Day 23: STA205 - Introduction to Statistical Methods

Today:

• Return Quiz (basics of the t-distribution)
  • "bell-shaped" rather than "normal" curve
  • Table areas are irrelevant. It's a consequence of who made the table!

• Quiz next time over 9.1/9.2

• Section 9.1: Inferences comparing two populations
  • Our over-riding objective is to decide whether two populations share a parameter or not: e.g. is the same for both?

  • There are two kinds of samples of the populations we'll compare:
    • Dependent (paired) samples
    • Independent (unpaired) samples

  • Dependent (paired) samples:
    • husband versus wife hours in the kitchen
    • male and female graduates matched on major and GPA
    • "pre" and "post" intervention scores

  • Independent (unpaired) samples:
    • 100 students randomly chosen from sta205 at NKU versus 100 students randomly chosen from the equivalent of sta205 at EKU
    • 1000 voters randomly chosen in KY versus 900 voters randomly chosen in OH.

  • One thing that separates the two pretty clearly is when the sample sizes are different: it's almost a sure sign that the samples are independent.

  • Examples:
    • #3, p. 328
    • #5, p. 328

• Section 9.2: Inferences about two population means: independent samples
  • Notation:

    Population 1	Mean:	Variance:
    Sample 1	Mean:	Variance:	Sample size:
    Population 2	Mean:	Variance:
    Sample 2	Mean:	Variance:	Sample size:

  • We're going to be making inferences about

  • Conditions under which we may do so (p. 330):
    • Two independent random samples must be selected
    • both sample means must be normally distributed

  • Standard error of :

  • Confidence interval: usual story: point estimate plus or minus t times the standard error to where the degrees of freedom are really horrible (p. 331) -- use technology!

  • Test of hypothesis: usual story: Decision rule: Accept H_a if p-value < Test Statistic: with appropriate degrees of freedom (p. 331 -- use technology!)

  • Examples (using StatCrunch):
    • For the following exercises, do not "pool" the variances (that is done when there is reasonable suspicion that the variances of the two samples is the same -- we have no such suspicions)
    • Example 9.3, p. 332 (http://www.nku.edu/~statistics/data/exam09-03.xls)
    • #6, p. 341 (http://www.nku.edu/~statistics/data/c09s02e06.xls)
    • #10, p. 343 (http://www.nku.edu/~statistics/data/c09s02e10.xls)

• Section 9.3: Paired comparisons in estimating mean differences
  • More good news: paired comparisons will turn out to be exactly like our first encounter with estimating in a single sample. So if you understood that, great!

  • From our pairs of quantitative data, we create a new variable: the difference. So we form differences

    d1=x1-y1,

    d2=x2-y2,

    ...,

    dn=xn-yn

    and then we work with this data, this random variable d.

  • Notation:

    : Population mean

    : Population Standard Deviation

    : Sample Mean

    : Sample Standard Deviation

    : Sample size (number of pairs)

  • Then Confidence Intervals and Hypothesis Testing are just as they were for single population samples (see pp. 346-347). One modest difference is that hypothesis tests are essentially always of the form

  • Examples (using StatCrunch):
    • Example 9.8, p. 352 (http://www.nku.edu/~statistics/data/exam09-08.xls)
    • #1, p. 354 (http://www.nku.edu/~statistics/data/c09s03e01.xls)
    • #4, p. 354 (http://www.nku.edu/~statistics/data/c09s03e04.xls)

• t-table
• Z-table
• Scientific calculator