Let Y1 follow a normal distribution with mean 50 and varianc

Let Y₁ follow a normal distribution with mean 50 and variance 12. Let Y₂ follow a normal distribution with mean 60 and variance 9. A random sample of size 4 is drawn from Y₁ and another independent random sample of size 9 is drawn from Y₂. Let sample mean of the first sample be denoted by M₁ and sample mean of the second sample be denoted M₂. Then Z = M₁ - M₂ follows a normal distribution

A. mean 10 and standard deviation 2

B. mean -10 and standard deviation 2

C. mean -10 and standard deviation 2

D. mean 10 and standard deviation 2

E. None of the above

Solution

The mean of Z = M1 - M2 = Y1bar - Y2bar = 50 - 60 = -10.

The standard deviation of their difference is

s(M1 - M2) = sqrt(var1/n1 + var2/n2)

Thus,

s(M1 - M2) = sqrt(12/4 + 9/9) = sqrt(4) = 2.

Thus, the answer is OPTION B: mean = -10, standard deviation = 2. [ANSWER]