Quiz 2 Solutions | Applied Statistical Methods I | STAT 381, Quizzes of Data Analysis & Statistical Methods

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STAT 381 - 13605 Quiz 2 2006-9-

Solution

1. Consider the weight x (in 1000 pounds) and the fule consumption y (gallons per 100

miles) of 4 cars in the following table.

Car weight x^^3 1 2 Fuel Consumption y 3 1 2 8

(1) (3pt) Draw the scatter plot of x and y.

X

Y

1 2 3 4 5 6

8 7 6 5 4 3 2 1 0

Fuel Assumption vs. Cars Weight

(2) Compute the sample correlation coefficient.

x y ( x x )( y y )

i −^ i −

r

( )

2 x (^) i x

r

− ( )

2 y (^) i − y

Car 1 3 3 0 0. Car 2 1 1 5 4 6. Car 3 2 2 1.5 1 2. Car 4 6 8 13.5 9 20.

Total Sum 12 14 20 14 29

= ∑ = = 3 ,

xi n

x ( ) 2. 160

− = =

−

= ∑ x x

n

s (^) x i

= ∑ = = 3. 5

yi n

y , ( ) 3. 109

− = =

−

= ∑ y y

n

sy i

Sample correlation coefficient

( )( )

( ) ( )

14 * 29

2 2

= =

− −

− −

=

∑ ∑

∑

x x y y

x x y y r

i i

i i r

r

(3) By the method of least squares, fit y = α + β x to the data given above.

Estimate of slope

( )( )

( )

ˆ 20

2 = =

−

− −

=

∑

∑

x x

x x y y

i

i i β ,

or 1. 429

2. 160

ˆ = ⋅ = 0. 993 ⋅ =

x

y

s

s

β r

Estimate of intercept

αˆ = y − βˆ⋅ x = 3. 5 − ( 1. 429 * 3 ) =− 0. 787

Hence the best linear fitted line is y = − 0. 787 + 1. 429 ⋅ x