6. Yu Ming travels to work and returns home once each day. The times, in minutes

Question

6. Yu Ming travels to work and returns home once each day. The times, in minutes, that he takes to travel to work and to return home are represented by the independent random variables W and H with distributions N(22.4, 4.82) and N(20.3,5.222) respectively. Find the probability that, on a day, Yu Ming takes longer to return home than he takes to travel to work. a. b. The total daily time that Yu Ming takes for these two activities is denoted by T minutes. (T-W+T) i. Find the mean and variance of T. ii. Yu Ming notes the value of T on each day in a random sample of 100 days and calculates the sample mean. Find the probability that the sample mean is between 41 and 43.

Explanation / Answer

a. Let A be the random variable, A = H-W

We need to find A >0

A is a normal variable with mean = 20.3 - 22.4 = -2.1

Standard Deviation = Sqrt (4.82+5.222) = sqrt(50.2884) = 7.0914

P(A>0) = P(Z > (0-(-2.1)/7.0914)) = P(Z>0.2961) = 0.3836

b. T = W + H

i. Mean of T = 22.4 + 20.3 = 42.7

Variance of T =4.82 + 5.222 = 50.2884

Standard Deviaton = 7.0914

P (41 T 43) = P ((41-42.7)/7.0914 Z (43-42.7)/7.0914 ) = P( -0.2397 Z 0.0423) = 0.5169-0.4053 = 0.1116

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