Assessmet ad Modelig of Forests FR 48 Sprig Assigmet Solutios. The first part of the questio asked that you calculate the average, stadard deviatio, coefficiet of variatio, ad 9% cofidece iterval of the huter success data for Oracle Juctio ad Piacle Peak. Start with calculatio of the followig statistics: Statistic Oracle Juctio Piacle Peak 7 7 9.3.57 38.358 7.543 Use,, ad to calculate the followig statistics: Statistic Formula Oracle Juctio Piacle Peak Average or sample mea Stadard deviatio s 4.9 ( ) s.6 3.8. Coefficiet of variatio CV 38.43 % 3.55 % Stadard error of the mea Cofidece iterval s s t ± ; where t is s.943 from Appedi Table 6.6 3. to 5.37.38.34 to 3.8 All sigificat digits were carried throughout the calculatios. Roudig occurred at the ed whe the values were reported. A t-value of.943 was used i the calculatio of the cofidece iterval. Degrees of freedom (df) were 6 (sample size ). The probability was. (.9). The value was foud i Appedi Table 6. The secod part of the questio dealt with estimatig mea success ad stadard error for two huters i Oracle Juctio. The tetbook i sectio -7 Epasio of Meas ad Stadard Errors gives directio o how to do the calculatios. The rule to remember is that epasio of sample meas must be accompaied by a similar epasio of stadard errors. (p. ) Multiply the mea ad stadard error for oe huter by two to estimate the mea ad stadard error of hutig parties ( huters).
Statistic Mea Stadard error Hutig parties i Oracle Juctio 8.38. The calculatios doe by had were checked usig Ecel. Formulas used to calculate Oracle Juctio statistics are show i colum D. Ecel fuctios used iclude COUNT, SUM, AVERAGE, STDEV, ad TINV. The TINV fuctio provides t-values for a give probability ad degrees of freedom.. The first part of the questio asked for calculatio by had of the itercept ad slope, correlatio coefficiet, ad stadard error about the regressio of visitor hours (Y) o traffic couter readig (X). Start by calculatig the followig 5 statistics: Statistic Value 6 759 X X Y Y XY 57737 4 5687 963698
Use, X, X, Y, ad cross products. Y, ad XY to calculate the corrected sum of squares Statistic Formula Value Corrected sum of squares for Y Corrected sum of squares for X Corrected sum of cross products SP y ( Y ) SS y Y 758674.4375 ( X ) SS X 776.9375 ( X )( Y ) XY 384547.35 Regressio estimates are the estimated usig the follow formulas: Regressio estimates Formula Value Slope b Itercept b Coefficiet of determiatio (correlatio coefficiet) Stadard error about the regressio The fitted equatio is b SPy b SS Y b ( SPy ) r SS SS.77 X -6.9.9 SS Visitor hours -6.9 +.77 traffic readig couter S y y y ( SP ) SS y 6.9 Visitor hours for a traffic readig couter of 35 is -6.9 +.77*35 493 visitor hours. These calculatios were checked agaist Ecel regressio results. First a scatter plot of visitor hours versus traffic couter readig was made. This is geerally a good first step whe eplorig the relatioship betwee two variables. 3
6 4 Visitor Hours 8 6 4 5 5 5 3 Traffic Couter Readig The assumptio of a liear relatioship betwee visitor hours ad traffic couter readig appears to be valid. More data poits i ecess of traffic couter readig would be beeficial to icrease cofidece that a liear model is appropriate. The regressio of visitor hours o traffic couter readig i Ecel produced the followig results: SUMMARY OUTPUT Regressio Statistics Multiple R.9598 R Square.9464 Adjusted R Square.89746 Stadard Error 6.899 Observatios 6 ANOVA df SS MS F Sigificace F Regressio 68799 68799 3.354.6E-8 Residual 4 7765.3 5483.4 Total 5 758674 Coefficiets Stadard Error t Stat P-value Itercept -6.88 9.48595 -.3644.9397 Traffic Couter.775.53967.49937.6E-8 Estimates for the slope, itercept, stadard error about the regressio, ad coefficiet of determiatio are cosistet with the estimates calculated by had (a good sig!). Stadardized residuals versus predicted values are graphed below: 4
Stadardized Residuals.5.5.5 -.5 - -.5 - -.5 5 5 5 3 Predicted Values The residuals rage betwee ad ; a good sig as a very large residual is idicative of a outlier. More data poits would be beeficial i testig the assumptios of liearity ad costat variace. The secod part of the questio asked that the simple liear regressio assumig a slope of zero be calculated. The slope is calculated from the followig equatio: b XY X.6 Ruig the regressio assumig a itercept of zero i Ecel produces the followig results: SUMMARY OUTPUT Regressio Statistics Multiple R.944 R Square.89534 Adjusted R Square.84867 Stadard Error 33.348 Observatios 6 ANOVA df SS MS F Sigificace F Regressio 6768 6768 3.99.5E-8 Residual 5 8666.8 5444.45 Total 6 758674 Coefficiets Stadard Error t Stat P-value Itercept #N/A #N/A #N/A Traffic Couter.64599.977 6.5398 4.9E- 5
Number of visitor hours for a traffic readig couter of 35 is.6*35 56 visitor hours. The first model (with itercept) estimated 493 visitor hours, 69 visitor hours less tha the secod model (itercept of ). Stadard error about the regressio was slightly lower i the first model (with itercept) tha the secod model (itercept of ), which suggests that the first model is a better fit. However, closer ispectio of the first model reveals that the itercept is ot sigificat. The models are similar eough that it is difficult to distiguish oe as better. 6