1. The following table provides a probability distribution for the random variable x.

X f(x)

20 .20

25 .15

30 .25

35 .40

1. What is the probability that X = 28
2. What is the probability that X = 30
3. What is the probability that X is less than or equal to 25.
4. What is the expected value of X
5. Compute the variance of X
6. Compute the standard deviation of X .
7. Is this a discrete or random variable or a continuous random variable? Why?

2. The label on a bottle of liquid detergent shows the content to be 12 ounces per bottle. The production operation fills the bottle uniformly according to the following probability density function:

f(x) = 8 for 11.975 ≤ x ≤ 12.100, and

f(x) = 0 elsewhere

1. Plot the graph of the probability density function using Excel 2007.
2. What is the probability that a bottle will be filled with between 12 and 12.05 ounces?
3. What is the probability that a bottle will be filled with 12.02 or more ounces?.

3. The average amount of precipitation in Dallas, Texas, during the month of April is 3.5 inches (the World Almanac, 2000). Assume that a normal distribution applies and that the standard deviation is .6 inches.

Answer the following questions using Excel.

1. What percent of the time does the amount of rainfall in April Exceed 5 inches?
2. What percent of time is the amount of rainfall in April less than 3 inches?
3. A month is classified as extremely wet if the amount of rainfall is in the upper 10% per the month. How much precipitation must fall in April for it to be classified as extremely wet?

4. Given that z is a standard normal random variable. Using Excel, compute the following probabilities.

a. P(z ≤ 1)

b. P(-.50 < z ≤ 1.25)

c. P(z > 1.58)

5. Given that z is a standard normal random variable. Using Excel, find the value of z such tat

a. the area to its right is .10.

b. the area to its right is .025.

6. The probability distribution for the random variable x follows:

X f(x)

20 .20

25 .15

30 .25

35 .40

a. Is this probability distribution valid? Explain.

b. What is the probability that X = 30?

c. What is the probability that X is less than or equal to 25?

d. What is the probability that X is greater than 30?

7. The random variable X is known to be uniformly distributed between 1.0 and 1.5.

a. Show the graph of the probability density function.

b. Compute P( X = 1.25)

c. Compute P(1.0 ≤ X ≤ 1.25)

d. Compute P(1.20 < X < 1.5).

8. The random variable x is known to be uniformly distributed between 10 and 20.

a. Show the graph of the probability density function.

b. Compute P( X < 15)

c. compute P(12 ≤ X ≤ 18).

d. compute E( X )

e. Compute Var ( X )

9. Delta Airlines quotes a flight time of 2 hours, 5 minutes for its flights from Cincinnati to Tampa. Suppose we believe that actual flight times are uniformly distributed between 2 hours and 2 hours, 20 minutes.

a. Show the graph of the probability density function for flight time.

b. What is the probability that the flight will be no more than 5 minutes late?

c. What is the probability that the flight will be more than 10 minutes late?

d. What is expected flight time?

10. Give that z is a standard normal random variable, compute the following probabilities.

a. P(0 ≤ z ≤ .83) =

b. P(-1.57 ≤ z ≤ 0) =

c. P(z > .44) =

d. P(z ≥ -.23) =

e. P(z <1.20) =

f. P(z ≤ -.71) =

11. Given that z is a standard normal random variable, compute the following probabilities.

a. P(-1.98 ≤ z ≤ .49) =

b. P (.52 ≤ z ≤ 1.22) =

c. P(-1.75 ≤ z ≤ -1.04) =

12. Given that z is a standard normal random variable, find z for each situation.

a. The area to the left of z is .9750.

b. The area between 0 and z is .4750.

c. The area to the left of z is .7291.

d. The area to the right of z is .1314.

e. The area to the left of z is .6700

f. The area to the right of z is .3300.

13. Given that z is a standard normal random variable, find z for each situation.

a. The area to the left of z is .2119.

b. The area between –z and z is .9030.

c. The area between –z and z is .2052.

d. The area to the left of z is .9948

e. The area to the right of z is .6915.

14. Given that z is a standard normal random variable, find z for each situation.

a. That area to the right of z is .01

b. The area to the right of z is .025

c. The area to the right of z is .05

d. The area to the right of z is .10

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