Respuesta :
Answer:
The proportion of measurements in the given interval is 88.89% .
Step-by-step explanation:
We are given that a distribution of measurements has a mean of 65 and a standard deviation of 7 i.e.; [tex]\mu[/tex] = 65 and [tex]\sigma[/tex] = 7
Also, we know nothing about the size or shape of the data. So, here we will use the Tchebysheff's Theorem.
According to this theorem, the lower bound formula for any population data set is given by;
Lower bound = [tex]\mu - k*\sigma[/tex]
On the other hand, upper bound formula for any population data set is given by;
Upper bound = [tex]\mu + k*\sigma[/tex]
In the question we have to find the proportion of measurements in the given interval of 44 to 86 i.e. Lower bound = 44 and Upper bound = 86
Therefore, 44 = [tex]\mu - k*\sigma[/tex] and 86 = [tex]\mu + k*\sigma[/tex]
44 = 65 - k * 7 and 86 = 65 + k * 7
k = [tex]\frac{65-44}{7}[/tex] = 3 and k = [tex]\frac{86-65}{7}[/tex] = 3
Now, proportion of measurements in any given interval is given by;
P = [tex](1-\frac{1}{k^{2} } )*100[/tex]
P = [tex](1-\frac{1}{3^{2} } )*100[/tex] = [tex]\frac{8}{9} *100[/tex] = 88.89%
Therefore, the proportion of measurements in the given interval of between 44 and 86 is 88.89%.
