Stat 462 Midterm Exam #1 Study Guide

You should:

    
    * Be able to distinguish between a functional relation and a
      statistical relation.
    * Know how to specify the simple linear regression model and the normal 
      error regression model (including
      the assumptions and their implications on the response).
    * Know why we moved from the simple linear regression model to the normal 
      error regression model.
    * Be able to distinguish between population parameters and sample
      statistics. 
    * Be able to distinguish between the true parameters (beta0 and
      beta1) and the estimated parameters (b0 and b1) of a linear
      regression.
    * Know Minitab output from Fitted Line Plot command (s is square
      root of MSE which estimates true standard deviation sigma;
      estimates b0 and b1; scatter plot with fitted line)
    * Know Minitab output from Regression command (especially the table
      used for inference for beta0 and beta1). Know how the formulas we
      learned are connected to the output.
    * Know that the least squares estimates minimize the sum of the
      squared distances between the observed data and the estimated
      regression line.
    * Know how we derived the least squares estimators b0 and b1 using least squares
      criterian. (be prepared to derive at the least squares estimators from other models,
      recall the hw. problem with y_i=beta0+epsilon_i)
    * Be able to calculate, by hand, the least squares estimates for b0
      and b1 for a small data set.
    * Be able to calculate, by hand, MSE for a small data set.  
    * Know that the Gauss-Markov Theorem says that the estimators b0 and
      b1 are unbiased estimators of beta0 and beta1, and they have
      minimum variance among all possible linear unbiased estimators of
      beta0 and beta1.
    * Be able to derive that the least squares estimators are unbiased.
    * Know how we derived the distribution of b0 and b1.
    * Know what a confidence interval is, know the general form of a
      confidence interval (sample estimate +/- multiplier *standard error), and
      know the special interpretation of the CI for the regression setting.
    * Know what a hypothesis test is, know how to draw a conclusion
      about a hypothesis by using a P-value, know the difference between
      the two types of errors that are possible whenever performing a
      hypothesis test.
    * Know how to calculate a confidence interval for beta0 and beta1,
      and know how to use it to draw conclusions about the true beta0
      and beta 1 values.
    * Know how to perform a hypothesis test for beta0 and beta1.
    * Know the interpretation of the intercept and slope parameters.
    * Be able to distinguish between predictor x_i, observed response
      y_i, fitted (estimated) response y-hat_i, residual e_i, error epsilon_i, and the
      average response E(Y_i).
    * Be able to estimate the fitted response from an estimated regression
      equation, that is, calculate y-hat_i and e_i given the estimated
      regression equation y-hat_i = b0 + (b1*x_i).
    * Recognize that beta1 = 0 means there is no linear association
      between X and Y. Recognize that beta1 > 0 means there is a
      positive linear association between X and Y. Recognize that beta1
      < 0 means there is a negative (inverse) linear association between
      X and Y.
    * Know that association between X and Y does not imply a causal
      relationship between X and Y.
    * Know the properties of the fitted regression line (such as the
      residuals sum to 0, the sum of the squared residuals is a minimum,
      the sum of the observed responses is the same as the sum of the
      estimated responses, etc.)
    * Know how the spread of the predictor X values affects the variance
      of the b0 and b1 estimates (and hence the preciseness of the
      estimation of beta0 and beta1).
    * Understand how a large error variance (estimated by MSE) affects 
      the variance of the b0 and b1 estimates and the preciseness of predictions.
    * Be able to distinguish between estimating a mean response,
      predicting a new observation, and predicting the mean of m new
      observations
    * Know what factors affect the width of the confidence inteval for
      the mean response (the interval becomes narrower as the sample
      size increases, as the spread of the X's increases, and as the
      specific Xh value gets close to the sample mean of the X's).
    * Know how to get the confidence interval for a mean response and
      the prediction interval for a new observation and for the mean of m new observations
      from Minitab output.
    * Know that (and why) a prediction interval for a new observation is
      wider than a confidence interval for the mean response.
    * Know the distinction between a confidence interval and a
      prediction interval.