Day 23: STA205 - Introduction to Statistical Methods

• Announcements:
  • Quiz over T-Test

  • What's in store for the rest of the course: three topics, namely
    • F-tests (comparing means of more than two populations)
    • Chi-square statistics
    • Linear regression
    Today: the F-test

• 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 made inferences about :

  • By the way: the T-test for two independent samples is a special case of the F-test (if we use pooled variances -- that is, if we can assume that ).

• 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
    • #1, p. 354
    • #4, p. 354

• Section 10.2: the F test
  • Analysis of Variance (ANOVA) F-test: A comparison of k population means (p. 403)
    1. 

    2. 

    3. Decision rule: Accept H_a if

       

    4. 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!;)
      • The numerator degrees of freedom refer to the number k of populations (e.g. treatments) under consideration (df=k-1)
      • The denominator degrees of freedom refer to the difference in the number N of samples of all the populations under consideration, minus the number k of populations (df=N-k)
      You won't have to worry about this if you use technology.... The curves above correspond to different pairs of degrees of freedom given, 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:
    • on-line example with nice graphs
    • Example 10.2, using StatCrunch
    • #2, p. 413 (with handout)
    • M&Ms and the F-test
    • #1, p. 413
    • A t-test is equivalent to an F-test (gives the same p-value): #6, p. 341

• Section 10.3: Multiple Comparisons
  • The F-test indicates that there is at least one pair of means that differ: what now? Multiple comparisons....

  • We do a t-test on each pair (see page 417), although you won't be required to do these. The analysis table will be presented, as in the following example.

  • Example: #2, p. 413 (with handout)

  • Important point: the more tests (comparisons) that are carried out, the more likely that one will be significant by chance (probability of a type I error)

Links

• A nice example of two-way anova (we're not going to study this, however....)
• t-table
• Z-table
• Scientific calculator