Question

The height, in inches, of a randomly chosen American woman is a normal random variable with mean μ = 64 and variance 2 = 7.84. (a) Calculate the probability that the height of a randomly chosen woman is between 59.8 and 68.2 inches. (b) Given that a randomly chosen woman is tall enough to be an astronaut (i.e., at least 59 inches tall), what is the conditional probability that she is at least 67 inches in height? (c) Four women are chosen at random. Calculate the probability that their total height is at least 260 inches. (d) Suppose that Z ∼ N(0, 1). Prove that E(Z19) = 0.

Answer 1

Answer:

Step-by-step explanation:

Given that the height in  inches, of a randomly chosen American woman is a normal random variable with mean μ = 64 and variance 2 = 7.84.

X is N(64, 2.8)

Or Z = $\frac{x-64}{2.8}$

a)  the probability that the height of a randomly chosen woman is between 59.8 and 68.2 inches.

$=P(59.8<X<68.2)\\= P(|Z|<1.5)\\=0.8664$

b) $P(X\geq 59)\\= P(X\geq -1.78)\\ \\=0.9625$

c) For 4 women to be height 260 inches is equivalent to

4x will be normal with mean (64*4) and std dev (2.8*4)

4x is N(266, 11.2)

$P(4x>260)= \\P(Z\geq -0.53571)\\=0.7054$

d) Z is N(0,1)

E(Z19) = $P(Z>19)\\= 0.000$

since normal distribution is maximum only between 3 std deviations form the mean on either side.