STA 3032 Homework 2
1. (10 points) Of a population of consumers, 20% are known to prefer a particular
brand A of toothpaste. A series of interviews is conducted on randomly selected
customers. Assume each person’s preference is independent of any other person.
(a) Find the probability that exactly six people have to be interviewed to en-
counter the first consumer who prefers brand A.
(b) On average, how many people do we expect to interview to encounter the first
consumer who prefers brand A?
2. (10 points) For a certain section of a pine forest, there are on average 9 diseased
trees per acre. Assume the number of diseased trees per acre has a Poisson dis-
tribution. The diseased trees are sprayed with an insecticide at a cost of $3 per
tree, plus a fixed overhead cost for equipment rental of $50. Let C denote the total
spraying cost for a randomly selected acre.
(a) Find the expected value of C and std deviation of C.
(b) Using Chebyshev’s inequality, find an interval where we would expect C to lie
with probability at least 0.75.
3. (10 points) Let two random variables X and Y are jointly distributed with finite
means and variances. Show that, for any 4 constants a, b, c, d, e, f ;
Cov(aX +bY +e, cX +dY +f) = ac·V ar(X)+bd ·V ar(Y)+ (ad +bc)·C ov(X , Y )
[Hint: Recall Cov(X, Y ) = E(XY )−E(X)E(Y) for any two random variables
X and Y. Try to write down the Covariance term in Expectation of product of
certain quantities, then expand the term inside expectation and use linearity of
expectation]
4. (10 points) Let the random variable X is normally distributed with mean µ= 18
and standard deviation σ= 2.5. Calculate
(a) P(X < 15)
(b) P(17 < X < 21)
(c) Value of k such that P(X > k)=0.1814
5. (5 points) A traffic control engineer reports that 75% of the vehicles passing through
a checkpoint are from within the state. What is the probability that fewer than 4
of the next 9 vehicles are from out of state?
6. (5 points) The heights of a random sample of 50 college students showed a mean
of 174.5 centimeters and a standard deviation of 6.9 centimeters. Construct a 95%
confidence interval for the mean height of all college students. (t0.025,49 = 2.01)
