This post covers Introduction to probability from Statistics for Engineers and Scientists by William Navidi.

Basic Ideas

• Interpreting the Slope of the Least-Squares Line

  • If the x-values of two points on a line differ by $1$, their y-values will differ by an amount

    equal to the slope of the line.

  • If the values of the explanatory variable for two individuals differ by $1$, their predicted values will differ by $ \hat \beta_1$.

  • If the values of the explanatory variable differ by an amount $d$, then their predicted values will differ by $ \hat \beta_1 d$.

• The Estimates Are Not the Same as the True Values

  • It is important to understand the difference between the least-squares estimates $ \hat \beta_0$ and $ \hat \beta_1$, and the true values $\beta_0$ and $\beta_1$.
  • The true values are constants whose values are unknown.
  • The estimates are quantities that are computed from the data. We may use the estimates as approximations for the true values.
  • Therefore $ \hat \beta_0$ and $ \hat \beta_1$ are random variables, since their values vary from experiment to experiment.
  • To make full use of these estimates, we will need to be able to compute their standard deviations.

• The Residuals Are Not the Same as the Errors

• Don’t Extrapolate Outside the Range of the Data

  • For many variables, linear relationships hold within a certain range, but not outside it.
  • If we extrapolate a least squares line outside the range of the data, therefore, there is no guarantee that it will properly describe the relationship.
  • If we want to know how the spring will respond to a load of 100 lb, we must include weights of 100 lb or more in the data set.

• Don’t Use the Least-Squares Line When the Data Aren’t Linear

  • When the scatterplot follows a curved pattern, it does not make sense to summarize

    it with a straight line.

• Measuring Goodness-of-Fit

  • A goodness-of-fit statistic is a quantity that measures how well a model explains a given

    set of data.

  • r is a goodness-of-fit statistic for the linear model.

    

  • The points on the scatterplot are $(x_i, y_i)$ where $x_i$ is the height of the ith man and $y_i$ is the length of his forearm. \(r^2 = \frac{Regression ~ sum ~ of ~ squares}{ Total ~ sum ~ of ~ squares}\)

  • The sums of squares appearing in this discussion are used so often that statisticians have given them names. They call

    $ \Sigma^n_{i=1}\Sigma (y_i − \hat y_i)^2$ the error sum of squares and $ n_i =\frac {1}{(y_i − y)^2}$ the total sum of squares. Their difference

    $ \Sigma^n_{i=1}(y_i − y)^2 − \Sigma^n_{i =1}(y_i - \hat y_i)^2$

    is called the regression sum of squares.