Day 29: STA205 - Introduction to Statistical Methods

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• Announcements:
  • Final exam schedule:
    • Section 006: Monday, May 1st, 1:00-3:00
    • Section 007: Wednesday, May 3rd, 3:10-5:10

  • Your Packet 13 IMath homework is due Friday, Cinco de Mayo (5th of May). It's optional, but will replace a weaker IMath score if you do it. You should do it anyway, to prepare for the final!
  • All your scores should be on-line now, in IMath

  • I'll be dropping your three lowest quizzes, and your four lowest homeworks.

  • If you do better on the "old" section of the final than you did on your worst midterm, then the better grade will replace the worser grade.

• Today we finish up regression, then review!
  • Packet 13: Regression

    What so important about regression? We use it all the time. For example, to model CO₂ in the atmosphere:

    Here's a model of the Keeling data (xlispstat)

  • So: what have we learned together this semester?
    • What is statistics?
      • The science of reasoning from data.
      • Also the names of various descriptive measures of data.

    • What is statistical inference?

      Generalization about a population based on information contained in a sample from that population.

    • We started with data:
      • Qualitative or categorical versus quantitative
      • Sample versus population data
      • How sampled? Random, convenient, biased, etc.
      • Measures of central tendency -- sample mean, median, mode (statistics), which seek to get at population mean, median, mode (parameters)
      • Visual displays of data -- histogram, boxplot
        • shape
        • skew
        • central tendency (center)
        • dispersion (spread)
      • Measures of spread, especially standard deviation (which occurs at the inflection point of the normal), but also variance, range, IQR.
      • All these help us get at how a variable is distributed.

    • Distributions we have known (if not loved):
      • normal
        • completely characterized by mean and standard deviation
        • empirical rule
        • z-statistic and z-table: computing probabilities as areas
        • means look normal for large n (distribution of the sample mean)
        • sample proportion normal if np>10 and n(1-p)>10
      • chi-square -- skewed right, requires degrees of freedom
      • t -- bell-shaped like a normal, but varies by degrees of freedom (need when we don't have sigma); behaves pretty much like a normal (more spread out).
      • f -- requires degrees of freedom (numerator and denominator; $f=\frac{MST}{MSE}$
      • Chebyshev's rule works for any distribution

    • Analysis:
      • Confidence Intervals:
        • Objective: we seek to estimate a parameter
        • A point estimate isn't enough, because we don't have a good sense of how bad it can be; so we give an interval, with a certain amount of confidence. (We build a "box" in which we believe the parameter lives.)
        • Proportions and means
        • Estimating sample size needed for fixed width of MOE with given confidence
        • More confidence, bigger box
        • bigger sample, smaller box
      • Tests of hypothesis:
        • null and alternative
        • decision rule (significance)
        • test statistic
        • interpretation
        • type I and type II errors
      • Differences in means
        • independent versus dependent samples (unpaired versus paired t-tests)
        • multiple means simultaneously (ANOVA)
          • f-test
          • multiple comparisons, if appropriate
      • Chi-square test:
        • Testing for dependence between qualitative variables
        • Build contingency table
        • Find expected counts, and compared to observed
        • Build into chi-square statistic
        • Do test of hypothesis -- independence versus dependence

      • Linear Regression:
        • scatterplot -- look at the data first!
        • Testing for linear dependence between quantitative variables
        • correlation coefficient
        • test for significance of linear relationship (r=0?)
        • linear equation for model (slope, intercept)
        • use for prediction
          • of mean of y for a given value of x
          • of a value of y for a given value of x

    • Any last question?

  • Links:
    • The Standard Normal Table of probabilities.