Respuesta :
Answer:
a) [tex] z = \frac{30-50}{4}= -5[/tex]
[tex] z = \frac{70-50}{4}= 5[/tex]
[tex] 1- \frac{1}{5^2} = 0.96 = 96\%[/tex]
b) [tex] z = \frac{35-50}{4}= -3.75[/tex]
[tex] z = \frac{65-50}{4}= 3.75[/tex]
[tex] 1- \frac{1}{3.75^2} = 0.929 = 92.9\%[/tex]
c) [tex] z = \frac{41-50}{4}= -2.25[/tex]
[tex] z = \frac{59-50}{4}= 2.25[/tex]
[tex] 1- \frac{1}{2.25^2} = 0.8025 = 80.25\%[/tex]
d) [tex] z = \frac{38-50}{4}= -3[/tex]
[tex] z = \frac{62-50}{4}= 3[/tex]
[tex] 1- \frac{1}{3^2} = 0.889 = 88.89\%[/tex]
e) [tex] z = \frac{33-50}{4}= -4.25[/tex]
[tex] z = \frac{67-50}{4}= 4.25[/tex]
[tex] 1- \frac{1}{4.25^2} = 0.9446 = 94.46\%[/tex]
Step-by-step explanation:
Data given
[tex]\mu =50[/tex] reprsent the population mean
[tex]\sigma=4[/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]
Part a
For this case we can find the deviations from the mean like this:
[tex] z = \frac{30-50}{4}= -5[/tex]
[tex] z = \frac{70-50}{4}= 5[/tex]
[tex] 1- \frac{1}{5^2} = 0.96 = 96\%[/tex]
Part b
For this case we can find the deviations from the mean like this:
[tex] z = \frac{35-50}{4}= -3.75[/tex]
[tex] z = \frac{65-50}{4}= 3.75[/tex]
[tex] 1- \frac{1}{3.75^2} = 0.929 = 92.9\%[/tex]
Part c
For this case we can find the deviations from the mean like this:
[tex] z = \frac{41-50}{4}= -2.25[/tex]
[tex] z = \frac{59-50}{4}= 2.25[/tex]
[tex] 1- \frac{1}{2.25^2} = 0.8025 = 80.25\%[/tex]
Part d
For this case we can find the deviations from the mean like this:
[tex] z = \frac{38-50}{4}= -3[/tex]
[tex] z = \frac{62-50}{4}= 3[/tex]
[tex] 1- \frac{1}{3^2} = 0.889 = 88.89\%[/tex]
Part e
For this case we can find the deviations from the mean like this:
[tex] z = \frac{33-50}{4}= -4.25[/tex]
[tex] z = \frac{67-50}{4}= 4.25[/tex]
[tex] 1- \frac{1}{4.25^2} = 0.9446 = 94.46\%[/tex]
