2.
The modern Olympic Games are a modified revival of the Greek Olympian Games that came to be largely through the efforts of the French sportsman and educator Baron Pierre de Coubertin. The Games are an international athletic competition that has been held at a different site every four years since their inauguration in 1896, with occasional interruptions in the times of world wars.
The data for the gold medal performances in long jump, high jump, discus throw are given below (in inches). Year is coded to be zero in 1900.
Year Long Jump High Jump Discus Throw
-4 249.75 71.25 1147.5
0 282.875 74.8 1418.9
4 289 71 1546.5
8 294.5 75 1610
12 299.25 76 1780
20 281.5 76.25 1759.25
24 293.125 78 1817.125
28 304.75 76.375 1863
32 300.75 77.625 1948.875
36 317.3125 79.9375 1987.375
48 308 78 2078
52 298 80.32 2166.85
56 308.25 83.25 2218.5
60 319.75 85 2330
64 317.75 85.75 2401.5
68 350.5 88.25 2550.5
72 324.5 87.75 2535
76 328.5 88.5 2657.4
80 336.25 92.75 2624
84 336.25 92.5 2622
88 343.25 93.5 2709.25
92 342.5 92 2563.75
a) Predict the Olympic performance for each event for the Athens 2004 games.
b) Compare each of your predictions to the actual results in the Athens 2004 games. Be sure to appropriately convert units.
Here are the Athens 2004 Olympic Results:
The long jump result in Athens was 338.1883 inches.
The high jump result in Athens was 92.9132 inches.
The discus result in Athens was 2751.5693 inches.
working on the same two questions tonight!
a) To predict the Olympic performance for each event for the Athens 2004 games, we can use a linear regression model based on the given historical data.
For the long jump event, we can fit a linear regression line to the data points Year vs. Long Jump:
-4 249.75
0 282.875
4 289
8 294.5
12 299.25
20 281.5
24 293.125
28 304.75
32 300.75
36 317.3125
48 308
52 298
56 308.25
60 319.75
64 317.75
68 350.5
72 324.5
76 328.5
80 336.25
84 336.25
88 343.25
92 342.5
By fitting a regression line to these data points, we can estimate the long jump performance for the Athens 2004 games.
b) To compare with the actual results in the Athens 2004 games, we need to convert the actual results from inches to the same unit as the historical data.
The actual long jump result in Athens was 338.1883 inches.
We can use the regression line equation to estimate the long jump performance for the Athens 2004 games.
To predict the Olympic performance for each event in Athens 2004, we can use regression analysis. Regression analysis helps us find a mathematical relationship between the independent variable (in this case, the year) and the dependent variable (the performance in each event).
a) To make predictions, we need to find the regression equations for each event based on the given data. We will use the year as the independent variable and the performance measurements as the dependent variables.
For the Long Jump event, we can calculate the regression equation as follows:
1. Calculate the mean of the year and long jump measurements:
Mean of Year = (Sum of Years) / (Number of Data Points) = (0 - 4 + 8 + 12 + 20 + 24 + 28 + 32 + 36 + 48 + 52 + 56 + 60 + 64 + 68 + 72 + 76 + 80 + 84 + 88 + 92) / 21 = 40
Mean of Long Jump = (Sum of Long Jump Measurements) / (Number of Data Points) = (282.875 + 289 + 294.5 + 299.25 + 281.5 + 293.125 + 304.75 + 300.75 + 317.3125 + 308 + 298 + 308.25 + 319.75 + 317.75 + 350.5 + 324.5 + 328.5 + 336.25 + 336.25 + 343.25 + 342.5) / 21 = 314.94345
2. Calculate the deviations from the means for both the year and long jump measurements:
Deviation of Year = Year - Mean of Year
Deviation of Long Jump = Long Jump - Mean of Long Jump
3. Calculate the products of the deviations:
Product of Deviations = Deviation of Year * Deviation of Long Jump
4. Calculate the squared deviations of the year:
Squared Deviation of Year = Deviation of Year^2
5. Find the sum of the products, the sum of the squared deviations of the year, and the number of data points:
Sum of Products = Sum of Product of Deviations
Sum of Squared Deviations of Year = Sum of Squared Deviation of Year
Number of Data Points = 21
6. Calculate the regression coefficients:
Regression Coefficient (b) = Sum of Products / Sum of Squared Deviations of Year
Intercept (a) = Mean of Long Jump - (b * Mean of Year)
Using these calculations, we can find the regression equation for the Long Jump event.
Repeat this process for the High Jump and Discus Throw events to find their respective regression equations.
b) To compare the predictions with the actual results in Athens 2004, we can substitute the year 104 (2004-1900) into each regression equation to obtain the predicted performance in inches. Then, we can convert the units from inches to the appropriate unit used in the Athens 2004 games.
For example, to compare the predicted and actual results for the Long Jump event:
1. Use the Long Jump regression equation: Long Jump = (b * Year) + Intercept
2. Substitute Year = 104 into the equation and calculate the predicted value in inches.
3. Convert the inches to the unit used in the Athens 2004 games.
Repeat this process for the High Jump and Discus Throw events to compare the predicted and actual results in the appropriate units.