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The table below shows the time (in seconds, rounded to ONE decimal place) taken by 12 athletes to run the 100 metre sprint and the distance (in metres, rounded to ONE decimal place) of their best long jump - NSC Mathematics - Question 1 - 2017 - Paper 2
Question 1
The table below shows the time (in seconds, rounded to ONE decimal place) taken by 12 athletes to run the 100 metre sprint and the distance (in metres, rounded to ON... show full transcript
Worked Solution & Example Answer:The table below shows the time (in seconds, rounded to ONE decimal place) taken by 12 athletes to run the 100 metre sprint and the distance (in metres, rounded to ONE decimal place) of their best long jump - NSC Mathematics - Question 1 - 2017 - Paper 2
Step 1
1.1 Determine the values of a and b.
Answer
To find the values of a and b in the least squares regression line ar{y} = a + bx, we first calculate the necessary statistics from the given data.
-
Calculate the means:
- Mean Time (ar{x}): Calculate the average of the times for the 100m sprint.
- Mean Distance (ar{y}): Calculate the average of the distances for the best long jump.
-
Calculate the slope (b):
b = rac{n( ext{sum of }xy) - (sum of x)(sum of y)}{n( ext{sum of }x^2) - (sum of x)^2}
where:
- n is the number of data points.
-
Calculate the y-intercept (a):
a = ar{y} - bar{x}
Using these calculations with the provided data, we find:
Step 2
1.2 An athlete runs the 100 metre sprint in 11.7 seconds. Use y = a + bx to predict the distance of the best long jump of this athlete.
Answer
To find the predicted distance of the best long jump for an athlete who runs the 100m sprint in 11.7 seconds, we can use the previously calculated values of a and b in the equation:
ar{y} = a + bx
Substituting in the values:
We compute:
ar{y} = 14.343 + (-0.642)(11.7)
ar{y} ext{ gives us approximately } 6.85 ext{ meters}
Thus, the predicted distance of the best long jump is approximately 6.85 metres.
Step 3
1.3 Another athlete completes the 100 metre sprint in 12.3 seconds and the distance of his best long jump is 7.6 metres. If this is included in the data, what is the gradient of the least squares regression line in increase or decrease?
Answer
To determine how the gradient of the least squares regression line changes with the new data point (12.3 seconds, 7.6 meters), we observe that the additional point may cause the regression line to shift.
- With the previous data, the original slope was calculated before including this new athlete.
- Adding the new point will change the sums used in calculating the slope, potentially increasing or decreasing the gradient.
Generally, if the new point follows the trend (i.e., it lies close to the line), the gradient will likely remain similar. If the point diverges significantly, the gradient could decrease.
Thus, the response should indicate that the gradient may change, depending on the correlation between the variable values. We conclude that the gradient could either increase or decrease, depending on further calculations and context.