Respuesta :
Answer:
The mean total luggage weight is 2,360 lb.
The standard deviation of total luggage weight is 505.16 lb.
Step-by-step explanation:
Let the random variables be defined as follows:
X = weight of luggage checked by tourist-class passengers
Y = weight of luggage checked by business-class passengers
The mean and standard deviation of random variable X are:
[tex]E(X)=40\ \text{lb}\\SD(X)=10\ \text{lb}[/tex]
The mean and standard deviation of random variable Y are:
[tex]E(Y)=30\ \text{lb}\\SD(Y)=6\ \text{lb}[/tex]
In the flight there were 12 business-class passengers and 50 tourist-class passengers.
Te equation representing the total luggage weight is:
T = 50 X + 12 Y
Compute the mean total luggage weight as follows:
[tex]E(T)=E(50X+12Y)[/tex]
[tex]=50E(X)+12E(Y)\\=(50\times 40)+(12\times30)\\=2000+360\\=2360[/tex]
Thus, the mean total luggage weight is 2,360 lb.
The two random variables X and Y are independent. This is because the two random variables describes values from two different population, one being weight of tourist class passenger and the other being business class passengers.
So, the covariance between X and Y is 0.
Compute the standard deviation of total luggage weight as follows:
[tex]SD(T)=\sqrt{V(T)}[/tex]
[tex]=\sqrt{V(50X+12Y)}\\=\sqrt{50^{2}V(X)+12^{2}V(Y)+2\cdot50\cdot12\cdot Cov(X, Y)}\\=\sqrt{(2500\times 10^{2})+(144\times 6^{2})+0}\\=505.16[/tex]
Thus, the standard deviation of total luggage weight is 505.16 lb.
