1. Let X be binomial random variable with parameters n = 100 and p = 0.65. Use normal approximation with the continuity correction to compute the following probabilities:

(a) P(X ? 50); (b) P(60 ? X ? 70); (c) P(X < 75).

2. Let X be the number of cavities that develop in a 6-month
period in the mouth of a child that uses the new brand of toothpaste
“Cavifree”. The distribution of X is shown below.c 0 1 2 3 P(X = c) 0.4
0.3 0.2 0.1

a) A family has three children and they all use Cavifree. Assuming
that the number of cavities acquired by any one child is independent of
the number acquired by any other child, ?nd the probability that between
them they acquire at most one cavity in a 6-month period.

b) Find the expected value and the standard deviation of X.

c) A boarding school has 150 students and they all use Cavifree.
Use the CLT to approximate the probability that the students acquire
more than a total of 200 cavities in a 6-month period. (Again, you may
assume that the number of cavities acquired by the di?erent students are
independent.)3. The waiting time T, in minutes, for the green light at a
certain intersection is a random variable with the following
probability density function: fT(t) =½3t2, 0 < t < 1, 0, otherwise
Using the CLT, ?nd the approximate probability that, after driving
through the intersection 60 times, you will have spent the total of more
than 45 minutes waiting for the green light. 4. Using either a computer
or pencil, paper and your knowledge, draw the normal probability plots
for the following distributions. Then use your knowledge to explain why
the graphs look the way they do. f1(x) = 1 2??e?x2/4; f2(x) = 1 ?(x2 +1)
; f3(x) = cx2e?x2; f4(x) = cx4e?x; f5(x) = cx?1/2e?x3. For f3,f4, and
f5, we assume the function to be zero for x?0 and choose c so that the
function is a probability density.