The average life of Canadian women is 73.75 years and the standard deviation of the women's life expectancy in Canada is 6.5 years.
Using the Chebyshev's theorem, determine the minimum percentage of women in Canada whose life expectancy is between 64 and 83.5 years.
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to the Z scores
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To use Chebyshev's theorem, we need to calculate the minimum percentage of data that falls within a certain range of values based on the standard deviation.
According to Chebyshev's theorem, at least (1 - 1/k^2) 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 minimum percentage of women whose life expectancy is between 64 and 83.5 years. Let's calculate the number of standard deviations away from the mean for both values:
For 64 years:
(64 - 73.75) / 6.5 = -1.5 standard deviations
For 83.5 years:
(83.5 - 73.75) / 6.5 = 1.5 standard deviations
Since the range extends to both sides of the mean, the total standard deviation needed is |1.5| = 1.5 standard deviations.
Now, let's find the minimum percentage of data that falls within this range using Chebyshev's theorem:
k = 1.5
Percentage = 1 - 1/k^2
Percentage = 1 - 1/(1.5)^2
Percentage = 1 - 1/2.25
Percentage = 1 - 0.4444
Percentage = 0.5556 or 55.56%
Therefore, the minimum percentage of women in Canada whose life expectancy is between 64 and 83.5 years is 55.56%.
To determine the minimum percentage of women in Canada whose life expectancy is between 64 and 83.5 years using Chebyshev's theorem, we need to calculate how many standard deviations these values are from the mean.
First, let's calculate the z-scores for both values using the formula:
z = (x - μ) / σ
For x = 64:
z1 = (64 - 73.75) / 6.5
z1 ≈ -1.5
For x = 83.5:
z2 = (83.5 - 73.75) / 6.5
z2 ≈ 1.5
The next step is to use the Chebyshev's theorem to determine the minimum percentage. Chebyshev's theorem states that for any distribution (regardless of its shape), the proportion of values within k standard deviations of the mean is at least 1 - 1/k^2.
In this case, we are interested in the range between -1.5 and 1.5 standard deviations from the mean.
Minimum proportion between -1.5 and 1.5 standard deviations = 1 - 1/(1.5)^2
≈ 1 - 1/2.25
≈ 1 - 0.4444
≈ 0.5556
Therefore, according to Chebyshev's theorem, the minimum percentage of women in Canada whose life expectancy is between 64 and 83.5 years is 55.56%.