MATH 1342 Ch. 24 April 25 and 27, 2013 Page 1 f 5 CHAPTER 24: INFERENCE IN REGRESSION Chapters 4 and 5: Relatinships between tw quantitative variables. Be able t Make a graph (scatterplt) Summarize the frm (linear r curved r ther) If it is linear, the directin (psitive r negative) the strength (hw clse the pints are t the line r curve that summarizes the pattern) If it is at least smewhat linear, cmpute and interpret a numerical summary f the strength (crrelatin cefficient) If the relatinship is at least smewhat linear, find the equatin f the best fitting line (the regressin line) Use that regressin line t make predictins f y when we are given the value fr x. Chapter 24: Make inferences abut the ppulatin frm which the sample data came. 1. Find and interpret a cnfidence interval fr the slpe cefficient. 2. Test the hypthesis f n linear relatinship between the variables (test the slpe and / r the crrelatin cefficient) 3. Find and interpret interval estimatrs fr the predicted value f y fr a given value f x. (Tw different estimatrs) 4. Discuss the cnditins needed t d regressin inference. Discuss whether these cnditins are met. D that by using (1) infrmatin in the descriptin, (2) a residual plt f the residuals versus the explanatry variable and (3) a histgram f the residuals. 5. Nte: Each f the prcesses in this chapter has an SE assciated with it. We will find ALL f these we need by reading them frm the utput. Despite the fact that ur text has frmulas fr these, we will nt cmpute them frm the frmulas. Review f Chapters 4 and 5, mainly using sftware. Activity 1: Lk at Example 24.1 in Mre s text n Crying and IQ. 1. Decide which variable is the explanatry variable and which is the respnse variable. 2. Use Minitab t a. Make a scatterplt. b. Find the crrelatin cefficient by using Stat > Basic Statistics > Crrelatin c. Find the equatin f the regressin line. 3. Interpret the slpe in cntext.
MATH 1342 Ch. 24 April 25 and 27, 2013 Page 2 f 5 4. Interpret the intercept in cntext. 5. Interpret the 2 r value in cntext. 6. Use the equatin f the regressin line t predict the IQ fr a child whse CryCunt was 10. 7. Use the Optins in the Minitab Stat > Regressin > Regressin cmmand t predict the IQ fr a child whse CryCunt was 10. Ntice that it gives yu sme intervals. We will learn t interpret these. Cpy the predictin and the intervals (with the labels CI and PI n the intervals) belw. Cnditins: Discuss the ppulatin mdel and the linear regressin equatin in the sample data. Ppulatin Sample Data (Repeated respnses f y are independent f each ther.) line µ ŷ = α + βx ŷ = a+ bx variability (standard deviatin) σ = Variatin f y values fr a given x value. It is the same fr all x values. (Fig. 24.2 n p. 590 in Estimating the parameters. ) s = sample standard deviatin. (Estimate f σ using residuals.) shape Ppulatin: y values fr a given x are nrmally distributed, with center n the line.
MATH 1342 Ch. 24 April 25 and 27, 2013 Page 3 f 5 Activity 2: In the utput frm Stat > Regressin > Regressin, find the value f s which is the estimatr f σ and the SE fr the slpe cefficient. (Example in text: Exercise 24.2 See Minitab utput. S = 7.90308, SE = 33.02) Nw, fr the CryCunt and IQ data, estimate σ and find the SE b f the slpe cefficient. (Answers: s =17.50 and SE b = 0.4870. ) Testing hypthesis f n linear relatinship between the variables. Text: See Ch. 24 Sectin Testing the hypthesis f n linear relatinship Steps: 1. Understand why the null hypthesis f beta = 0 means n linear relatinship. 2. Write hyptheses and chse yur significance level. 3. Use Stat > Regressin > Regressin t find the equatin f the regressin line and the table f values including the slpe cefficient, the SE next t the slpe cefficient, the t scre, and the P value (tw sided p value). 4. Find the t scre and P value: (Practice bth ways) Use Minitab t see these. Use the sample slpe cefficient and the SE b frm the utput. Als use df = n 2 and the t table t find the P value. When the df yu need is nt in the table, yu may use the clsest ne. Or yu may use the next smaller ne, as the textbk says. 5. Write yur cnclusin in cntext. Activity 3: D Exercise 24.5. Text: See Ch. 24 Sectin Testing lack f crrelatin. This tests exactly the same questin as the test f the slpe cefficient. But it uses the crrelatin cefficient fr the test statistic instead. When yu use Minitab t cmpute the crrelatin cefficient, it als gives yu a P value, which is fr these hyptheses: H: ppulatin crrelatin cefficient is 0 Ha: ppulatin crrelatin cefficient is nt equal t 0. Activity 4: D Exercise 24.7b using Minitab instead f using Table E. (Yu will nt be expected t use Table E at all in this class.) Use the fur step prcess.
