1. A certain diagnostic test for a rare form of cancer detects the cancer | Complete Solution

MOCK IN-CLASS FINAL

1. A certain diagnostic test for a rare form of cancer detects the cancer if it is present 98 % of the time. If cancer is not present it detects it incorrectly (false positive) 2 % of the time. It is known from a prior study that 0.25 % of the population have this rare disease.
a. (3 pts) Find the probability that a person chosen at random from this population will test positive.
b. (3 pts) If a person chosen at random tests positive, find the probability that the person in fact has the cancer.

2. The distribution of brood size X (number of young in a nest) of a certain species of bird is given as follows:
Brood Size
X
P (X = x)
1
0.082
2
0.108
3
0.378
4
0.189
5
0.243
a. (3 pts) Find the Pr ( 2 ≤ X ≤ 4 )
b. (3 pts) Find the population mean of X
c. (3 pts) Find the population standard deviation of X

3. (3 pts) Five infants are chosen at random from a population where the male:female ratio of newborns is known to be 3:5. What is the probability that of the five selected 3 are male and 5 are female?

4. (3 pts) The brain weight of a certain species of elephants follow a normal distribution with mean 1200 grams and standard deviation 75 grams. What percentage of these species will have brain weights between 1125 and 1275 grams?

5. Repeated red blood cell counts X of the same blood sample indicate that the standard deviation is .008μ where μ is the true mean count. If X is normally distributed with mean μ
a. (3 pts) Find the probability Pr (0.98μ ≤ X ≤ 1.02μ).
b. (3 pts) What percentage of the readings will differ from the correct value by at least 2 %?
6. (3 pts) When a computer produced repeated random samples of size n from a population that is N(50, 9), it was found that about 62 % of sample means lie between the values 44 and 56. Find n to the closest integer value. Assume n to be large enough that the Central Limit Theorem is applicable.

7. (3 pts) The average repair cost for a microwave oven has a normal distribution with a mean of
$55, and with a standard deviation of $8. If 12 ovens are sent for repairs, find the probability that
the cost of repairs for 6 or more ovens will not exceed $55.

8. Given that X has a binomial distribution with values of n and p as given below. Use the
normal approximation to the binomial for each of the corresponding events described:
a. (3 pts) N = 50, p = .7 P40  X  60
b. (3 pts) N = 100, p = .4 PX  50
9. (3 pts) If X is distributed as   X X N  , and Y is distributed as   Y Y N  , and X and Y are independent then the distribution of aX + bY is  2 2 2 2  , X Y X Y N a  b a   b  . Suppose further that  5, 10,  2, 1 X Y X Y     . Find the P(35 ≤ 2X + 3Y ≤ 45)

10. (3 pts). A random sample of size 20 was drawn from a Normal population with unknown
mean μ and known standard deviation σ = 2. Suppose the sample average is 10, find a 95%
confidence interval for μ and interpret what it means.