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Linear Regression & Least Squares Method quiz #1

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  • Which type of plot will produce a straight line when the relationship between two variables is linear?
    A scatter plot of two variables with a linear relationship will produce a straight line when the data points are plotted and the line of best fit (regression line) is drawn.
  • What is the slope in the regression equation 'performance = 9.32 + 0.52 * aptitude score'?
    The slope of the regression equation is 0.52.
  • Given a least squares regression equation in the form y = a + b*x, how do you use it to make predictions?
    To make predictions, substitute the desired x value into the equation y = a + b*x to calculate the predicted y value.
  • Which statement accurately describes linear regression?
    Linear regression models the relationship between two variables by fitting a straight line (the regression line) that minimizes the sum of the squared residuals (vertical distances between data points and the line).
  • If a linear regression equation has a slope b = 3 and intercept a = –6, what is the predicted value of y when x = 4?
    The predicted value of y is y = –6 + 3*4 = 6.
  • For a linear regression equation with b = 3 and a = –6, what is the predicted value of y when x = 4?
    The predicted value of y is y = –6 + 3*4 = 6.
  • Why is the regression line called the line of best fit?
    The regression line is called the line of best fit because it minimizes the sum of the squared residuals, making it the line that best represents the trend in the data.
  • What is the main goal of regression analysis?
    The main goal of regression analysis is to model the relationship between two variables and use the regression equation to make predictions.
  • Which of the following is not a purpose of a regression line: modeling relationships, making predictions, minimizing residuals, or displaying raw data?
    Displaying raw data is not a purpose of a regression line; the regression line is used to model relationships, make predictions, and minimize residuals.
  • In simple linear regression analysis, what is the general form of the regression equation?
    The general form of the regression equation is y = a + b*x, where a is the y-intercept and b is the slope.
  • Which expression is used to calculate the slope of the trend line in a scatterplot?
    The slope b of the trend line is calculated as b = (Σ(xy) – n * x̄ * ȳ) / (Σ(x^2) – n * (x̄)^2), where x̄ and ȳ are the means of x and y, respectively.
  • How can you estimate the slope of the line of best fit for a set of data points?
    The slope of the line of best fit can be estimated by calculating the change in y divided by the change in x between two points on the line, or more accurately using the least squares formula for slope.
  • How do you determine the slope of the line of best fit in a scatterplot?
    The slope of the line of best fit is determined by the least squares method, which calculates the value that minimizes the sum of squared residuals between the observed y values and the predicted y values.
  • What is the general form of the line of best fit in linear regression?
    The general form of the line of best fit is y = a + b*x, where a is the y-intercept and b is the slope.
  • In regression analysis, what does the y-intercept represent in the equation y = a + b*x?
    The y-intercept (a) represents the predicted value of y when x is zero.
  • Given the least squares regression equation ŷ = 1,202 + 1,133x, what is the predicted value of ŷ when x = 3?
    The predicted value is ŷ = 1,202 + 1,133*3 = 4,601.
  • The least squares method minimizes which quantity in regression analysis?
    The least squares method minimizes the sum of the squared residuals (the squared vertical distances between the observed data points and the regression line).
  • What does the least squares regression line minimize?
    The least squares regression line minimizes the sum of the squared residuals between the observed y values and the predicted y values from the regression line.
  • What does the least squares regression line minimize in linear regression analysis?
    The least squares regression line minimizes the sum of the squared residuals, which are the vertical distances between each data point and the regression line.
  • What information does the graph of a regression model display?
    The graph of a regression model shows the relationship between two variables, typically with data points plotted as a scatter plot and the regression line indicating the best fit through the data, revealing patterns, trends, and the strength of correlation.
  • What is the general equation for the least squares regression line in linear regression?
    The general equation for the least squares regression line is ŷ = ax + b, where 'a' is the slope and 'b' is the y-intercept.
  • In regression analysis, what does SSR (sum of squared residuals) represent?
    SSR represents the sum of the squared differences between the observed values and the values predicted by the regression line; it measures how well the regression line fits the data.
  • What is the main purpose of regression analysis in statistics?
    The main purpose of regression analysis is to model the relationship between two variables and use the resulting equation to make predictions about one variable based on the value of the other.
  • Why is a regression line used in statistical analysis?
    A regression line is used to provide the best linear estimate of the relationship between two variables, allowing for prediction of values and understanding of trends within the data.