Midterm III Review Sheet for Math 156.3, Fall 2008

Words/concepts you should know (i.e.,their definitions, examples, their relationships,...)

  1. The Least Squares Regression Line
    1. the general intent — a line "going through the middle of the data", which can be used to "predict" values of the response variable (the y coordinate) given a particular value of the explanatory variable (the x coordinate)
    2. the precise definition in terms of the residuals — which are the differences between the observed y-values and the predicted y-values for each of the data points — as that line which minimizes the sum of the squares of all the residuals
    3. the equation: we use the form y-hat = a+bx, where:
      1. y-hat is a predicted y-value for a given x-value
      2. b is the slope coefficient with the value b=r sy/sx, where r is the correlation coefficient of the x- and y-data, sx is the standard deviation of the x-data by itself and sy is the standard deviation of the y-data by itself
      3. a is the intercept coefficient with value a=y-bar - b x-bar where b is as above and x-bar and y-bar are the means of the x- and y-data, each separately
    4. using it, particularly
      1. to predict a y-value when only the x is known
      2. cautions about prediction with the LSRL:
        1. we must be very cautious about extrapolation, which is when we predict a y-value corresponding to an x-value which is very different from the mass of the x-values of the data which was used to compute the coefficients of the LSRL
        2. the LSRL is based upon a linear model of the data, which can be inappropriate if the data has another form; this can be seen if a graph of the residuals vs the x's of all the data shows some pattern — in which case, one tries an appropriate algebraic transformation of one of the coordinates until the residual plot no longer has any pattern at all, which indicates that a better model has been found
  2. The Normal distribution
    1. general features of its shape:
      1. the center (being the population mean μ)
      2. the spread (being the population standard deviation σ) &mdash which is the distance out from the center to where the curve turns from being concave down to concave up
      3. its symmetry
    2. interpreting the area under the Normal curve for a certain range of x-values as the probablity that the X lies in that range
    3. computing Normal probabilities with the calculator or by standardizing the x to a z and then using the standard Normal table
  3. the idea of a sampling distribution (for a proportion): the distribution of the RV which is formed by taking a sample out of some population and computing the proportion (denoted p-hat) of the sample which says "yes" to the original survey question (or which makes some other particular choice in the original survey categorical RV)
  4. The Central Limit Theorem (CLT) for proportions, which says that the sampling distribution for p-hat, based on samples of size n from a population with population proportion π is approximately Normal, if n is large enough, with mean π and standard deviation sqrt(π(1-π)/n)
  5. Confidence Intervals for proportions; for which, here are vocabulary, computations, and cautions:
    1. the standard error of some sample data is sqrt(p-hat (1 - p-hat)/n)
    2. the interval itself, given by p-hat ± z* sqrt(p-hat (1 - p-hat)/n), where
    3. the critical value z* comes from the standard Normal distribution (usually by looking it up in a table of critical values)
    4. that whole piece z* sqrt(p-hat (1 - p-hat)/n) is called the margin of error (MoE) of the confidence interval
    5. that "... ± ..." formula means we are computing the interval (p-hat - MoE, p-hat + MoE) [where the interval notation (a,b) is shorthand for "the set of all numbers x such that a < x < b"]
    6. CIs are only reliable if the sample size and proportion are such that both n p-hat ≥ 10 and n (1 - p-hat) ≥ 10 and sample data was from an SRS
    7. the meaning of the confidence level of a CI

Things to know how to do:

  1. Most importantly: the homework problems which were assigned, and similar problems from the same chapters, which may not have been assigned. I.e., make sure you are comfortable with the assigned problems, then look through those sections and try the unassigned next problem after, or the problem right before, some of the assigned ones.
  2. Use your calculator to ...
    1. generate and graph the residuals for a LSRL
    2. compute the coefficients of a LSRL, and graph it on top of a scatterplot of the data itself
    3. compute a normal probability
  3. Compute a ..
    1. probability for a sample proportion in some range when the population mean and standard deviation (and the sample size) are known (CLT)
    2. confidence intervals
    3. margin of error
    4. sample size in order to get a particular MofE
  4. Discuss (succinctly, accurately and completely) ..
    1. the meaning of the confidence level of a confidence interval
    2. the conditions under which it makes sense to use our approach to confidence intervals