Answer :
Answer:
[tex] E(T) = E(X+Y) = E(X) + E(Y) = 50 + 20 = 70[/tex]
And the variance can be founded like this:
[tex] Var(T) = Var(X+Y) = Var(X) + Var(Y) + 2 Cov (X,Y) = 50+30 + 2*10 =100[/tex]
[tex] Sd(T) = \sqrt{100}=10[/tex]
[tex] z = \frac{T/n -\bar T}{\frac{\sigma}{\sqrt{n}}}[/tex]
And if we use this formula we got:
[tex] z= \frac{700/100 -70}{\frac{10}{\sqrt{100}}}= 0[/tex]
So then we have this:
[tex] P(T>700) = P(Z >0) = 0.5[/tex]
Step-by-step explanation:
For this case we have the following data given:
[tex] E(X) = 50, E(Y)= 20 , Var(X) = 50, Var(Y) = 30, Cov (X,Y) = 10[/tex]
We define the random variable T as the total number of hours that the people watch Games of Thrones or star Wars. So then we can define [tex] T = X+Y[/tex]
We can find the expected value of T like this:
[tex] E(T) = E(X+Y) = E(X) + E(Y) = 50 + 20 = 70[/tex]
And the variance can be founded like this:
[tex] Var(T) = Var(X+Y) = Var(X) + Var(Y) + 2 Cov (X,Y) = 50+30 + 2*10 =100[/tex]
[tex] Sd(T) = \sqrt{100}=10[/tex]
We know that the sampel mena is defined as:
[tex] \bar T = \frac{\sum_{i=1}^n T_i}{n}= \frac{T}{n}[/tex]
And for this case since the sampel large is enough n>30, we have that the distribution for T can be approximated with the normal distribution using the central limit theorem:
[tex] \bar T \sim N (T/n , \frac{\sigma}{\sqrt{n}})[/tex]
We want to calculate the following probability:
[tex] P(T>700) [/tex]
We can use the z score formula given by:
[tex] z = \frac{T/n -\bar T}{\frac{\sigma}{\sqrt{n}}}[/tex]
And if we use this formula we got:
[tex] z= \frac{700/100 -70}{\frac{10}{\sqrt{100}}}= 0[/tex]
So then we have this:
[tex] P(T>700) = P(Z >0) = 0.5[/tex]