Respuesta :
Answer:
a) [tex] 1- \frac{1}{2.5^2} = 0.84[/tex]
And that represent 84% of the data within 2.5 deviations from the mean
b) For this case we can assume that the limits between 39 and 59 are given by:
[tex] 39 =\mu -\sigma= 49-\sigma[/tex]
[tex] 59 =\mu +\sigma= 49+\sigma[/tex]
Because within one deviation from the mena we have at least 68% of the data.
And we can solve for the deviation and we got:
[tex] \sigma = 49-39 = 10[/tex]
[tex]\sigma= 59-49 = 10[/tex]
Step-by-step explanation:
Part a
Data given
[tex]\mu=49 [/tex] reprsent the population mean
[tex]\sigma[/tex] represent the population standard deviation
The Chebyshev's Theorem states that for any dataset
• We have at least 75% of all the data within two deviations from the mean.
• We have at least 88.9% of all the data within three deviations from the mean.
• We have at least 93.8% of all the data within four deviations from the mean.
Or in general words "For any set of data (either population or sample) and for any constant k greater than 1, the proportion of the data that must lie within k standard deviations on either side of the mean is at least: [tex] 1-\frac{1}{k^2}"[/tex]
And if we use the value of k=2.5 we got:
[tex] 1- \frac{1}{2.5^2} = 0.84[/tex]
And that represent 84% of the data within 2.5 deviations from the mean
Part b
For this case we can assume that the limits between 39 and 59 are given by:
[tex] 39 =\mu -\sigma= 49-\sigma[/tex]
[tex] 59 =\mu +\sigma= 49+\sigma[/tex]
Because within one deviation from the mena we have at least 68% of the data.
And we can solve for the deviation and we got:
[tex] \sigma = 49-39 = 10[/tex]
[tex]\sigma= 59-49 = 10[/tex]
