Class 40: Topic 29: Inference for Correlation and Regression

Held: Wednesday, 7 May 2008

Summary: We conclude our exploration of statistics by examining what regression lines between two variables from a sample of a population tell us about the relationships between the variables within the populations from which the sample was drawn.

Notes:

• I still hope to distribute tentative grades on Friday, but, given the number of administrative tasks that have been thrown at me this week, I'm less hopeful than I was.
• The final remains Thursday, May 15, 2-5 p.m.
• Katherine will hold a review session on Wednesday evening, May 14.
• I should be around on the morning of May 15 to answer questions. I can also hold an earlier review session.
• For the final, you may bring your own cheat sheet: One hand-written, double-sided, 8.5x11 inch (or A4) sheet of notes.
• We will stop at 29-3. (You are not responsible for 29-4 or 29-5.)
• And yes, you can ask questions before the exam-lette.
• I will be leaving early again today to pick up my youngest after school.
• There is no homework for Friday! However, I do expect you to show up.

• Handouts: Applet: Sampling Regression Lines; R Notes for Topic 29.
• Due: 28-5, 28-10, 28-24, 28-25.

Overview:

• Making inferences about populations (once again).
• Important R.

The Progression Continues: What Samples Tell us About Populations

• We've been studying the relationship between two quantitative variables taken from one sample.
• Of course, once we start to work with samples, we should ask ourselves whether or not they represent the population.
• What are our normal strategies?
• A test of significance
• A confidence interval

• Test of significance:
  • Choose a null hypothesis.
  • Choose an alternate hypothesis.
  • Using the two hypotheses and the sample, generate a test statistic.
  • Look up the test statistic to get a p-value.
• For regression models, we do all of this in terms of the slope of the regression line.
• The null hypothesis: There is no relationship between the variables (so the slope is 0).
• The alternatives: It's negative, it's positive, or it's just not zero.
• The test statistic: b/SE(b)
• The table: Use the t-table with n-2 degrees of freedom. (Why n-2? It seems to work well.)
• What should we use as the standard error? We let the computer figure it out for us.
  • (Another thing in which almost no one wants to see the formula)

• Confidence interval
  • b +/- t^*SE(b)
• How do we compute the critical value? We look it up in the t-table.
• How do we compute the standard error? We rely on our statistical software.

Getting Information from R

• So, how do we get R to tell us the standard error (and other things)?
• We continue to use lm.
• We filter the results of lm through summary.
• For example,

> TBP = read.csv("/home/rebelsky/Stats115/Data/TextbookPrices.csv")
> summary(lm(TBP$Price ~ TBP$Pages))

Call:
lm(formula = TBP$Price ~ TBP$Pages)

Residuals:
     Min       1Q   Median       3Q      Max 
-65.4748 -12.3241  -0.5838  15.3037  72.9909 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) -3.42231   10.46374  -0.327    0.746    
TBP$Pages    0.14733    0.01925   7.653 2.45e-08 ***
---
Signif. codes:  0 "***" 0.001 "**" 0.01 "*" 0.05 "." 0.1 " " 1 

Residual standard error: 29.76 on 28 degrees of freedom
Multiple R-Squared: 0.6766,	Adjusted R-squared: 0.665 
F-statistic: 58.57 on 1 and 28 DF,  p-value: 2.452e-08