Day 25: STA205 - Introduction to Statistical Methods

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Today:

Announcements:
- Quiz over inferences about two population means returned.
- Quiz over the F-test and ANOVA
- Note: Tutoring Workshops (review for final)
  - Sun April 26: 1pm-4pm, SU107B
  - Thu April 30: 2:30pm-4:30pm, SU108
Section 10.2: the F test
- Analysis of Variance (ANOVA) F-test: A comparison of k population means (p. 403)
  1. ${H_0:\mu_1=\mu_2=\ldots=\mu_k}$
  2. ${H_a: \mbox{at least two of the population means differ}}$
  3. Decision rule: Accept H_a if
    ${\mbox{p-value}<\alpha}$
  4. Test Statistic:
    ${F=\frac{\mbox{Variability among the sample means}}{\mbox{Variability expected by chance}}}$
- 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:
  - Let's review our example of 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). The "Tukey HSD with confidence level" option in StatCrunch effectively gives us the same information as the "multiple comparisons" results in the handout (HSD stands for "Honestly Significant Differences").
- The Up-shot: the more tests (comparisons) that are carried out, the more likely that one will be significant by chance (probability of a type I error)
Section 12.1: Comparing two qualitative variables: the chi-square test
- How could we determine if men, women, and children have the same opinion of four breakfast cereals? We might do a survey, collect data, and then make a statistical answer. What would lead us to say that they have the same opinion?
- One approach:
  Data is summarized in a contingency table, with one variable's levels across the top (e.g. men, women, children) and with the other variable's levels across the side (e.g. Cereal A, Cereal B, Cereal C, Cereal D). The cells might contain the counts of those who pick that cereal as the best (ties not permitted).
  
  Men Women Children
  
  Cereal A 9 12 15
  
  Cereal B 25 30 19
  
  Cereal C 35 25 20
  
  Cereal D 31 33 46
- We might also include row and column totals in the table. There will be some dependency in the counts.
  
  Men Women Children Total
  
  Cereal A 9 12 15 36
  
  Cereal B 25 30 19 74
  
  Cereal C 35 25 20 80
  
  Cereal D 31 33 46 110
  
  Total6.4 100 100 100 300
- Chi-square test: A test for dependence of two variables (p. 495)
  ${H_0: \mbox{The two variables are independent.}}$
  ${H_a: \mbox{The two variables are dependent.}}$
  Decision rule: Accept H_a if ${\mbox{p-value}<\alpha}$
  Test Statistic: ${\chi^2=\sum \frac{\mbox{(Observed Counts-Expected Counts)}^2}{\mbox{Expected Counts}}}$
- Conditions of using a chi-square test (p. 496):
  - The sampled data must be obtained by
    1. Selecting a random sample of the population of interest, or
    2. Selecting random and independent samples for each level of one of the variables.
  - All expected cell counts must be greater than or equal to 5.
- The chi-square distribution
  - Is skewed right:
  - Calculation requires knowing degrees of freedom ("k" in the plot above; df=(R-1)(C-1), where R is the number of levels of one variable and C is the number of levels of the other)
  - Large values of chi-square signify that it's unlikely that the the variables are independent (reject the null!). As you can see, p-values are in the right tail area:
  - To get an idea of what is "large", the mean of the chi-square is ${\mu=df}$ , and the standard deviation is ${\sqrt{2*df}}$
- Definition (p. 492): the qualitative variables A and B are independent if the percentage of responses at each level of Variable A is the same for every level of Variable B. Otherwise the variables are dependent.
- Let's look at the web example to get started: example from website (use this data to reproduce the results in StatCrunch)
- Here's how to do a chi-square in StatCrunch (using this data)
- Here's the cereal example data
- Important note: dependency does not mean that one variable determines the other.
- A look at Example 12.1, p. 496, in StatCrunch.
Section 12.2: Describing the relationship: sample percentages
- The web example above leads us into section 12.2. Once the chi-square test indicates that there is dependence in the variables, we want to have an easy way to describe it, and sample percentages is the easiest way.
- Note: we do not perform the chi-square test on the percentages, but rather on the actual data collected instead. If we do not detect a dependency, then we do not move to the discussion of the differences in percentages.
- Back to the web example: example from website (use this data to reproduce the results in StatCrunch)
- Let's return to Exercise 1, page 503/509: http://www.nku.edu/~statistics/data/c12s01e01.xls