The gold-medal-winning heights (meters) for the men\'s high jump for 1948 to 199

Question

The gold-medal-winning heights (meters) for the men's high jump for 1948 to 1992 are shown here:

Year Height Year Height 1948 1.98 1972 2.23 1952 2.04 1976 2.25 1956 2.12 1980 2.36 1960 2.16 1984 2.35 1964 2.18 1988 2.38 1968 2.24 1992 2.34 The gold-medal-winning heights (meters) for the men's high jump for 1948 to 1992 are shown here: Year Height Year Height 1948 1.98 1972 2.23 1952 2.04 1976 2.25 1956 2.12 1980 2.36 1960 2.16 1984 2.35 1964 2.18 1988 2.38 1968 2.24 1992 2.34 Use the trend projection to estimate the constant rate of change and the constat percentage rate of change for the period 1948 to 1992. Use the estimated trend projection equations to forecast the winning height for 1996. Use exponential smoothing with a smoothing constant of infinite = 0.8 to forecast the winning height for 1996. Which approach (trend projection or exponential smoothing) provides the more accurate forecast? explain. infinite = 0.2 and

Explanation / Answer

XY19481.9819522.0419562.1219602.1619642.1819682.2419722.2319762.2519802.3619842.3519882.3819922.34

The equation of the regression line is:

y = ?14.708 + 0.009?x

The graph of the regression line is:

y = -14.708 + 0.009 x

1940

1945

1950

1955

1960

1965

1970

1975

1980

1985

1990

1995

2000

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

3.4

Explanation

We will find an equation of the regression line in 4 steps.

Step 1: Find X?Y and X2 as it was done in the table below.

Step 2: Find the sum of every column:

?X=23640 , ?Y=26.63 , ?X?Y=52480.76 , ?X2=46573088

Step 3: Use the following equations to find a and b:

ab=?Y??X2??X??XYn??X2?(?X)2=26.63?46573088?23640?52480.7612?46573088?236402??14.708=n??XY??X??Yn??X2?(?X)2=12?52480.76?23640?26.6312?46573088?(23640)2?0.009

Step 4: Substitute a and b in regression equation formula

y = a + b?x= ?14.708 + 0.009?x

Thus constant rate of change is slope = 0.009

For 1996 we substitute x=1996

y = -14.708+0.009(1996) = 3.256

Least squares method of finding linear regression gives the most reliable result provided the scatter plot shows a linear relationshiup

T

X Y X?Y X?X 1948 1.98 3857.04 3794704 1952 2.04 3982.08 3810304 1956 2.12 4146.72 3825936 1960 2.16 4233.6 3841600 1964 2.18 4281.52 3857296 1968 2.24 4408.32 3873024 1972 2.23 4397.56 3888784 1976 2.25 4446 3904576 1980 2.36 4672.8 3920400 1984 2.35 4662.4 3936256 1988 2.38 4731.44 3952144 1992 2.34 4661.28 3968064

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