Problem 1

The weights of a certain item are shown in the histogram below. Which statement(s) below is(are) true about the distribution of weight data?

• [ ] More than 75% of the weights fall between 330 and 370 grams
• [ ] The mean is between 370 and 380 grams
• [x] The value of the ratio is greater than 1
• [ ] None of the above

Problem 2

The side-by-side boxplots below show the distributions of ages for male and female Academy Award winners from 1977 to 2009. Which statement(s) below is(are) true about the distribution of age?

• [x] The median age for winning male actors is higher than the median age for winning female actresses.
• [ ] The distribution for winning female actresses is left skewed.
• [ ] Approximately 75% of winning male actors are between 37 and 50 years old.

Problem 3

Let , and . Then the events A and B are:

• [ ] I) Disjoint
• [x] II) Independent
• [ ] III) Complements
• [ ] IV) None of the above

Problem 4

The number of defective parts produced each hour by a certain production line has the following probability distribution:

Number of Defective Parts ()	0	1	2	3	4
	0.15	0.30	0.25	0.20	0.10

Suppose it is known that there were more than 2 defective parts produced in a particular hour. What is the probability that the number of defective parts was fewer than 4?

• [ ] I) 0.10
• [ ] II) 0.36
• [x] III) 0.67
• [ ] IV) 0.90
• [ ] V) None of the above

Problem 5

The length of a metal rod is a random variable with mean 150m and standard deviation 2m. The mean (in m) and variance (in m) of the total length of five randomly chosen metal rods will be:

• [ ] I) Mean = 150, Variance = 10
• [ ] II) Mean = 750, Variance = 10
• [ ] III. Mean = 150, Variance = 20
• [x] IV. Mean = 750, Variance = 20
• [ ] V. Mean = 750, Variance = 100

Problem 6

Consider tossing a coin 3 times, and define the following events: and . Choose one of the following answers.

• [ ]
• [x]
• [ ]
• [ ] Not enough information to tell.

Problem 7

A process for making a particular type of alloy yields up to 1 ton of alloy a day. The actual amount produced, is a random variable because of machine breakdowns and various slowdowns. Suppose Y has the following pdf

The company is paid $300 per ton of alloy, but there is also a fixed overhead cost of $100 per day. Let be the company’s daily profit (in hundreds of dollars).

(a)

Find the probability density function of .

(b)

What is the company’s expected daily profit?

Problem 8

A manufacturing company performs risk assessments to try to prevent worker injuries. Workers’ tasks are classified as low risk, medium risk and high risk. From previous records, 32% of tasks are low risk, 47% are medium risk and 21% are high risk. In a given year, the probability of a worker having an accident is 0.11 for a low risk task, 0.23 for a medium risk task and 0.44 for a high risk task.

(a)

What is the probability that a randomly selected worker will have an accident?

(b)

What is the probability that a randomly selected worker performs a medium risk task and does not have an accident?

(c)

If a randomly selected worker is known to have had an accident, what is the probability that they were performing a low risk task?

Problem 9

A computer manufacturer offers a one year warranty. If the computer fails for any reason during this period, it is replaced. The time to failure is modelled by an exponential distribution with a mean of 4 years.

(a)

Find the median time to failure.

(b)

What percentage of computers will fail within the warranty period?

Problem 10

Suppose and are two independent random variables, where and . Let . Find .