Respuesta :
Answer:
The interval (meassured in Inches) that represent the middle 80% of the heights is [64.88, 75.12]
Step-by-step explanation:
I beleive those options corresponds to another question, i will ignore them. We want to know an interval in which the probability that a height falls there is 0.8.
In such interval, the probability that a value is higher than the right end of the interval is (1-0.8)/2 = 0.1
If X is the distribuition of heights, then we want z such that P(X > z) = 0.1. We will take W, the standarization of X, wth distribution N(0,1)
[tex] W = \frac{X-\mu}{\sigma} = \frac{X-70}{4} [/tex]
The values of the cumulative distribution function of W, denoted by [tex] \phi [/tex] , can be found in the attached file. Lets call [tex] y = \frac{z-70}{4} [/tex] . We have
[tex]0.1 = P(X > z) = P(\frac{X-70}{4} > \frac{z-70}{4}) = P(W > y) = 1-\phi(y)[/tex]
Thus
[tex] \phi(y) = 1-0.1 = 0.9 [/tex]
by looking at the table, we find that y = 1.28, therefore
[tex]\frac{z-70}{4} = 1.28\\z = 1.28*4+70 = 75.12[/tex]
The other end of the interval is the symmetrical of 75.12 respect to 70, hence it is 70- (75.12-70) = 64.88.
The interval (meassured in Inches) that represent the middle 80% of the heights is [64.88, 75.12] .
The interval that represent the middle 80% of the heights (inches) is [64.88, 75.12].
Step-by-step explanation:
Given :
Mean -- [tex]\rm \mu = 70 \; inches[/tex]
Standard Deviation -- [tex]\rm \sigma = 4 \; inches[/tex]
Calculation :
We want to know an interval in which the probability that a height falls there is 0.8.
In such interval, the probability that a value is higher than the right end of the interval is
[tex]\rm P(x>z) = \dfrac {1-0.8}{2} = 0.1[/tex]
If x is the distribuition of heights, then we want y such that P(x > y) = 0.1.
[tex]Z = \dfrac{x-\mu}{\sigma}[/tex]
Now, let
[tex]U = \dfrac{y-70}{4}[/tex]
We have
[tex]\rm 0.1 = P(x>y)= P(\dfrac{x-70}{4} > \dfrac{y-70}{4})=P(Z>U)=1-\phi(U)[/tex]
[tex]\phi (U) = 1-0.1=0.9[/tex]
by looking at the table, we find that U = 1.28, therefore
[tex]\dfrac{y-70}{4}=1.28[/tex]
[tex]1.28\times 4 + 70 = y[/tex]
[tex]y=75.12[/tex]
The other end of the interval is the symmetrical of 75.12 respect to 70, hence it is
70- (75.12-70) = 64.88.
The interval that represent the middle 80% of the heights (inches) is [64.88, 75.12].
For more information, refer the link given below
https://brainly.com/question/10729938?referrer=searchResults
