1. The Better Baby Buggy Company has just come out with a new model, the Turbo. The market research department predicts that the demand equation for Turbos is given by ???? = −2???? + 320, where ???? is the number of buggies it can sell in a month if the price is \$???? per buggy.
2. At what price should it sell the buggies to get the largest revenue? (2 points)

q = -2P + 320

The total revenue depends on the price and is given by, R = pq

R = p (-2p + 320)

= -2p² + 320p.

The above expression is a quadratic function of the form R (p) = ap² + bp + c, where a = -2, b =320, and c = 0. Since is a negative the graph of the above function is a parabola. Therefore, the vertex of the graph is the highest point.

P = –         = – 320   = \$80

2a                 -4

1. What is the largest monthly revenue? (1 point)

R = -2p² + 320p

= -2(80)² + 320(80)

= -12,800 + 25,600

= \$12,800

1. Investment in Gold. The following are approximate values of the Amex Gold BUGS index (x=0 represents 2000).

 Year x 0 5 10 Index y 50 200 500

1. a) Complete the following table (2 points)

 Year (x) Index (y) xy x² 0 50 0 0 5 200 1000 25 10 500 5000 100 Sum 15 750 6000 125

1. b) Obtain the associated regression line. (Round coefficients to the nearest whole number) (2 points)

^y = mx + b

Where, n = 3

m= n(∑xy) – (∑x) (∑y)

n(∑x²) – (∑x)²

= 3(6000) – (15) (750)   = 18,000 – 11,250   = 45

3(125) – 15²                375 – 225

b= ∑y – m(∑x)

n

= 750 – 45(15)   = 750 – 675     = 25

3                      3

y = 45x + 25

1. c) Use your regression equation to project the index in 2011 (2 points)

y = 45x + 25

The index for 2011 is given by substituting x = 11.

Y = 45(11) + 25

= 520

1. A common perception is that airline profits are strongly correlated with the price of oil. The Following are annual net incomes of Continental Airlines together with the approximate price of oil in the period 2005-2010.

 Year Barrel Oil (\$) (x) Continental Net Income (\$ million) (y) xy x² Y² 2005 56 -70 -3,920 3,136 4,900 2006 63 370 23,310 3,969 136,900 2007 67 430 28,810 4,489 184,900 2008 92 -590 -54,280 8,464 348,100 2009 54 -280 -15,120 2,916 78,400 210 71 150 10,650 5,041 22,500 ∑ 403 10 -10,550 28,015 775,700

(Part a – d are 1 point each) a. What is n = 6                 b. What is ∑xy = -10,550

1. What is (∑????²) = 28,015 d. What is (∑????)² = 162,409

1. Obtain a regression line showing continental’s net income as a function of the price of oil (3 points)

^y = mx + b

m= n(∑xy) – (∑x) (∑y)

n(∑x²) – (∑x)²

m= 6(-10,550) – (403) (10)    = -63,330 – 4030        = -11.857

6(28,015) – 403²            168,090 – 162,409

b= ∑y – m(∑x)

n

b= 10 – (-11.857) (403)     = 10 + 4778.371   = 798.06

6                                  6

y= -11.857x + 798.06

1. Obtain the coefficient of r (3 points)

r= n(∑xy) – (∑x) (∑y)

√n(∑x²) – (∑x)² × √n(∑y²) – (∑y)²

r= 6(-10,550) – (403) (10)                            = -63,300 -4030

√6(28,015) – 403² × √6(775,700) – 10²           √5681 × √4654100

r= –67330

75.37 × 2157.34

r= -67330                     = -0.4141

162,598.716

1. What does the value of r suggest about the relationship of Continental’s net income to the price of oil? (1 point)

The coefficient of r is negative; therefore, it suggests that an increase in the prices of oil is associated to a decrease in the Continental’s net income.

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