Probability and Statistics Course Midterm Exam

Name___________________________________ Due by 10/12/10 class time

Show your work. Only the answer will not be evaluated. Solve any 8 of 10 questions. 2 questions are bonus.

Find the mean for the given sample data. Unless indicated otherwise, round your answer to one more decimal place than is present in the original data values.

1)

The students in Hugh Logan's math class took the Scholastic Aptitude Test. Their math scores are shown below. Find the mean score, median, and the variance.

1)

_____________

Find the indicated probability. Round to the nearest thousandth.

2)

In a batch of 8,000 clock radios 5% are defective. A sample of 14 clock radios is randomly selected without replacement from the 8,000 and tested. The entire batch will be rejected if at least one of those tested is defective. What is the probability that the entire batch will be rejected?

2)

_____________

Answer the question.

3)

If an apple is hanging from a string and three flies land on it, find the probability that all three are on points that are within the same hemisphere.

3)

_____________

Use Bayes' theorem to find the indicated probability.

4)

Shameel has a flight to catch on Monday morning. His father will give him a ride to the airport. If it rains, the traffic will be bad and the probability that he will miss his flight is 0.05. If it doesn't rain, the probability that he will miss his flight is 0.02. The probability that it will rain on Monday is 0.30.

Suppose that Shameel misses his flight. What is the probability that it was raining?

4)

_____________

Find the indicated probability. Round to three decimal places.

5)

A machine has 12 identical components which function independently. The probability that a component will fail is 0.2. The machine will stop working if more than three components fail. Find the probability that the machine will be working.

5)

_____________

Provide an appropriate response.

6)

If a procedure meets all the conditions of a binomial distribution except that the number of trials is not fixed, then the geometric distribution can be used. The probability of getting the first success on the xth trial is given by where p is the probability of success on any one trial. Assume that the probability of choosing a yellow piece of candy in a bag of hard candy is 0.231. Find the probability that the first yellow candy is found in the tenth inspected. Round your answer to the nearest thousandth.

6)

_____________

Use the Poisson Distribution to find the indicated probability.

7)

In one town, the number of burglaries in a week has a Poisson distribution with a mean of 4.6. Find the probability that in a randomly selected week the number of burglaries is at least three.

7)

_____________

Solve the problem.

8)

On a multiple choice test with 19 questions, each question has four possible answers, one of which is correct. For students who guess at all answers, find the mean and standard deviation for the number of correct answers.

8)

_____________

Provide an appropriate response.

9) A ssume that a probability distribution is described by the discrete random variable x

that can assume the values 1, 2, . . . , n; and those values are equally likely. This

probability has mean and standard deviation described as follows:

μ = and σ =

Show that the formulas hold for the case of n = 7.

9) _ _ ___________

10) The joint density function of the random variables X and Y is

f (x, y) =

0, .

, 1 3, 1 2,

9

3

otherwise

x y

x y

a) Find marginal density functions of X and Y

b) Are X and Y are idependent?

c) Find P(X>2).

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