x Where a is the intercept and b is the slope coefficients of the regression equation. y is the (dependent or response) variable and x is the (independent or explanatory) variable. Step 2 Answer and explanation As we know, the Regression analysis useful for prediction or forecasting. In Regression line, one Variable is dependent variable and other variable is independent variable. The dependent variable being predicted on the basis of independent variable (Explanatory variable) Thus, the dependent variable is being predicted .i.e., Option b is correct. Answer 2 In a regression analysis, the variable that is being predicted is the dependent variable. The dependent variable, also known as the response variable or outcome variable, is the variable of interest that is being explained or predicted by the regression model. It is the variable that researchers want to understand and analyze in relation to other variables in the study. In a regression equation, the dependent variable is typically denoted by Y. The goal of the regression analysis is to find the relationship between the dependent variable y and one or more independent variables, denoted by X₁, X₂ , ..., Xₙ. The regression model seeks to estimate the relationship between the dependent variable Y and the independent variables X₁, X₂, ..., Xₙ. This relationship is often expressed in the form of a mathematical equation, such as: Y = β₀ + β₁X₁ + β₂X₂ + ... + βₙXₙ + ε where β₀ is the intercept term, β₁, β₂ ,..., βₙ are the coefficients corresponding to the independent variables, and ε represents the error term. The regression analysis aims to estimate the values of the coefficients β₀, β₁, ..., βₙ, in order to create the best-fitting line or curve that represents the relationship between the dependent variable y and the independent variables X₁, X₂ , ..., Xₙ. Therefore, in a regression analysis, the dependent variable is the variable that is being predicted or explained, making the correct answer to your question {(b) Dependent variable}. Answer 3 Result: (b) Dependent variable Solution: The dependent variable is the outcome or response variable that we are interested in understanding or predicting. It is the variable that we believe is influenced by one or more independent variables. In a regression analysis, we use the values of the independent variables to estimate or predict the value of the dependent variable. The other options can be explained as follows: a. Intervening variable: An intervening variable is a variable that comes between the independent and dependent variables and affects the relationship between them. It is not the variable being predicted in a regression analysis. b. Dependent variable: As explained above, the dependent variable is the variable being predicted or explained in a regression analysis. c. None: This option is incorrect because in a regression analysis, there is always a dependent variable that is being predicted. d. Independent variable: The independent variable is a variable that is believed to have an effect on the dependent variable. It is not the variable being predicted, but rather the variable used to explain or predict the dependent variable. Therefore, the correct answer is b. Dependent variable. Question 2 What is the difference between a relative maximum and an absolute maximum on an interval ? Answer Step 1 Suppose a function given by y = f(x) is defined in the interval (a,b). In order to find the maximum or minimum. value of the function, we make use of the derivative dy/dx as it gives the rate of change of quantity y with respect to quantity x. First we need to find the critical points. These are the points at which the function either attains maximum or minimum. This is done by equating dy/dx = 0. If dy/dx changes from +ve to -ve then we have a maximum value of f (x). If dy/dx changes from -ve to +ve then we have a minimum value of f (x). In the curve, the point of maxima is called a Peak, and the point of minima is called a Valley. Relative Maximum It refers to all the peak values of the function, the points at which the term dy/dx changes its sign from +ve to -ve. However, it does not give the real picture of the scenario, as there may be many other points at which dy/dx changes its sign from +ve to -ve. (Refer the graph) Step 2 Absolute Maximum It refers to the value of the function which is maximum in the entire interval. It gives a much better analysis of the function. function is said to have Absolute maximum at a point z = c if and only if f(c) ≥ f(x) for all values of x in the given interval. Step 2 Absolute Maximum It refers to the value of the