- Quiz over the F-test and ANOVA (returned)
- In general, as a group, you have no idea what ANOVA is, why it's useful, or when it can be carried out.
- Every statistic, every test, tells a story: your job is to figure out what the story is. Last time in class I sang the song "One of these things is not like the others", from Sesame Street: this begins to tell the story. You need to fill in the details.
- A student came to me with a statistic: that 83.4/1000
Americans is hearing impaired. I asked what the standard error
is, and it wasn't given. Without that, we only have a point
estimate. We can say that someone thinks that about
25 million = 83.4/1000*300 million Americans are hearing-impaired. But if we knew the sample size of the study that produced that estimate, then we'd be able to give a range of Americans. For example, if it were n=1000 people, then the 95% confidence interval would be(0.066,0.101) meaning that, with 95% confidence, somewhere between about 20-30 million Americans are hearing impaired. - Quiz over Anova results
- Note: Tutoring Workshops (review for final)
- Sun April 26: 1pm-4pm, SU107B
- Thu April 30: 2:30pm-4:30pm, SU108
- I'm planning on doing review on Monday night, 5/4. 7:00 pm-9:00 pm.
- 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)
Decision rule: Accept Ha if
Test Statistic:
- Conditions of using a chi-square test (p. 496):
- The sampled data must be obtained by
- Selecting a random sample of the population of interest, or
- 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 sampled data must be obtained by
- 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
, and the standard deviation is
- Is skewed right:
- 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.
- 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
- In contrast to chapter 12, where we attempted to determine if two
qualitative variables were dependent, in chapter 11 we look at the relationship
between two quantitative variables, and attempt to decide if it is
linear.
- We use a "scatterplot" to represent the two variables:
- Interpreting a scatterplot:
- 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"
- We use the correlation coefficient (denoted "r") to
characterize the strength of the linear relationship:
- See the image above!
- Always between -1 and 1
-
- correlation coefficient test: A test for linear dependence of two
variables (p. 446)
Decision rule: Accept Ha if
Test Statistic:
where n is the number of pairs, and df=n-2.
- Conditions for using the test for correlation (p. 447)
- 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.
- Examples:
- 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
- STA205 on-line correlation and regression example (here's the data)
