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- Olympic Pole Vault The graph in Figure 7 indicates that in recent years the winning Olympic men’s pole vault height has fallen below the value predicted by the regression line in Example 2. This might have occurred because when the pole vault was a new event there was much room for improvement in vaulters’ performances, whereas now even the best training can produce only incremental advances. Let’s see whether concentrating on more recent results gives a better predictor of future records. (a) Use the data in Table 2 (page 176) to complete the table of winning pole vault heights shown in the margin. (Note that we are using x=0 to correspond to the year 1972, where this restricted data set begins.) (b) Find the regression line for the data in part ‚(a). (c) Plot the data and the regression line on the same axes. Does the regression line seem to provide a good model for the data? (d) What does the regression line predict as the winning pole vault height for the 2012 Olympics? Compare this predicted value to the actual 2012 winning height of 5.97 m, as described on page 177. Has this new regression line provided a better prediction than the line in Example 2?Consider the following computer output of a multiple regression analysis relating annual salary to years of education and years of work experience. Regression Statistics Multiple R 0.73520.7352 R Square 0.54050.5405 Adjusted R Square 0.52050.5205 Standard Error 2131.18202131.1820 Observations 4949 ANOVA dfdf SSSS MSMS F� Significance F� Regression 22 245,793,126.4218245,793,126.4218 122,896,563.2109122,896,563.2109 27.058227.0582 1.7E-081.7E-08 Residual 4646 208,929,085.5374208,929,085.5374 4,541,936.64214,541,936.6421 Total 4848 454,722,211.9592454,722,211.9592 Coefficients Standard Error t� Stat P-value Lower 95%95% Upper 95%95% Intercept 14268.6823614268.68236 2,521.08442,521.0844 5.65975.6597 0.0000009340.000000934 9194.00279194.0027 19,343.362119,343.3621 Education (Years) 2352.26982352.2698 337.1115337.1115 6.97776.9777 0.000000010.00000001 1673.69951673.6995 3030.84013030.8401 Experience (Years) 832.2096832.2096…Consider the following computer output of a multiple regression analysis relating annual salary to years of education and years of work experience. Regression Statistics Multiple R 0.73650.7365 R Square 0.54240.5424 Adjusted R Square 0.52250.5225 Standard Error 2124.60962124.6096 Observations 4949 ANOVA dfdf SSSS MSMS F� Significance F� Regression 22 246,127,958.1791246,127,958.1791 123,063,979.0896123,063,979.0896 27.262927.2629 1.6E-081.6E-08 Residual 4646 207,642,442.8821207,642,442.8821 4,513,966.14964,513,966.1496 Total 4848 453,770,401.0612453,770,401.0612 Coefficients Standard Error t� Stat P-value Lower 95%95% Upper 95%95% Intercept 14256.268814256.2688 2,513.30952,513.3095 5.67235.6723 0.0000008950.000000895 9197.23929197.2392 19,315.298419,315.2984 Education (Years) 2353.85412353.8541 336.0719336.0719 7.00407.0040 0.0000000090.000000009 1677.37651677.3765 3030.33173030.3317 Experience (Years) 832.8371832.8371…
- Consider the following computer output from a multiple regression analysis relating the price of a used car to the variables: age of car, mileage, and safety rating. Coefficients Coefficients Standard Error t� Stat P-value Intercept 42465.6942465.69 5320.545320.54 7.9817.981 0.00000.0000 Age (Year) −21096.02−21096.02 2551.522551.52 −8.268−8.268 0.00000.0000 Mileage(in Thousands) −1312.73−1312.73 103.02103.02 −12.743−12.743 0.00000.0000 Safety Rating 1533.821533.82 165.72165.72 9.2559.255 0.00000.0000 Does the sign of the coefficient for the variable safety rating make sense?The following data is a regression model where the U.S. Department of Transportation has tried to relate the rate of fatal traffic accidents (per 1000 licenses) to the percentage of motorists under the age of 21. Data has been collected for 42 major cities in the United States. SUMMARY OUTPUT Regression Statistics Multiple R 0.83938748 R Square 0.70457134 Adjusted R Square 0.69718562 Standard Error 0.58935028 Observations 42 ANOVA df SS MS F Regression 1 33.13441764 33.1344 95.3964 Residual 40 13.89335048 0.34733 Total 41 47.02776812 Coefficients Standard Error t Stat P-value Intercept -1.5974138 0.371671454 -4.2979 0.00010 Percent Under 21 0.28705317 0.029389769 9.76711 3.79E-9…If the standard error of the estimate for a regression model fitted to a large number of paired observations is 1.75, approximately 95% of the residuals would lie within ______. −3.50 and +3.50 −1.75 and +1.75 −0.95 and +0.95 −0.68 and +0.68 −0.97 and +0.97
