Consider the following table, which gives the winning Olympic pole vaults in the 20th century.
Year Gold Medalists Height(ft)
1900 Irving Baxter 10.83
1904 Charles Dvorak 11.84
1908 A. Gilbert 12.17
1912 harry Babcock 12.96
1920 Frank Ross 13.42
1924 Lee Barnes 12.96
1928 Sabin Carr 13.77
1932 William Miller 14.15
1936 Earl meadows 14.27
1948 Guinn Smith 14.10
1952 Robert Richards 14.92
1956 Robert Richards 14.96
1960 Don Bragg 15.42
1964 Fred Hansen 16.73
1968 Bob Seagren 17.71
1972 W. Nordwig 18.04
1976 Tadeusz S. 18.04
1980 W. K. 18.96
1984 Pierre Quinon 18.85
1988 Sergei Bubka 19.77
1992 M. Tarassob 19.02
1996 Jean Jalfione 19.42
a. After analyzing the data, find the regression line that best fits.
b. What does the model predict for the Olympic pole vault record in the year 2000?
a. To find the regression line that best fits the given data, we can use linear regression.
Step 1: Create a scatter plot of the data to visualize the relationship between the Year and the Height.
Step 2: Calculate the mean of the Year (x) and the mean of the Height (y).
Step 3: Calculate the differences between each Year (x) and the mean Year, and each Height (y) and the mean Height.
Step 4: Calculate the product of the differences calculated in step 3 for each data point.
Step 5: Calculate the squared differences for the Year (x) and sum them.
Step 6: Calculate the squared differences for the Height (y) and sum them.
Step 7: Calculate the product sum from step 4 and sum it.
Step 8: Calculate the slope (b) using the formula: b = sum of product / sum of squared differences of x.
Step 9: Calculate the intercept (a) using the formula: a = mean of y - (slope * mean of x).
Step 10: The regression line equation is given by: y = a + bx.
b. To predict the Olympic pole vault record in the year 2000, we can substitute the year (x) value of 2000 into the regression line equation and calculate the corresponding height (y).
To find the regression line that best fits the data, we will use the method of least squares. This involves finding the line that minimizes the sum of the squared distances between the observed data points and the predicted values on the line.
Step 1: Calculate the mean of the x-values (Year) and the mean of the y-values (Height).
Mean of x:
(1900 + 1904 + 1908 + ... + 1996) / 20 = 1948
Mean of y:
(10.83 + 11.84 + 12.17 + ... + 19.42) / 20 = 14.2675
Step 2: Calculate the differences between each x-value and the mean of x (Year - x-mean) and the differences between each y-value and the mean of y (Height - y-mean).
Step 3: Calculate the product of the differences (Year - x-mean) * (Height - y-mean) for each data point.
Step 4: Calculate the squared differences between each x-value and the mean of x ((Year - x-mean)^2) and the squared differences between each y-value and the mean of y ((Height - y-mean)^2).
Step 5: Calculate the sum of the product of differences (Σ(Year - x-mean) * (Height - y-mean)) and the sum of the squared differences for x ((Σ(Year - x-mean)^2)), and the sum of the squared differences for y ((Σ(Height - y-mean)^2)).
Step 6: Calculate the slope (b) of the regression line using the formula:
b = Σ(Year - x-mean) * (Height - y-mean) / Σ(Year - x-mean)^2
Step 7: Calculate the y-intercept (a) of the regression line using the formula:
a = y-mean - b * x-mean
Step 8: The regression line equation is: y = a + bx
Step 9: Use the equation to predict the Olympic pole vault record in the year 2000 by substituting the value of x (Year) into the equation.
Let's go through the steps to find the regression line and make the prediction:
Step 1:
x-mean = 1948
y-mean = 14.2675
Step 2:
Year - x-mean: (-48, -44, -40, ..., 48)
Height - y-mean: (-3.4375, -2.4275, -2.0975, ..., 4.1525)
Step 3:
(-48 * -3.4375, -44 * -2.4275, -40 * -2.0975, ..., 48 * 4.1525)
Step 4:
((-48)^2, (-44)^2, (-40)^2, ..., (48)^2)
Step 5:
Σ(Year - x-mean) * (Height - y-mean) = -29433.375
Σ(Year - x-mean)^2 = 784800
Σ(Height - y-mean)^2 = 145.07425
Step 6:
b = -29433.375 / 784800 ≈ -0.0375
Step 7:
a = 14.2675 - (-0.0375) * 1948 ≈ 25.9275
Step 8:
The regression line equation is: y = 25.9275 - 0.0375x
Step 9:
To predict the Olympic pole vault record in the year 2000, substitute x = 2000 into the equation:
y = 25.9275 - 0.0375 * 2000
y ≈ 25.9275 - 75
y ≈ -49.0725
Therefore, the model predicts that the Olympic pole vault record in the year 2000 is approximately -49.0725 ft.