Day 28: STA205 - Introduction to Statistical Methods

• Thu April 30: 2:30pm-4:30pm, SU108

• Posted.
• You are to work alone.
• You will be asked to sign that you carried out the analysis entirely on your own.
• The results will be incorporated into your final exam, and constitute 25% of your exam. It is due with your final.
• The take-home component

• positive linear relationship (as x increases, y increases)
• negative linear relationship (as x increases, y decreases)
• no linear relationship apparent
• Your conclusions should be drawn only for values of the independent variable (X) between the limits of the data values for X.
• Write your interpretation in terms of the variables of interest, not "X" and "Y"

• See the image above!
• Always between -1 and 1
• 

Decision rule: Accept H_a if

Test Statistic:

where n is the number of pairs, and df=n-2.

• At each value of x, the distribution of the y values in the population is normal, with standard deviation independent of x (see Figure 11.5, p. 447)
• The sampled values of y are independent of one another.

• Something too cool not to share: Gapminder
  • Life Expectancy versus Income per person
  • Life Expectancy versus Children per Woman

• STA205 on-line correlation and regression example (here's the data)

• Independent variables should give a "shotgun" appearance to the scatterplot: example, two independent normal variables.

• Let's look at the book example, Example 11.2, p. 437, in StatCrunch: http://www.nku.edu/~statistics/data/exam11-02.xls. Is that relationship "not extremely strong"?

• Book exercise, #1, p. 444 and p. 451: http://www.nku.edu/~statistics/data/c11s01e01.xls

• 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

  Influential points in a regression

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

• 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).

• 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

• 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?