Worksheet 23 ( ) Introduction to Simple Linear Regression (continued)

Transcription

1 Worksheet 3 ( ) Itroductio to Simple Liear Regressio (cotiued) This worksheet is a cotiuatio of Discussio Sheet 3; please complete that discussio sheet first if you have ot already doe so. This worksheet cotais material from Sectios 11.5 through 11.8 of the textbook. Please read those sectios. Cosider two variables: x, the temperature 10's of degrees Fahreheit for a maufacturig process, ad y, a measure of the yield of the process. Suppose that the followig sample of data for x ad s provided: x y A scatter plot of these data with the SLR regressio lie looks like the followig: Y X Cofidece Iterval for β 1 A 100( 1 α ) % cofidece iterval for β 1 is of the form s b 1 ± t ν, α SS xx where ν, s is the residual mea square error from the ANOVA table, ad SS xs what has already bee computed i fidig b Fid a 95 percet cofidece iterval for the β 1 of the data with which we have bee workig. 1

2 The F-distributio Hypothesis Test (F-Test) for the Sigificace of a SLR Model The followig is a test at the 100( 1 α ) % sigificace level for whether or ot a SLR model is statistically sigificat, that is, that is does or does ot happe by chace because of the particular sample that is chose: H 0 : The model is ot sigificat. H 1 : The model is sigificat. Test statistic : F MS reg s Rejectio regio : F > F ν,ν d,α where ν df reg umber of predictor variables 1 ad ν d df residual.. Perform the F-test at the 95% sigificace level to see if the SLR model for the data that we have bee usig is statistically sigificat or ot. 3. What does this hypothesis test tell us? Two-sided Hypothesis Test for β 1 A 100( 1 α ) % hypothesis test of whether a parameter β 1 equals a certai umerical value, call it β 10, has the form: H 0 : β 1 β 10 H 1 : β 1 β 10 Test statistic : t b 1 β 10 s SS xx Rejectio Regio : t > t ν, α (where ν ) Coclusio : Not reject H 0 or reject H 0 (accept H 1 as ew workig hyp.) Probability that the coclusio is correct : 1- a 4. Perform a hypothesis test at the 95% cofidece level to see whether or ot β 1 equals 0.

3 5. What would it mea i terms of the SLR model if β 1 equals 0? What would this tell you about whether or ot x helps to reduce the ucertait predictig y? 6. How ca this be used as a test of whether the SLR model is statistically sigificat or ot? Relatio of F-test for the SLR Model ad the t-test for β 1 0 The hypotheses for the two tests for the SLR model are equivalet. Further F t ad the F or t values for the rejectio regios are similarly related. 7. Verify that the relatio betwee F ad t give i the box above is true usig the F from 7 ad the t from 9. 3

4 Correlatio of Two Variables Recall that the defiitio of correlatio for a populatio 1s: Cov(x, y) ρ XY σ x σ y For a sample the two variables would have a correlatio defied as: cov(x, y) r xy s x s y x ) y x ) 1 x ) y x ) ( ) 1 ( y ) ( ) ( y ) 1 SS xy SS xx SS yy 8. Calculate the sample correlatio, r xy, for the data we have bee usig. 9. What does this correlatio coefficiet, r XY sigify? 4

5 Relatio of the Correlatio Coefficiet ad the Coefficiet of Determiatio for SLR Models For SLR models i which r xs the sample correlatio coefficiet of the two variables ivolved ad R is the coefficiet of determiatio for the SLR model the we have ( r xy ) R Note: This is true oly for SLR models. With more tha oe predictor variable (more tha oe X) as i multiple regressio, the coefficiet of determiatio, R, is calculated as before but there is o sigle r XY to be related to it. 10. You have calculated r x #9 above ad you have calculated R i a previous worksheet for the data that we have bee usig. Verify that r xy ( ) R withi roud-off error for these data. Relatio of Slope Coefficiet b 1 ad Correlatio Coefficiet r XY There is a systematic relatioship betwee the slope coefficiet b 1 for a SLR model ad for the sample correlatio coefficiet r xy for the two variables ivolved. It ca be show that: b 1 s y s x r xy where s s the sample stadard deviatio of y ad s s the sample stadard deviatio of x. 11. You have foud b 1 for the data with which we have bee workig ad you foud r xy. Verify that the relatio i the box above holds for these data. Suggested Homework: 11.18abcd, 11.19, 11.5, 11.6, 11.8, 11.35abcd, 11.37, Solutios to be Posted: 11.18ac, 11.5, 11.8, 11.35ac,