Question

Use an appropriate normal random variable.

-Find the value of Z such that the area to the right of the Z is 0.72.

-The middle 99% of the standard normal distribution is contained between −Z and Z. Find these values.

-Suppose that the area that can be painted using a single can of spray paint is slightly variable and follows a normal distribution with a mean of 25 square feet and a standard deviation of 3 square feet. What is the probability that the area covered by a can of spray paint is between 21 and 26 square feet?

Answer 1

1. Let, P(Z > z) = 0.72 i.e. P(Z < z) = 1 - 0.72 = 0.28 i.e. (z) = 0.28 [(.) is the cdf of N(0,1)] i.e. z = (0.28) = - 0.583.

2. Let, P(-a < Z < a) = 0.99 i.e. (a) - (-a) = 0.99 i.e. 2(a) - 1 = 0.99 i.e. (a) = 0.995 i.e. a = (0.995) = 2.576.

3. Let X be the random variable denoting the area.

X ~ N(25, 3) i.e. (X - 25)/3 ~ N(0,1)

Hence, P(21 < X < 26) = P[(21 - 25)/3 < (X - 25)/3 < (26 - 25)/3] = P[-1.3333 < (X - 25)/3 < 0.3333] = (0.3333) - (-1.3333) = 0.6305 - 0.0912 = 0.5393.