Question 651584: PART I. Use Chebyshev's theorem to find what percent of the values will fall between 166 and 346 for a data set with a mean of 256 and standard deviation of 18.
Part II. Use the Empirical Rule to find what two values 95% of the data will fall between for a data set with a mean of 272 and standard deviation of 14.
Failure to show ALL work will result in loss of point!
Tutors.... I hope you can help me.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! chebyshev's rule is this:
as long as k is > 1, then AT LEAST (1- 1/k^2) of the data will fall within k standard distributions from the mean.
the percentage could be higher, but, according to chebyshev, it will not be less.
chebyshev's theorem applies to any distribution, not just a normal distribution.
it's applicability is therefore more usable for distributions that are not normal than the z-score which is only used for normal distributions.
see the following reference for additional information if you require it.
http://free-l-ramswriter.hubpages.com/hub/Chebyshevs-Theorem-and-Empirical-Rule
general formula to find out how many standard deviations a score is from the mean is:
z = (x-m)/s, where:
z = the number of standard deviations from the mean.
x is the data score.
m is the mean
s is the standard deviation.
this formula will be used below.
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PART I. Use Chebyshev's theorem to find what percent of the values will fall between 166 and 346 for a data set with a mean of 256 and standard deviation of 18.
mean is 256
standard deviation is 18.
number of standard deviations from the mean for 166 is based on the formula:
z = (166-256)/18 which comes out to be z = -5
number of standard deviations from the mean for 346 is based on the formula:
z = (346-256)/18 which comes out to be be z = 5
the 2 points of 166 and 346 are both 5 standard deviations from the mean.
this means that k is equal to 5.
using chebyshev's theorem, you get:
1-(1/k^2) = 1-(1/5^2) = 1-(1/25) = 24/25 = .96
based on chebyshev's theorem, this means that:
at least .96 of the data will fall within 5 standard deviations from the mean.
.96 is equivalent to 96%.
use of the z-score for normal distributions indicates that the percentage for a normal distribution between the values of 166 and 346 is in excess of 99% which agrees with chebyshev's theorem since he states that the percentage will be at least 95%.
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Part II. Use the Empirical Rule to find what two values 95% of the data will fall between for a data set with a mean of 272 and standard deviation of 14.
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using chebyshev's formula of:
R = 1 - (1/k^2) where R represents the rate of the data.
Note that the rate of the data is equivalent to the percent of the data divided by 100.
R% will be equal to R*100%.
since the percent we are looking for is 95%, then R = 95%/100% = .95.
the formula becomes:
.95 = 1 - (1/k^2)
if we add 1/k^2 to both sides of this equation and subtract .95 from both sides of this equation, we get:
1/k^2 = 1 - .95 which becomes:
1/k^2 = .05
if we multiply both sides of this equation by k^2, we get:
1 = .05*k^2
this is equivalent to:
.05*k^2 = 1
if we divide both sides of this equation by .05, we get:
k^2 = 1/.05 which becomes:
k^2 = 20
we take the square root of both sides of this equation to get:
k = sqrt(20).
that's the value of k that we can use to find the min and the max values.
we start again:
the mean is 272 and the standard deviation is 14.
chebyshev's theorem states that at least 95% of the value will lie between k standard deviations from the mean.
we know that k is equal to sqrt(20).
this means that the minimum value will be:
272 - sqrt(20)*14 = 209.39
this means that the maximum value will be:
272 + sqrt(20)*14 = 334.61
your minimum and maximum values are:
minimum = 209.39
maximum = 334.61
to confirm that this is accurate, we apply chebyshev's theorem to this as follows:
k = sqrt(20)
R = rate of values that will be within k standard deviations above or below the mean.
chebyshev's formula states that:
R = 1 - (1/k^2)
since k = sqrt(20), the formula becomes:
R = 1 - (1/(sqrt(20)^2) which becomes:
R = 1 - 1/20 which becomes:
R = 1 - .05 which becomes:
R = .95
that's the rate.
the percent is equal to 100 times that to get:
R% = .95*100 = 95%
At least 95% of the data will be between 209.39 and 334.61 according to chebyshev.
i checked this range using the normal distribution curve and found that, for a normal distribution, over 99.999% of the data falls within these limits. that is at least 95% so chebyshev's formula is accurate.
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note that 95% of the data falling within 2 standard deviations of the mean only applies to a normal distribution.
chebyshev's formula applies to any distribution, not just a normal one.
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