Question

The Condé Nast Traveler Gold List for 2012 provided ratings for the top 20 small cruise ships (Condé Nast Traveler website, March 1, 2012). 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 Itineraries/Schedule, 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 94.6, the highest component of which is 97.8 for food/dining.

Ship	Overall	Shore Excursions	Food/Dining
Seabourn Odyssey	94.6	90.7	97.8
Seabourn Pride	93.2	84.4	96.5
National Geographic Endeavor	92.7	99.9	88.4
Seabourn Sojourn	91.1	94.8	97.0
Paul Gauguin	90.4	87.7	91.1
Seabourn Legend	90.5	82.2	98.6
Seabourn Spirit	90.1	86.3	92.0
Silver Explorer	89.9	92.6	89.0
Silver Spirit	89.6	85.7	90.9
Seven Seas Navigator	89.4	83.3	90.6
Silver Whisperer	89.2	82.1	88.6
National Geographic Explorer	88.9	92.9	89.6
Silver Cloud	88.5	78.2	91.1
Celebrity Xpedition	87.0	91.9	73.7
Silver Shadow	87.1	75.0	89.9
Silver Wind	86.8	78.3	91.6
SeaDream II	86.2	77.2	91.1
Wind Star	86.1	76.5	91.6
Wind Surf	86.0	72.3	89.1
Wind Spirit	85.4	77.2	92.0

a. Determine an estimated regression equation that can be used to predict the overall score given the score for Shore Excursions (to 3 decimals).

Overall = ____ + _____ Shore Excursions

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 (to 3 decimals).

Overall = _____ + _____ Shore Excursions + _____ Food/Dining

c. Predict the overall score for a cruise ship with a Shore Excursions score of 80 and a Food/Dining Score of 90.

(to 2 decimals)

Answer 1

a.

X - Mx	Y - My	(X - Mx)2	(X - Mx)(Y - My)
6.24	5.465	38.9376	34.1016
-0.06	4.065	0.0036	-0.2439
15.44	3.565	238.3936	55.0436
10.34	1.965	106.9156	20.3181
3.24	1.265	10.4976	4.0986
-2.26	1.365	5.1076	-3.0849
1.84	0.965	3.3856	1.7756
8.14	0.765	66.2596	6.2271
1.24	0.465	1.5376	0.5766
-1.16	0.265	1.3456	-0.3074
-2.36	0.065	5.5696	-0.1534
8.44	-0.235	71.2336	-1.9834
-6.26	-0.635	39.1876	3.9751
7.44	-2.135	55.3536	-15.8844
-9.46	-2.035	89.4916	19.2511
-6.16	-2.335	37.9456	14.3836
-7.26	-2.935	52.7076	21.3081
-7.96	-3.035	63.3616	24.1586
-12.16	-3.135	147.8656	38.1216
-7.26	-3.735	52.7076	27.1161
		SS: 1087.808	SP: 248.798

Sum of X = 1689.2
Sum of Y = 1782.7
Mean X = 84.46
Mean Y = 89.135
Sum of squares (SSX) = 1087.808
Sum of products (SP) = 248.798

Regression Equation = ŷ = bX + a

b = SP/SSX = 248.8/1087.81 = 0.229

a = MY - bMX = 89.14 - (0.23*84.46) = 69.818

Overall = 0.229*Shore Excursions + 69.818

b.

X1-Mx1	X2-Mx2	Y-My	(X1-Mx1)2	(X2-Mx2)2	SPx1y	SPx2y	SPx1x2
6.24	6.79	5.465	38.938	46.104	34.102	37.107	42.37
-0.06	5.49	4.065	0.004	30.14	-0.244	22.317	-0.329
15.44	-2.61	3.565	238.394	6.812	55.044	-9.305	-40.298
10.34	5.99	1.965	106.916	35.88	20.318	11.77	61.937
3.24	0.09	1.265	10.498	0.008	4.099	0.114	0.292
-2.26	7.59	1.365	5.108	57.608	-3.085	10.36	-17.153
1.84	0.99	0.965	3.386	0.98	1.776	0.955	1.822
8.14	-2.01	0.765	66.26	4.04	6.227	-1.538	-16.361
1.24	-0.11	0.465	1.538	0.012	0.577	-0.051	-0.136
-1.16	-0.41	0.265	1.346	0.168	-0.307	-0.109	0.476
-2.36	-2.41	0.065	5.57	5.808	-0.153	-0.157	5.688
8.44	-1.41	-0.235	71.234	1.988	-1.983	0.331	-11.9
-6.26	0.09	-0.635	39.188	0.008	3.975	-0.057	-0.563
7.44	-17.31	-2.135	55.354	299.636	-15.884	36.957	-128.786
-9.46	-1.11	-2.035	89.492	1.232	19.251	2.259	10.501
-6.16	0.59	-2.335	37.946	0.348	14.384	-1.378	-3.634
-7.26	0.09	-2.935	52.708	0.008	21.308	-0.264	-0.653
-7.96	0.59	-3.035	63.362	0.348	24.159	-1.791	-4.696
-12.16	-1.91	-3.135	147.866	3.648	38.122	5.988	23.226
-7.26	0.99	-3.735	52.708	0.98	27.116	-3.698	-7.187
			SSX1: 1087.808	SSX2: 495.758	SPX1Y: 248.798	SPX2Y: 109.813	SPX1X2: -85.392

Sum of X1 = 1689.2
Sum of X2 = 1820.2
Sum of Y = 1782.7
Mean X1 = 84.46
Mean X2 = 91.01
Mean Y = 89.135
Sum of squares (SSX1) = 1087.808
Sum of squares (SSX2) = 495.758
Sum of products (SPX1Y) = 248.798
Sum of products (SPX2Y) = 109.813
Sum of products (SPX1X2) = -85.392

Regression Equation = ŷ = b1X1 + b2X2 + a

b1 = ((SPX1Y)*(SSX2)-(SPX1X2)*(SPX2Y)) / ((SSX1)*(SSX2)-(SPX1X2)*(SPX1X2)) = 132720.75/531997.72 = 0.249

b2 = ((SPX2Y)*(SSX1)-(SPX1X2)*(SPX1Y)) / ((SSX1)*(SSX2)-(SPX1X2)*(SPX1X2)) = 140700.82/531997.72 = 0.264

a = MY - b1MX1 - b2MX2 = 89.14 - (0.25*84.46) - (0.26*91.01) = 43.994

Overall = 0.249*Shore Excursions + 0.264*Food/Dining + 43.994

c. For Shore Excursion=80 and Food/Dining=90

Overall=0.249*80+0.264*90+43.994=19.92+23.76+43.994=87.67