Question

The following table gives the gold medal times for every other Summer Olympics for the women's 100 meter freestyle (swimming).

Year	Time (seconds)
1912	82.2
1924	72.4
1932	66.8
1952	66.8
1960	61.2
1968	60.0
1976	55.65
1984	55.92
1992	54.64
2000	53.8
2008	53.1

A- Calculate the least squares line. Put the equation in the form of: ŷ = a + bx. (Round your answers to three decimal places.)
ŷ = ____ + _____x

B- Find the correlation coefficient r. (Round your answer to four decimal places.)
r =

C- Find the estimated gold medal time for 1924. (Use your equation from part (a). Round your answer to two decimal places.)

Find the estimated gold medal time for 1984. (Use your equation from part (a). Round your answer to two decimal places.)

D- Why are the answers from part (c) different from the chart values?

Does it appear that a line is the best way to fit the data? Why or why not?

E- Use the least squares line to estimate the gold medal time for the 2016 Summer Olympics. (Use your equation from part (a). Round your answer to two decimal places.)

Answer 1

A.

Sum of X = 21608
Sum of Y = 682.51
Mean X = 1964.3636
Mean Y = 62.0464
Sum of squares (SS_X) = 10062.5455
Sum of products (SP) = -2773.2655

Regression Equation = ŷ = bX + a

b = SP/SS_X = -2773.27/10062.55 = -0.276

a = M_Y - bM_X = 62.05 - (-0.28*1964.36) = 603.430

ŷ = -0.276X + 603.430

B.

X Values
∑ = 21608
Mean = 1964.364
∑(X - M_x)² = SS_x = 10062.545

Y Values
∑ = 682.51
Mean = 62.046
∑(Y - M_y)² = SS_y = 844.805

X and Y Combined
N = 11
∑(X - M_x)(Y - M_y) = -2773.265

R Calculation
r = ∑((X - M_y)(Y - M_x)) / √((SS_x)(SS_y))

r = -2773.265 / √((10062.545)(844.805)) = -0.9512

c. For x=1924,

ŷ = (-0.276*1924) + 603.430=72.41

For x=1984

ŷ = (-0.276*1984) + 603.430=55.85

D. As this are estimated values so they are different

As most of the points lie near the line, so it appears to be best fit

E. For x=2016, ŷ = (-0.276*2016) + 603.430=47.01