Day 24: STA205 - Introduction to Statistical Methods

Today:

• Quiz over 9.1/9.2

• Amazingly enough, our next test is next week! It will cover section 8.2 through 10.2/3 -- the F test (which we're working on today)

• 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!

    (Compare pages 346-347 with 290.)

  • 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)

• Section 10.2: the F test
  • Analysis of Variance (ANOVA) F-test: A comparison of k population means (p. 403) Decision rule: Accept H_a if p-value < Test Statistic:

  • Conditions of using an F-test (p. 404):
    • Completely randomized design to collect the k samples
    • All k populations are normally distributed
    • All k populations have the same (unknown) standard deviation

    Even so, ANOVA is robust -- even if some of the conditions above aren't met, it's liable to produce reasonable p-values.

  • The F-distribution
    • Is skewed right:
    • Calculation requires knowing degrees of freedom, numerator and denominator -- yikes!;) You won't have to worry about this if you use technology.... The curves above correspond to different pairs of df, and the .05 rejection region.
    • Large values of F signify that it's unlikely that the means are the same (reject the null!). As you can see, p-values are in the tail area:

  • Examples:
    • Example 10.2, using StatCrunch: http://www.nku.edu/~statistics/data/exam10-02.xls
    • #2, p. 413 (with handout): http://www.nku.edu/~statistics/data/c10s02e02.xls
    • M&Ms and the F-test
    • #1, p. 413: http://www.nku.edu/~statistics/data/c10s02e01.xls
    • A t-test is equivalent to an F-test (gives the same p-value): #6, p. 341 (http://www.nku.edu/~statistics/data/c09s02e06.xls)

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