AP STATISTICS LESSON 14 – 1 (DAY 2)

1. AP STATISTICSLESSON 14 – 1(DAY 2) CONFIDENCE INTERVALS FOR THE REGRESSION LINE

2. ESSENTIAL QUESTION: How are confidence intervals and significance tests made and used for inference when working with regression lines? Objectives: • To create significance tests for regression. • To create confidence intervals for regression lines. • To make inferences dealing with regression situations.

3. Technology Tips: Here’s a quick way to calculate s: • With x-values in L1 / list1 and the y-values in L2 / list 2. • Perform least-squares regression. • The calculator creates or updates a list named RESID. Store the results in L3 /list 3. Then specify 1 – var Stat L3 and look at the value ∑ x2. • Dividing this number by (n – 2) to get s2 . • Take the square root to obtain s.

4. Confidence Intervals for the Regression Slope The slope β of the true regression line is usually the most important parameter in a regression problem. The slope is the rate of change of the mean response as the as explanatory variable increases.

5. Confidence Interval for Regression Slope A level C confidence interval for the slope the slope β of the true regression line is b ± t* SEb In this recipe, the standard error of the least-squares slope b is SEb = s √∑ ( x – x )2 And t* is the upper ( 1 – C)/2 critical value with n – 2 degrees of freedom.

7. Testing the Hypothesis of No Linear Relationship We can also test hypothesis about the slope β. The most common hypothesis is Ho : β = 0 A regression line with a slope of 0 is horizontal. That is, there is no true linear relationship between x and y. You can use the test for zero slope to test the hypothesis of zero correlation between any two quantitative variables.

8. Significance Tests for Regression Slopes To test the hypothesis Ho : β = 0, compute the t statistic t = b Seb To test the hypothesis Ho : β = 0, compute the t statistic t = b/ SEb

9. Significance tests for Regression Slopes (continued…) In terms of a random variable T having the t(n – 2) distribution, the P-value for a test of Ho against Ha : β > 0 is P( T ≥ t ) Ha : β < 0 is P( T ≤ t ) Ha : β ≠ 0 is 2P( T ≥ l t l) This test is also a test of the hypothesis that the correlation is 0 in the population.

10. Testing the Hypothesis of No Linear Relationship H0 : β = 0 A regression line with slope 0 is horizontal. That is, the mean of y does not change at all when x changes. Put another way Ho says that straight-line dependence on x is of no value for predicting y. Regression output from statistical software usually gives t and its two-sided P-value. For a one-sided test, divide the P-value in the output by 2.

11. Example 14.4 Page 789Regression Output: Crying and IQ You can find a confidence interval for the intercept a of the true regression line in the same way, using a and SEafrom the “Constant” line of the printout. We rarely need to estimate a. Minitab regression output fo the crying and IQ data

12. Example 14.5 Page 791Testing Regression Slope The hypothesis H0 : β = 0 says that crying has no straight-line relationship with IQ. Figure 14.1 shows that relationship, so it is not surprising that the computer output in Figure 14.4 gives t = 3.07 with two-sided P-value 0.004. There is very strong evidence that IQ is correlated with crying. Figure 14.4 Figure 14.1

13. Example 14.6 Page 791Beer and Blood Alcohol How well does the number of beers a student drinks predict his or her blood alcohol content? • 16 volunteers drank a randomly assigned number of beers • 30 minutes later, their blood alcohol content (BAC) was measured. The scatterplot shows one unusual point: student number 3 may also be influential , though the point is not extreme in the x direction (solid line). To verify that our results are not too dependent o n this one observation, do the regression again omitting student # 3 (dashed line).