Transcript 12.1
Chapter 12 More About Regression ο Letβs look at the Warm-Up first to remind ourselves what we did with regression! Remember FODS! Section 12.1 Inference for Linear Regression Least-Squares Regression fits a straight line of the form π¦ = π + ππ₯ to data to predict a response variable y from the explanatory variable x. ο Inference in this setting uses the sample regression line to estimate or test a claim about the population (true) regression line. ο ο Confidence intervals and significance tests about the slope of the population regression line are based on the sampling distribution of b, the slope of the sample regression line. Conditions - LINER ο ο ο ο ο Linear β the actual relationship between x and y is linear. For any fixed value of x, the mean response ππ₯ falls on the population (true) regression line ππ¦ = πΌ + π½π₯ β graph scatterplot Independent β individual observations are independent Normal β for any fixed value of x, the response y varies according to a Normal distribution β you will be graphing the Normal probability plot of the residuals here. Equal Variance β the standard deviation of y (call it π) is the same for all values of x β graph residual plot Random β the data are produced from a well-designed random sample or a randomized experiment. How it worksβ¦ The slope b and intercept a of the leastsquares regression line estimate the slope π½ and the intercept πΌ of the population (true) regression line. ο To estimate π, use the standard deviation of s of the residuals. ο Confidence Intervals Confidence intervals and significance tests for the slope π½ of the population regression line are based on a t distribution with n-2 degrees of freedom. ο n-2 because we have two lists β we need to allow another degree of freedom for the extra variableβ¦ ο The t interval for the slope π½ has the form π ± π‘ β ππΈπ ο The standard error of the slope is ππΈπ = π ο π π₯ πβ1 Letβs look at SEβ¦ ο ππΈπ = π π π₯ πβ1 πππ πππ’πππ 2 πβ2 ο π = ο On the AP formula sheet: ο π π = (π¦βπ¦)2 πβ2 (π₯βπ₯)2 = (π¦βπ¦)2 πβ2 Hypothesis Tests ο ο To test the null hypothesis, carry out a t test for the slope. Use π = πβπ·π πΊπ¬π (π»0 : π½ = βπ¦πππ‘βππ ππ§ππ π£πππ’π) ο The most common null hypothesis is π»0 : π½ = 0, which says that there is no linear relationship between x and y in the population. ο Letβs do a confidence interval! We examined data from a study that investigated why some people donβt gain weight even when they overeat. Researchers deliberately overfed a random sample of 16 healthy young adults for 8 weeks. They measured fat gain and change in energy use from activity other than deliberate exercise (non-exercise activity, NEA) β fidgeting, daily living, etc β for each subject. Here are the results: NEA Change (cal) -94 -57 -29 135 143 151 245 355 Fat Gain (kg) 4.2 3.0 3.7 2.7 3.2 3.6 2.4 1.3 NEA Change (cal) 392 473 486 535 571 580 620 690 Fat Gain (kg) 3.8 1.7 1.6 2.2 1.0 0.4 2.3 1.1 Construct and interpret a 90% confidence interval for the slope of the population regression line. Check conditions first! Type information into calculator! ο Linear β look at scatterplot and draw it to prove that you have checked this condition. ο Independent β ο Normal β look at Normal probability plot of residuals and draw it to prove you checked this condition. (find the LinReg first and then do NPP with RESID β 2nd list) Keep checking conditionsβ¦ ο Equal Variance β we want the standard deviation (the average distance from the mean β or 0) to be the same for all points β draw the residual plot to prove that you have looked at it. ο Random β Do: A Significance Testβ¦ Infants who cry easily may be more easily stimulated than others. This may be a sign of higher IQ. Researchers explored the relationship between crying infants 4 to 10 days old and their later IQ scores. The researchers flicked the infants with a rubber band and recorded the crying. They measured its intensity by the number of peaks in the most active 20 seconds. The table below contains data from a random sample of 38 infants. a) Here is a scatterplot of the data with the least-squares regression line added. Describe what this graph tells you about the relationship between these two variables. b) Using the min-tab output, what is the equation of the least-squares regression line? c) Interpret slope and y-intercept of the regression line in context d) Do these data provide convincing evidence that there is a positive linear relationship between crying counts and IQ scores in the population of infants? Homework ο Pg 759 (6, 8, 13-15, 18-26)