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?
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability between the two Z scores. Multiply by 100 to get the percentage.
To apply Chebyshev's theorem to this scenario, we need to know the range within which a certain percentage of the data falls based on the standard deviation. Chebyshev's theorem states that regardless of the shape of a distribution, at least (1 - 1/k^2) of the data will be within k standard deviations of the mean.
In this case, the mean age of the gym members is 44 years, and the standard deviation is 12 years. Let's determine the range within which the percentage of gym members between the ages of 26 and 62 falls by using Chebyshev's theorem:
First, we need to find the number of standard deviations away from the mean that each boundary value is:
Lower bound = (26 - 44) / 12 ≈ -1.5 standard deviations
Upper bound = (62 - 44) / 12 ≈ 1.5 standard deviations
According to Chebyshev's theorem, at least (1 - 1/k^2) of the data falls within k standard deviations of the mean. For this case, let's consider k = 1.5 standard deviations:
Percentage of data within 1.5 standard deviations of the mean = 1 - 1/1.5^2 ≈ 1 - 1/2.25 ≈ 1 - 0.444 ≈ 0.556
Therefore, we can conclude that at least 55.6% of the gym members' ages will fall between the ages of 26 and 62, according to Chebyshev's theorem.
Chebyshev's theorem states that for any set of data, regardless of its shape, at least (1 - 1/k^2) percentage of the data falls within k standard deviations from the mean, where k is any positive constant greater than 1.
In this case, we want to find the percentage of gym members aged between 26 and 62.
To apply Chebyshev's theorem, we first need to determine how many standard deviations these ages are from the mean.
The mean age is 44 years, and the standard deviation is 12 years.
For the lower bound, 26 years, the deviation from the mean would be (26 - 44) / 12 = -1.5 standard deviations.
For the upper bound, 62 years, the deviation from the mean would be (62 - 44) / 12 = 1.5 standard deviations.
Since the deviation of both bounds is the same (1.5) and Chebyshev's theorem applies to any positive constant greater than 1, we can choose k = 1.5.
The theorem guarantees that at least (1 - 1/1.5^2) percentage of the data fall within 1.5 standard deviations from the mean.
Calculating the value, (1 - 1/1.5^2) = (1 - 1/2.25) ≈ 1 - 0.4444 ≈ 0.5556.
So, we can conclude that at least 55.56% of the gym members' ages will fall between 26 and 62 years, according to Chebyshev's theorem.