STATISTICS QUESTIONS. Step by Step Solutions.

Transcription

1 STATISTICS QUESTIONS Step by Step Solutons 9//016

2 Problem 1: A researcher s nterested n the effects of famly sze on delnquency for a group of offenders and examnes famles wth one to four chldren. She obtans a sample of 16 famles, four of each sze, and dentfes the number of arrests per chld for delnquency. The data s as follows: Group 1 chldren n= Group 3 chldren n= Group 3 chldren n= Group 1 chld n= Famly Famly Famly Famly a) Calculate the total sum of squares. b) Calculate the mean square (between groups). c) Calculate the F-rato d) Use the Turkey HSD (alpha=0.05) to test for sgnfcance between groups. Whch groups dffered? e) Based on your results, wrte a 1- paragraph essay that descrbes your observatons obtaned from ths sample n regard to the effects of famly sze on delnquency for a group of offenders. Soluton: (a) The followng table wth descrptve statstcs s obtaned from the nformaton provded

3 Obs. Group 1 Group Group 3 Group Mean St. Dev We need to test H : H A : Not all the means are equal Wth the data found n the table above, we can compute the followng values, whch are needed to construct the ANOVA table. We have: Between 1 k SS n x x from whch we get

4 SS Between Now we also see that, k 1 SS n s Wthn 1 whch mples Wthn SS Hence, SS Total = 78+3 = 101 (b) Therefore MS Between SSBetween 78 6 k 1 3 Also, we obtan that

5 MS Wthn SSWthn N k 1 1 (c) Therefore, the F-statstcs s computed as F MSBetween MS Wthn The crtcal value for 0.05, df1 3 and df 1 s gven by F C and the correspondng p-value s p Pr F ,1 Observed that the p-value s less than the sgnfcance level 0.05, then we reject H 0. (d) The HSD dfference s computed as follows: MSE HSD Q*.0.91 n

6 The followng table s obtaned: Post hoc analyss Tukey smultaneous comparson t-values (d.f. = 1) Group Group 3 Group Group Group 3.3 Group Group Group crtcal values for expermentwse error rate: (e) Based on the above results, we have enough evdence to reject the null hypothess of equal means, at the 0.05 sgnfcance level. Summarzng, we have the followng ANOVA table: Source SS df MS F p-value Crt. F Between Groups Wthn Groups Total

7 The parwse dfferences that are sgnfcant are between Group 1 and Group, Group and Group, and Group 3 and Group. In fact, the mean for Group s sgnfcantly lower when compared to the means for groups 1, and 3, respectvely. Problem : Move Success. Usng the data n Table 7., make a scatter dagram for the relatonshp between producton budget and vewer ratng of moves. Estmate the correlaton coeffcent. Based on these data, do you thnk a large producton budget s lkely to result n a move wth a hgh vewer ratng? Explan. Soluton: The scatter plot s shown below. Scatterplot of Ratng vs Budget 9 8 Ratng Budget It seems lke there's a mld negatve lnear relatonshp between Budget and Ratng. The actual correlaton coeffcent s computed as

8 As predcted by the vsual trend, the correlaton s negatve, but snce t's very small, the relatonshp s farly weak. Ths means that s not certan that a larger budget wll produce a hgher ratng, as t's not certan that a larger budget wll produce a lower ratng, but there a nclnaton to have lower ratng wth hgher budgets. Problem 3: Whch of these models s a better representaton of the relatonshp between students age and startng salary? Explan your decson. Soluton: As mentoned n the prevous part, the model obtaned once the outler was elmnated s relatvely smlar to the model wth n=5 cases, as the regresson coeffcents don't change dramatcally. But stll ths relatvely small dfference n coeffcents makes a relatvely large dfference n R^. In fact, for the model wth n = 5 we get R = 0.33, and for the model wth n = 5 we get R = 0.7. Ths makes the second model (wth n = ) the preferred one. The preferred model s Startng Salary^ = -67, , * Age Problem :

9 Compute an mprsonment rate per 1000 populaton for 000. Introduce ths ncarceraton rate as an ndependent varable nto the model run n Part B. Test the hypothess that the R squared =0. Does ths model ft the data better than the model n Part B above? Explan. Does each of the ndependent varables have a statstcally sgnfcant effect on homcde? Explan. How strong s the effect of each of the ndependent varables? Explan. Whch of the ndependent varables has the stronger effect on the homcde rate? Explan. Soluton: The new varable s computed as ImprPer1000 = Prson0/pop0 (let us recall that pop0 s already gven n 1000 s). The followng s obtaned wth Excel: Regresson Analyss R² 0.99 Adjusted R² 0.66 n 9 R k 3

10 Std. Error Dep. Var. homrt0 ANOVA table Source SS df MS F p-value Regresson E-07 Resdual Total Regresson output confdence nterval varables coeffcents std. error t (df=5) p-value 95% lower 95% upper std. coeff. Intercept ImprPer sglmom unempl The model s Homcde Rate n 000 = * ImprPer * sglmom * unempl0 Notce that the model s sgnfcant overall, snce F(3, 5) = 1.95, p = < 0.05, so then R s sgnfcantly greater than zero.

11 Ths model fts only slghtly better than the prevous one, snce now Adj. R = 0.66, whch means that n ths case the amount of explaned varaton n the response varable by ths model s 6.6%. Notce that n ths model, the varable sglmom80 s ndvdually sgnfcant, wth t =.77 and p = < 0.05, but the varable uempl0 s not ndvdually sgnfcant, t = 1.35, p = > The varable ImprPer1000 s not sgnfcant ether, snce t = 1.3, p = > The effect of ImprPer1000 and uempl0 s qute moderate snce the standardzed coeffcents assocated to them are less than 0. (ths s, an ncrease n one standard devaton n ether of the varables brngs a change of less than 0. standard devatons n the response varable). The varable wth the strongest effect s sglmom80, wth a standardzed coeffcent of Problem 5: Usng the data below, answer the followng questons usng a table format. X y 5-1 a. x b. y c. x y d. Show that x. y x y x y x y x x e. f. g. h. Show that. Show that ( x x ) ( y y ) 0 1 1

12 Soluton: We have: X Y X^ Y^ X*Y Sum = (a) 1 x 0 (b) 1 y 10 (c) 1 x y 57 (d) Notce that x y 57, and x y x y x y n ths case., whch means that

13 (e) 1 x (f) 1 y 6 (g) x y (h) x , and 1 x 110, so then x 1 1 x () we get that X 5, Y.5. Observe that X Y X-Xbar Y-Ybar Sum = 0 0 so then

14 ( x x ) ( y y )