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/probability between the two Z scores.
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 the minimum proportion of data that falls within a certain number of standard deviations from the mean.
Chebyshev's theorem states that for any number k greater than 1, at least 1 - 1/k^2 of the data will fall within k standard deviations from the mean, where k is a positive constant.
First, we need to calculate the number of standard deviations between the range of 64 to 83.5 years using the given mean and standard deviation.
The mean of women's life expectancy in Canada is given as 73.75 years, and the standard deviation is 6.5 years.
To find the number of standard deviations, we can use the formula:
Z = (X - mean) / standard deviation
Where Z is the number of standard deviations, X is the value we want to convert, mean is the mean of the data distribution, and standard deviation is the standard deviation of the data distribution.
For the lower bound of 64 years:
Z_lower = (64 - 73.75) / 6.5 = -1.5
For the upper bound of 83.5 years:
Z_upper = (83.5 - 73.75) / 6.5 = 1.5
Now, we have the number of standard deviations for the given range.
Next, we apply Chebyshev's theorem formula:
1 - 1/k^2
In this case, k = 1.5, so we substitute it into the formula:
1 - 1/1.5^2 = 1 - 1/2.25 = 1 - 0.4444 = 0.5556
Therefore, at least 55.56% of women in Canada will have a life expectancy between 64 and 83.5 years, according to Chebyshev's theorem.