Question

Two cross-country running teams, called the Gazelles and the Cheetahs, participated in a (hypothetical) study in which 50% of the Gazelles and 65% of the Cheetahs used weight training to supplement a running workout. The remaining runners did not use weight training. At the end of the season, the mean improvement in race times (in seconds) for each team was as shown in the table below.

Describe how the results recorded in the table might seem surprising or paradoxical.

Answer 1

Consider the following table which shows the time improvement for two running teams with two types of training.

	Time Improvement with Weight Training	Time Improvement without Weight Training	Team Average Time Improvement
Gazelles	10s	2s	6.0s
Cheetahs	9s	1s	6.2s

 

The time improvement with and without weight training is greater for the team Gazelles but the team average time improvement is greater for Cheetahs.

 

This apparent inconsistency is caused due to the percentage of people that are in the two types of training program.

 

Let’s find the percentage of team in each training program by using the average time improvement provided in the table.

 

Suppose, there are a total of 100 members in each team, x are in weight training mode and 100 – x are in without weight training mode.

 

So, for team Gazelles, the definition of average value will give:

x × 10 + (100 – x) × 2/100 = 6

10x – 2x + 200 = 600

8x – 600 – 200

x = 400/8

 

So, the value of x is 50, which means that 50 out of 100 team members in Gazelles are in weight training mode and 50 are in without weight training mode.

Therefore, the percentage of team gazelles which are weight training is 50% and which are not weight training is 50%

 

For team Cheetah, the average value can be written as:

x × 9 + (100 – x) × 1/100 = 6.2

9x – x + 100 = 620

8x = 620 – 100

x = 520/8

 

So, the value of x is 65, which means that 65 out of 100 team members in Cheetah are in weight training mode and 35 are in without weight training mode.

 

Therefore, the percentage of team Cheetah which are weight training is 65% and which are not weight training is 35%.

Therefore, the percentage of team Cheetah which are weight training is 65% and which are not weight training is 35%.