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 = – b          = – 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

2. 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

 

3. 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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