5 Probability Problems: Bayesian Probability and Plots

1. The following data are monthly steel production figures, in millions of tons.

7.0, 6.9, 8.2, 7.8, 7.7, 7.3, 6.8, 6.7, 8.2, 8.4, 7.0, 6.7, 7.5, 7.2, 7.9, 7.6, 6.7, 6.6, 6.3, 5.6, 7.8, 5.5, 6.2, 5.8, 5.8, 6.1, 6.0, 7.3, 7.3, 7.5, 7.2, 7.2, 7.4, 7.6

a. Draw a stem-and-leaf display of these data.
b. Draw a box plot of these data.
c. Are there any outliers?
d. Is the distribution of the data symmetric or skewed?
e. If it is skewed, to what side?

2. A chemical plant has an emergency alarm system. When an emergency situation exists, the alarm sounds with a probability 0.95. When an emergency situation does not exist, the alarm system sounds with probability 0.02. A real emergency situation is a rare event with probability 0.004. Given that the alarm has just sounded, what is the probability that a real emergency situation exists?

3. A computer laboratory in a school has 33 computers. Each of the 33 computers has a 90% reliability. Allowing for 10% of the computers to be down, an instructor specifies an enrollment ceiling of 30 for his class. Assume that a class of 30 students is taken into the lab.

a. What is the probability that each of the 30 students will get a computer in working condition?
b. The instructor is shocked to see the low value of the answer to (a), and decides to improve it to 95% by doing one of the following:

i. Decreasing the enrollment ceiling.
ii. Increasing the number of computers in the lab.
iii. Increasing the reliability of all the computers.

To help the instructor, find out what the increase or decrease should be for each
of the three alternatives.

4. The time it takes an international telephone operator to place an overseas phone call is normally distributed with a mean of 45 seconds, and a standard deviation of 10 seconds.

a. What is the probability that my call will go through in less than 1 minute?
b. What is the probability that will get through in less than 40 seconds?
c. What is the probability that I will have to wait more than 70 seconds for my call to go through?

5. An economist wishes to estimate the average family income in a certain population. The population standard deviation is known to be $4,500, and the economist uses a random sample of size = 225. What is the probability that the sample mean will fall within $800 of the population mean?