Question

The Condé Nast Traveler Gold List provides ratings for the top 20 small cruise ships. The data shown below are the scores each ship received based upon the results from Condé Nast Traveler's Annual Readers' Choice Survey. Each score represents the percentage of respondents who rated a ship as excellent or very good on several criteria, including Shore Excursions and Food/Dining. An overall score was also reported and used to rank the ships. The highest ranked ship, the Seabourn Odyssey, has an overall score of 94.4, the highest component of which is 97.8 for Food/Dining.

Ship	Overall	Shore Excursions	Food/Dining
Seabourn Odyssey	94.4	90.9	97.8
Seabourn Pride	93.0	84.2	96.7
National Geographic Endeavor	92.9	100.0	88.5
Seabourn Sojourn	91.3	94.8	97.1
Paul Gauguin	90.5	87.9	91.2
Seabourn Legend	90.3	82.1	98.8
Seabourn Spirit	90.2	86.3	92.0
Silver Explorer	89.9	92.6	88.9
Silver Spirit	89.4	85.9	90.8
Seven Seas Navigator	89.2	83.3	90.5
Silver Whisperer	89.2	82.0	88.6
National Geographic Explorer	89.1	93.1	89.7
Silver Cloud	88.7	78.3	91.3
Celebrity Xpedition	87.2	91.7	73.6
Silver Shadow	87.2	75.0	89.7
Silver Wind	86.6	78.1	91.6
SeaDream II	86.2	77.4	90.9
Wind Star	86.1	76.5	91.5
Wind Surf	86.1	72.3	89.3
Wind Spirit	85.2	77.4	91.9

(a) Determine an estimated regression equation that can be used to predict the overall score given the score for Shore Excursions. (Round your numerical values to two decimal places. Let x₁ represent the Shore Excursions score and y represent the overall score.)

ŷ =

(b) Consider the addition of the independent variable Food/Dining. Develop the estimated regression equation that can be used to predict the overall score given the scores for Shore Excursions and Food/Dining. (Round your numerical values to two decimal places. Let x₁ represent the Shore Excursions score, x₂ represent the Food/Dining score, and y represent the overall score.)

ŷ =

(c) Predict the overall score for a cruise ship with a Shore Excursions score of 78 and a Food/Dining Score of 92. (Round your answer to one decimal place.)

Answer 1

We input the data set in Excel and use the "Regression" option under Data > Data Analysis to build two regression models - the first with y on x1 and second with y on x1 and x2. The screenshot of the data set and the outputs is shown below.

B С D E F H - J K N 0 0 P R S M SUMMARY OUTPUT SUMMARY OUTPUT 1 ly 2 A x1 94.4 93 92.9 x2 90.9 3 84.2 97.8 96.7 88.5 4 100 5 94.8 97.1 Regression Statistics Multiple R 0.70 R Square 0.49 Adjusted R Square 0.46 Standard Error 1.87 Observations 20 91.3 90.5 90.3 Regression Statistics Multiple R 0.86 R Square 0.74 Adjusted R Square 0.71 Standard Error 1.38 Observations 20 6 87.9 7 82.1 91.2 98.8 92 8 90.2 86.3 9 89.9 92.6 88.9 10 89.4 85.9 90.8 ANOVA ANOVA 11 89.2 83.3 90.5 SS MS F df SS MS F df 1 Significance F 0.00 Significance F 0.00 89.2 82 60.20 17.21 2 90.95 45.48 24.00 12 13 88.6 89.7 60.20 3.50 89.1 Regression Residual Total 93.1 18 62.96 Regression Residual Total 17 1.89 32.21 123.17 14 88.7 78.3 91.3 19 123.17 19 15 87.2 91.7 73.6 16 89.7 Lower 95% 75 78.1 Upper 95% 79.38 17 Upper 95% 59.85 91.6 Coefficients Standard Error t Stat P-value 69.30 4.80 14.44 0.00 0.23 0.06 4.15 0.00 87.2 86.6 86.2 86.1 Intercept Intercept x1 59.22 0.12 Coefficients Standard Error t Stat P-value 45.18 6.95 6.50 0.00 0.25 0.04 6.04 0.00 0.25 0.06 4.03 0.00 Lower 95% 30.51 0.16 18 77.4 90.9 0.35 x1 0.34 19 76.5 91.5 X2 0.12 0.38 20 86.1 72.3 89.3 21 85.2 77.4 91.9 22


(a) The regression line is: $\hat{y}$ = 69.30 + (0.23) $x_1$ .
(b) The regression line is: $\hat{y}$ = 45.18 + (0.25) $x_1$ + (0.25)
12
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(c) Using the regression line in part (b) and the values of x1 = 78 and x2 = 92, we get predicted overall score = 45.18 + (0.25 * 78) + (0.25 * 92) = 87.7.