Day 30: STA205 - Introduction to Statistical Methods

Today:

• Quizzes returned

• Final Exam prep from the Math Center

• Final Exam: December 11, 1:00-3:00. Be there!;)
  • comprehensive, but extra emphasis on chi-square and linear regression
  • Bring calculator, formula sheet; in addition, you may write on your formula sheet anything else you find useful.

• Section 11.3: fitting a straight line to sampled data: least squares
  • If we decide that there is significant linear correlation between two quantitative variables, then it makes sense to try to fit a line to the data.

  • Lines:
    • are determined by two parameters: slope and intecept -- often written y = mx+b
    • the parameters can be determined by two points on the line: e.g. celsius and fahrenheit temperature scales.
      • The slope measures the rate at which y changes as x changes. It's the change in y divided by the change in x.
      • The intercept b is the value of y when x=0.
    • In statistical parlance, the variable x is often called the predictor, and the y variable is often called the response.

  • Linear regression/ least squares line
    Fitting points to lines, in a "best way"

    Let's look at the VNP data from our worksheet with this applet

    Influential points in a regression

  • Examples:
    • Exercise #3, p. 461 (#1, p. 444, p. 451): http://www.nku.edu/~statistics/data/c11s01e01.xls

• Section 11.4: prediction
  • Figure 11.4, p. 447 gives a good picture to guide us.

  • The best point estimate of a response value for a give predictor value is given by the line, but, as in the past, we often prefer a confidence interval to a point estimate.

  • The software gives two kinds of confidence intervals: a confidence interval for the mean response , and a confidence interval for a response value y(x). The first is called the "CI" by StatCrunch (for Confidence Interval), whereas the latter is called the Prediction Interval, and hence denoted "PI".

  • Example:
    • Exercise #1, p. 470 (#3, p. 461; #1, p. 444, p. 451): http://www.nku.edu/~statistics/data/c11s01e01.xls

  • Summary:
    • If the objective is to estimate the mean value of y at a particular value of x, use the confidence interval.
    • If the objective is to estimate a value of y at a particular value of x, use the prediction interval.
    • Use these intervals only if a test of hypothesis indicates that the two variables are correlated.
    • Use the linear regression results only within the range of the x values used in the regression: outside of those values, the behavior of y may change dramatically (there's no assurance that it continues to show a linear relationship).

• Section 11.5: coefficient of determination
  • The last piece of the analysis of a linear regression is the value of r², aka the coefficient of determination.

  • r² is the fraction of variability in the values of the response explained by the linear relationship (always between 0 and 1).

  • Hence 1-r² is the fraction of variability not explained in the values of the response explained by the linear relationship.

  • Recall that 1-r² appeared in the t-statistic for r:

  • Example:
    • Exercise #1, p. 480 (#1, p. 470, #3, p. 461; #1, p. 444, p. 451): http://www.nku.edu/~statistics/data/c11s01e01.xls

• Let's put it all together for one: Exercise #8, p. 487: http://www.nku.edu/~statistics/data/c11r08.xls
  • The t-test versus the f-test (ANOVA) -- one last piece of the puzzle....

• So: what have we learned together this semester?
  • What is statistical inference?

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

  • We started with 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
      • skew
      • central tendency
      • dispersion
    • Measures of dispersion, especially variance and standard deviation.
    • 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
      • central limit theorem: means look normal for large n (sampling distribution of the mean, standard errors)
      • sample proportion normal if expected plus/minus 3 standard errors is within [0,1]
    • t -- bell-shaped like a normal, but varies by degrees of freedom (need when we don't have sigma)
    • f -- skewed right, requires degrees of freedom (numerator and denominator!)
    • chi-square -- skewed right, requires degrees of freedom

  • Analysis:
    • Confidence Intervals:
      • We seek to estimate a parameter with a statistic
      • 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.
      • Proportions and means, at the outset
      • Estimating sample size needed for fixed width with given confidence
    • 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 (t-tests)
      • multiple means simultaneously
        • 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:
    • Testing for linear dependence between quantitative variables
    • scatterplot
    • 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?

• Evaluations -- can I get a volunteer? The evaluations need to be returned to the math office, room 305 in this building.

• StatCrunch
• t-table
• Z-table
• Scientific calculator