- Calculate the R2of the following multivariate sample regression functions and interpret theanswers.3.1 Investment-hat = β1-hat + β2-hat*Interest rate + β3-hat*Exchange rateESS = 900RSS = 1003.2 Investment-hat = β1 + β2-hat*Interest rate + β3-hat*number of 311 studentsESS = 400RSS = 6003.3 Salary-hat = β1 + β2-hat*Frequency of blinking eyes + β3-hat*Colour of hairRSS = 950TSS = 1000A part of the output of a regression analysis of Y against X using Excel is given below:SUMMARY OUTPUTRegression StatisticsMultiple R 0.954704R Square 0.91146Adjusted R Square 0.896703Standard Error 28.98954Observations 8ANOVAdf SS MS F Significance FRegression 1 51907.64 51907.64Residual 6 5042.361 840.3936Total 7 56950Coefficients Standard Error t Stat P-valueIntercept 45.2159 39.8049Age 5.3265 0.6777a. State the estimated regression line and interpret the slope coefficient.The following is a partial computer output of a multiple regression analysis of a data set containing 20 sets of observations on the dependent variableThe regression equation isSALEPRIC = 1470 + 0.814 LANDVAL + 0.820 IMPROVAL + 13.5 AREA Predictor Coef SE Coef T P Constant 1470 5746 0.26 0.801 LANDVAL 0.8145 0.5122 1.59 0.131 IMPROVAL 0.8204 0.2112 3.88 0.0001 AREA 13.529 6.586 2.05 0.057 S = 79190.48 R-Sq = 89.7% R-Sq(adj) = 87.8% Analysis of Variance Source DF SS MS Regression 3 8779676741 2926558914 Residual Error 16 1003491259 62718204 Total 19 9783168000 For the problem above, we want to carry out the significance test about the coefficient of LANDVAL, what is the t-value for this test, and is it significant? 46.66, significant 2.05, significant 1.59, not significant 0.26, not significant
- Consider the following computer output from a multiple regression analysis relating the price of a used car to the variables: age of car, mileage, and safety rating. Coefficients Coefficients Standard Error t� Stat P-value Intercept 38356.1138356.11 4686.294686.29 8.1858.185 0.00000.0000 Age (Year) −18219.29−18219.29 2196.312196.31 −8.295−8.295 0.00000.0000 Mileage(in Thousands) 1149.561149.56 1897.651897.65 0.6060.606 0.54720.5472 Safety Rating 1396.751396.75 159.64159.64 8.7498.749 0.00000.0000 Does the sign of the coefficient for the variable mileage make sense?The Following data is given for the period 1999-2003 Interest rate Inflation i π 1999 4.7 4.4 2000 4.6 5.4 2001 6.3 5.7 2002 4.8 4.6 2003 2.9 2.4 Obtain residuals of the regression. Calculate variance of the residuals and the standard errors of the parametersA researcher notes that, in a certain region, a disproportionate number of software millionaires were born around the year 1955. Is this a coincidence, or does birth year matter when gauging whether a software founder will besuccessful? The researcher investigated this question by analyzing the data shown in the accompanying table. Complete parts a through c below. a. Find the coefficient of determination for the simple linear regression model relating number (y) of software millionaire birthdays in a decade to total number (x) of births in the region. Interpret the result. The coefficient of determination is 1.___? (Round to three decimal places as needed.) This value indicates that 2.____ of the sample variation in the number of software millionaire birthdays is explained by the linear relationship with the total number of births in the region. (Round to one decimal place as needed.) b. Find the coefficient of determination for the simple linear regression model…
