12.1 Inference for Linear Regression

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Howard Simpson
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1 12.1 Inference for Linear Regression Least Squares Regression Line y = a + bx You might want to refresh your memory of LSR lines by reviewing Chapter 3! 1

2 Sample Distribution of b p740 Shape Center Spread 2

3 Conditions for Regression Inference Suppose we have n observations over an explanatory variable x and a response variable y. Our goal is to study or predict the behavior of y for given values of x. Linear:The actual relationship between x and y is lenear. For any fixed value of x, the mean response μ y = α +βx. the slope β and the intercept α ar usually unknown parameters. Independent: Individual observations are independent of each other. Normal: For any fixed value of x, the response y varies according to a Normal distribution. Equal variance: The standard deviation of y (call it σ) is the same for all values of x. The common standard deviation σ is usually an unknown parameter. Random: The data come from a well designed random sample or randomized experiment. The acronym LINER should help you remember the conditions for inference about regression. Note that the symbols α and β here refer to the intercept and slope respectively, of the population regression line. These are not related to Type I or Type II errors. 3

4 How to Check the Conditions for Regression Inference Linear: Examine the scatterplot to check that the overall pattern is roughly linear. Look for curved patterns in the residual plot. check to see that the residuals center on the "residual =0" line at each x value in the residual plot. Independent Look at how the data were produced. Random sampling and random assignment help ensure the independence of individual observations. If sampling is done without replacement, remember to check that the population is at least 10 times as large as the sample (10% condition). Normal Make a stemplot, histogram, or Normal probability plot of the residuals and check for clear skewness or other major departures from Normality. Equal variance Look at the scatter of the residuals above and below the "residual =0" line in the residual plot. The amount of scatter should be roughly the same from the smallest to the largest x value. Random: See if the data were produced by random sampling or a randomized experiment. 4

Ex: The Helicopter Experiment p743 Check the Conditions: LINER Linear: The scatterplot shows a clear linear form. For each drop, the residuals are centered on the horizontal line 0.

5 Ex: The Helicopter Experiment p743 Check the Conditions: LINER Linear: The scatterplot shows a clear linear form. For each drop, the residuals are centered on the horizontal line 0. The residual plot shows a random scatter about the horizontal line. Independent: Released in a random order and none used twice. Normal: The histogram of the residuals is single peaked, unimodal, and somewhat bell shaped. The normal prob. plot is close to linear. Equal Variance: The residual plot shows a similar amount of scatter about the residual = 0 line. Random: Random assignment. 5

6 Estimating Parameters Ex: The Helicopter p745 The least squares regression line for these data is: flight time = (drop height) You will be asked to estimate from the above data α β σ 6

7 Constructing a Confidence Interval for the Slope t Interval for the Slope of a Least Squares Regression Line When the conditions for regression inference are met, a level C confidence interval for the slope β of the population (true) regression line is b ± t*se b In this formula, the standard error of the slope is SE b = s/(s x n 1) and t* is the critical value for the t distribution with df = n 2 having area C between t* and t* 7

8 Ex: p747 The Helicopter A confidence interval for β a) Identify the standard error of the slope SEb from SE coef column. If we repeated the random assignment many times, the slope of the sampe regression line would typically vary by about.0002 from the slope of the true regression line. b) Find the critical value for a 95% CI for the slope of the true regression line. Conditions are met. df = n 2 = 70 2 = 68. Using Table B and the conservative df = 60 t* = The 95% CI = b ± t*seb = ±2.000( ) = ± = ( , ) This interval is slightly narrower due to the more precise t*critical value. c) We are 95% confident that the interval ( , ) seconds per cm captures the slope of the true regression line relating the flight time y and the drop height x of paper helicopters. d) If we repeat the experiment many, many times, and use the method in part a to construct a conf. interval each time, about 95% of the resulting intervals will capture the slope of the true regression line relating flight time y and drop height x of paper helicopters. 8

9 Ex: Does Fidgeting Keep you Slim? p749 9

10 Performing a Significance Test for the Slope t Test for the Slope of the Population Regression Line Suppose the conditions for inference are met. To test the hypothesis H 0 : β = hypothesized value, compute the test statistic t = (b β 0 )/SE b Find the P value by calculating the probability of getting a t statistic this large or larger in the direction specified by the alternative hypothesis H a. Use the t distribution with df = n 2 H a : β > hypothesized value H a : β < hypothesized value H a : β hypothesized value 10

11 Ex: Crying and IQ p752 11

12 Ex: Magnetic Fields p755 Technology Corner p756 Regression Inference on the calculator. 12

13 Homework: P Odd P odd 13

Correlation and Regression

Correlation and Regression Scatterplots Correlation Explanatory and response variables Simple linear regression General Principles of Data Analysis First plot the data, then add numerical summaries Look