8. (6pts) Five automobiles of the same type are to be driven on a 300-mile trip.

Question

8. (6pts) Five automobiles of the same type are to be driven on a 300-mile trip. The first two will use an economy brand of gasoline, and the other three will use a name brand. Let X, X, X, X, and X, be the observed fuel efficiency (mpg) for the five cars. Suppose these variables are independent and normally distributed with u H-20, H ? ?-21, and ?-4 for the economy brand and 3.5 for the name brand. Define an random variable Y by 2 3 So that Y is a measure of the difference in efficiency between economy gas and name-brand gas. 8.1. (3pts) Compute P(Yz0) 8.2. (3pts) Compute P(-1 Y 1) [Hint Y-aX+ Hax, with a-1/2, , a.--13]

Explanation / Answer

Here is the answer which I solved earlier. Please modify your figures or let me know in case any clarification is required. If this helped you then please rate 5 stars first that is before you rate anyone else. #60 Five automobiles of the same type are to be driven on a300-mile trip. The first two will use an economy brand of gasoline,and he other three will use a name brand. Let X1, X2, X3, X4, andX5 be the observed fuel efficiencies(mpg) for the five cars.Suppose these variables are independent and normally distributedwith m1 = m2 = 20, m3 = m4 = m5 = 21, and o2 (omega square) = 4 forthe economy brand and 3.5 for the name brand. Define an rv Yby Y = ((X1+X2)/2) - ((X3+X4+X5)/3), so that Y is a measureof the difference in efficiency between economy gas and name-brandgas. Compute P(0<=Y) and P(-1<=Y<=1). [Hint: Y = a1*X1 +... + a5*X5, with a1 = 1/2, ..., a5 = -1/3. Property: A linear combination of independent normal variates is also anormal variate. Let Xi, (i=1,2,..,n) be 'n' independent normalvariates with mean ?i and variance?2i respectively. Then ?ai Xi ~ N [?ai ?i ,?a2i ?2i ] Given that X1, X2, X3, X4, and X5 be the observed fuelefficiencies(mpg) for the five cars. Suppose these variables areindependent and normally distributed with m1 = m2 = 20, m3 = m4 =m5 = 21, and ?2= 4 for the economy brand and 3.5for the name brand. And also given that Y = ((X1+X2)/2)- ((X3+X4+X5)/3) Let Y = a1*X1 + a2*X2 - a3*X3 - a4*X4 - a5*X5 where a1 = (1/2) ; a2 = (1/2) ; a3 = (-1/3) ; a4 = (-1/3) ; a5= (-1/3) From the above property Y ~ N[{(1/2)*20 + (1/2)*20 - (1/3)*21 - (1/3)*21 - (1/3)*21} ,{(1/2)2*4 + (1/2)2*4 +(-1/3)2*3.5 + (-1/3)2*3.5 +(-1/3)2*3.5}] This implies Y ~ N[-1 , 3.16667] P(0<=Y) = P(Y>=0) = P[((Y-(-1))/3.16667) >= ((0-(-1))/3.16667)] =P[Z >= 0.32] = 1 - P[Z<=0.32] = 1 - 0.6255 = 0.3745 P(-1 <= Y <= 1) = P[((-1-(-1))/3.16667) <=((Y-(-1))/3.16667) <= ((1-(-1))/3.16667)] = P[ 0 <= Z <= 0.63] = P[Z<=0.63] - P[Z<=0] = 0.7357 - 0.5 = 0.2357

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