MATH 1342 Ch. 24 April 25 and 27, 2013 Page 4 f 5 Finding and interpreting a cnfidence interval fr the slpe cefficient. Text: Cnfidence intervals fr the regressin slpe. See the frmula and Example 24.6. Ntice that the SE b needed here is the SAME ONE we used in the hypthesis test f the slpe cefficient. We use Table C t find t scres, nt sftware. When the df yu need is nt in the table, yu may use the clsest ne. Or yu may use the next smaller ne, as the textbk says. Pay careful attentin t the interpretatin. Activity 5: D bth exercises 24.9 and 24.11 in this sectin. Be sure t write an interpretatin f yur interval befre checking yur answer. Find and interpret interval estimatrs fr the predicted value f y fr a given value f x. Mre text: Chapter 23, sectin Inference abut predictin 1. Read example 23.7 abut Beer and bld alchl. 2. Lk at Minitab utput t find the cnclusin f this prblem. 3. Learn hw t distinguish between the tw intervals that are given. 4. Use Minitab t perfrm the regressin and cmpute the fitted value yu need (Yu tell Minitab the cnfidence level yu want.) That will always prvide tw intervals. (Crunch It claims t d this, but the answers dn t seem t always be accurate.) 5. Read the questin VERY CAREFULLY t see which f these is requested: a. the predictin fr ONE y value when x is that given value (wide interval, PI) b. the predictin fr the MEAN y value fr all pints where x is equal t that given value (narrw interval, CI) Activity 6: D the fllwing prblems. 24.13 and then 24.12a. Ntice that these DO NOT ask fr the same interval. Lk at the language very carefully t see hw they differ.
MATH 1342 Ch. 24 April 25 and 27, 2013 Page 5 f 5 Checking the cnditins Text: See Ch. 24 sectin Checking the cnditins fr inference There are fur things t check, listed n the first page f this sectin. The third (independent bservatins) is perhaps the trickiest ne. I will nt expect yu t d that n a test. The thers can all be checked using varius graphs. Read thse. Then review Example 24.9 (Climate change and fish) in this sectin. Activity 7: Exercise 24.15. Fr part a, d thse tw parts fr Exercise 24.2 abut this same data. Fr part b, d nt d a stemplt, but instead, chse the Graphs buttn in the Minitab Regressin Regressin and chse t d a histgram f the residuals. Fr part d, d a residual plt f the residuals versus the explanatry variable. In rder t becme skilled in interpreting all f this, yu wuld need t wrk with mre data sets than we have time t d at this pint in the curse. Yu are expected t knw what graphs are needed t check each cnditin, t prduce the apprpriate graphs, and t interpret them if the interpretatin is pretty clear. If it is brderline I will nt expect yu t necessarily interpret it in the way I wuld. Quiz 14: Due Wed. May 1 at the beginning f class. Submit a wrd prcessing dcument in this quiz assignment in Blackbard with yur slutins fr this quiz, as if it were a test questin. Include yur answers in the main part and yur cmputer utput in the Appendix. In the Appendix, identify the utput fr each f the individual prblems separately. Yu have ONE pprtunity t submit this, just as yu will fr the test. 24.44 (40 pints) In the fur step prcess, list tw questins in the state part. As yu wrk the prblem, make clear which parts are relevant t answering which questin. The fllwing parts f prblems are intended t make sure yu understand the varius types f intervals we learned in this chapter. These are NOT fur step prblems. In the slutin part f yur dcument, It is nly necessary that yu give the particular answers requested. 24.32c, 24.34, 24.38c, 24.40c