Solve the problem.
-The ages of the members of a gym have a mean of 44 years and a standard deviation of 12 years. What can you conclude from Chebyshev's theorem about the percentage of gym members aged between 26 and 62?
A) The percentage is at least 33.3%
B) The percentage is at least 55.6%
C) The percentage is at most 55.6%
D) The percentage is approximately 33.3%

Question

Quizplus · Accepted Answer

According to Chebyshev's theorem, at least 1 - (1/k^2) of the data will lie within k standard deviations from the mean, where k is any number greater than 1. 
In this problem, we want to find the percentage of gym members aged between 26 and 62, which is equivalent to finding the percentage of data that lie within 2 standard deviations from the mean. 
Using Chebyshev's theorem with k=2, we get: 
at least 1 - (1/2^2) = 0.75 or 75% of the data will lie within 2 standard deviations from the mean. 
Since we know that the mean is 44 and the standard deviation is 12, we can calculate the range of ages within 2 standard deviations of the mean: 
44 - 2(12) = 20 and 44 + 2(12) = 68 
Therefore, the percentage of gym members aged between 26 and 62 is at least 75% - the percentage that lies outside this range (i.e. below 20 or above 68). 
To find the percentage outside this range, we need to consider the extreme cases: 
- The percentage of gym members aged below 20 (more than 2 standard deviations below the mean) is at most 1/4 or 25% (since k=2, and 1-1/k^2=3/4, the remaining 1/4 of the data can be at most 1/2 below the mean). 
- Similarly, the percentage of gym members aged above 68 (more than 2 standard deviations above the mean) is also at most 25%. 
Therefore, the percentage of gym members aged between 26 and 62 is at least 75% - 25% - 25% = 25%. 
The only choice that reflects this conclusion is B.

Solve the Problem. -The Ages of the Members of a Gym Have a Gym