function which is maximum in the entire interval. It gives a much better analysis of the function. function is said to have Absolute maximum at a point z = c if and only if f(c) ≥ f(x) for all values of x in the given interval. Step 3 Relation between Absolute maximum and Relative maximum- It can be observed from the graph that, Abs max f(x) = largest of all the relative maximum in [a, b]. Question 3 (a) In a regression analysis, the sum of squares for the predicted scores is 100 and the sum of squares error is 200, what is R²? (b) In a different regression analysis, 40% of the variance was explained. The sum of squares total is 1000. What is the sum of squares of the predicted values? Answer 1 Step 1. Introduction: In a regression analysis, denote SST as the total sum of squares, SSR as the regression sum of squares or the sum of squares for the predicted values, and SSE as the error sum of squares. Then, the coefficient of determination is given as: R² = SSR/SST (1) Now, it is known that for a regression analysis, SST=SSR+SSE. Thus, SSR=SST-SSE. Using this relation, R² can be obtained as: R² = (SST - SSE) / SST R² = 1 - (SSE / SST) (2) Any of equation (1) or (2) may be used to find R². Step 2 a. Here, SSR=100, SSE=200. Thus, SST=SSR+SSE = 100 + 200 = 300. Substitute SSE=200, SST=300 in equation (2): R² = 1 - (SSE / SST) R² = 1 - (200 / 300) R² = 1 - (2 / 3) R² = 1 / 3 ≈ 0.3333. Thus, the value of R² is 0.3333. Sometimes, R² is expressed as a percentage, instead of a proportion. The proportion 0.3333 can be expressed as a percentage, by multiplying it with 100. Hence, in percentage form, the value of R² is 33.33%. It must be understood that both give the same value, just expressed in a different manner. Answer 2 Step 1 b. The R² value can be interpreted as the proportion or percentage of variation in the response variable that is explained by the variation in the predictor variables in the model. In the second regression model, since 40% of the variance in the response variable was explained by the variation in the model, the value of R² is 40% or 0.40 (obtained by dividing the percentage by 100). Further, it is also given that, SST=1,000. In order to find the value of the sum of squares of the predicted values, substitute SST=1,000, R² = 0.40 in equation (1) R² = SSR/SST 0.40 = SSR / 1000 SSR = 1,000 * (0.4) SSR=400. Hence, the sum of squares of the predicted values is 400. Question 4 How do you find critical points for function of two variables f(x, y) = 8x³ + 144xy + 8y³? Answer For two-variables function, critical points are defined as the points in which the gradient equals zero, just like you had a critical point for the single-variable function f(x) if the derivative f'(x) = 0. The matter is that you now can differentiate the function with respect to more than one variable (namely 2, in your case), and so you must define a derivative for each directions. The gradient is thus defined as the n-dimensional vector (again, in your case n = 2), and its coordinates are the derivatives with respect to each variable. So, the gradient of a two-variable function f(x, y) is the vector (∂f/∂x, ∂f/∂y), where deriving with respect to a variable means to consider the other as a constant. Let's compute the two derivatives: ∂f/∂x = 24x² + 144y ∂f/∂y = 144x + 24y² To find the critical points, we must find the values of x and y for which (∂f/∂x , ∂f/∂y) = (0,0) holds. In other words, we must solve 24x² + 144y = 0 24y² + 144x = 0 Simplifying both expression, we have x² + 6y = 0 y² + 6x = 0 An obvious solution is (x, y) = (0, 0), which is thus our first critical point, while the other solution are (x, y) = (-6, -6), which is the second and last critical point. Question 5 Find the critical points of the following multivariable function, and using the Second Derivative Test to find the maxima and minima of the equation below: a(x, y) = (8x - y) / (e^(x² + y²)) Answer Differentiating the function, a(x, y) = (8x - y) / (e^(x² + y²)) aₓ = (e^(x²+y²)(8) - (8x - y)e^(x²+y²) * 2x) / (e^(x²+y²))² aₓ = [8 - (8x - y)*2x] / e^(x²+y²) aₓ = [8(1 - 2x²)] / e^(x²+y²) aᵧ = (e^(x²+y²)(-1) - (8x - y)e^(x²+y²) * 2y) / (e^(x²+y²))² aᵧ = [-1 - (8x - y)*2y] / e^(x²+y²) aᵧ = [-1 + 2y²] / e^(x²+y²) For critical points, aₓ = 0, aᵧ = 0 aₓ = [8(1 - 2x²)] / (e^(x²+y²)) = 0 1 - 2x² = 0 2x² = 1 x = ± 1/√2 aᵧ = [-(1 - 2y²)] / (e^(x²+y²)) = 0 1 - 2y² = 0 2y² = 1 y = ± 1/√2 The critical points are (± 1/√2, ± 1/√2) aₓₓ = [e^(x²+y²)(-32x) - 8(1-2x²)e^(x²+y²) * 2x] / (e^(x²+y²))² aₓₓ = [8(-6x + 4x³)] / e^(x²+y²) aᵧᵧ = [-e^(x²+y²)(-4y) - (1 - 2y²)e^(x²+y²) * 2y] / (e^(x²+y²))² aᵧᵧ = [-(1-2y²)] / e^(x²+y²)