# MATHEMATICS

Find the following probabilities based on the standard normal variable Z. Use Table 1. (Round your answers to 4 decimal places.)

a. P(Z > 0.74)

b. P(Z ≤ −1.92)

c. P(0 ≤ Z ≤ 1.62)

d. P(−0.90 ≤ Z ≤ 2.94)

2, Find the following z values for the standard normal variable Z. Use Table 1. (Negative values should be indicated by a minus sign. Round your answers to 2 decimal places.)

a. P(Z ≤ z) =

b. P(Z > z) =

c. P(−z ≤ Z ≤ z)

d. P(0 ≤ Z ≤ z) =

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3. Let X be normally distributed with mean μ = 10 and standard deviation σ = 6. Use Table 1.

a. Find P(X ≤ 0). (Round “z” value to 2 decimal places and final answer to 4 decimal places.)
b. Find P(X > 2). (Round “z” value to 2 decimal places and final answer to 4 decimal places.)
c. Find P(4 ≤ X ≤ 10). (Round “z” value to 2 decimal places and final answer to 4 decimal places.)
d. Find P(6 ≤ X ≤ 14). (Round “z” value to 2 decimal places and final answer to 4 decimal places.)

4. Let X be normally distributed with mean μ = 120 and standard deviation σ = 20. Use Table 1.

a. Find P(X ≤ 86). (Round “z” value to 2 decimal places and final answer to 4 decimal places.)
b. Find P(80 ≤ X ≤ 100). (Round “z” value to 2 decimal places and final answer to 4 decimal places.)
c. Find x such that P(X ≤ x) = 0.40. (Round “z” value to 2 decimal places.)
d. Find x such that P(X > x) = 0.90. (Round “z” value to 2 decimal places and final answer to 1 decimal place.)

5.Let X be normally distributed with mean μ = 2500 and standard deviation σ = 800. Use Table 1.

a. Find x such that P(X ≤ x) = 0.9382. (Round “z” value to 2 decimal places, and final answer to nearest whole number.)
b. Find x such that P(X > x) = 0.025. (Round “z” value to 2 decimal places, and final answer to nearest whole number.)
c. Find x such that P(2500 ≤ X ≤ x) = 0.1217. (Round “z” value to 2 decimal places, and final answer to nearest whole number.)
d. Find x such that P(X ≤ x) = 0.4840. (Round “z” value to 2 decimal places, and final answer to nearest whole number.)

6. The time required to assemble an electronic component is normally distributed with a mean and a standard deviation of 16 minutes and 8 minutes, respectively. Use Table 1.

a. Find the probability that a randomly picked assembly takes between 10 and 20 minutes. (Round “z” value to 2 decimal places and final answer to 4 decimal places.)
b. It is unusual for the assembly time to be above 24 minutes or below 6 minutes. What proportion of assembly times fall in these unusual categories? (Round “z” value to 2 decimal places and final answer to 4 decimal places.)

7. Average talk time between charges of a cell phone is advertised as 4 hours. Assume that talk time is normally distributed with a standard deviation of 0.8 hour. Use Table 1.

a. Find the probability that talk time between charges for a randomly selected cell phone is below 3.5 hours.(Round “z” value to 2 decimal places and final answer to 4 decimal places.)
b. Find the probability that talk time between charges for a randomly selected cell phone is either more than 4.5 hours or below 3.5 hours. (Round “z” value to 2 decimal places and final answer to 4 decimal places.)
c. Twenty-five percent of the time, talk time between charges is below the 1st quartile value. What is this value? (Round “z” value to 2 decimal places and final answer to 3 decimal places.)

8. A construction company in Naples, Florida, is struggling to sell condominiums. In order to attract buyers, the company has made numerous price reductions and better financing offers. Although condominiums were once listed for \$300,000, the company believes that it will be able to get an average sale price of \$210,000. Let the price of these condominiums in the next quarter be normally distributed with a standard deviation of \$15,000. Use Table 1.

a. What is the probability that the condominium will sell at a price (i) below \$200,000?, (ii) above \$240,000? (Round “z” value to 2 decimal places and final answer to 4 decimal places.)

Probability
Below \$200,000

Above \$240,000

b. The company is also trying to sell an artist’s condo. Potential buyers will find the unusual features of this condo either pleasing or objectionable. The manager expects the average sale price of this condo to be the same as others at \$210,000, but with a higher standard deviation of \$20,000. What is the probability that this condo will sell at a price (i) below \$200,000?, (ii) above \$240,000? (Round your answers to 4 decimal places.)

Probability
Below \$200,000

Above \$240,000

9.A random variable X is exponentially distributed with a mean of 0.1.
a-1. What is the rate parameter λ? (Round your answer to 3 decimal places.)
Rate parameter λ

a-2. What is the standard deviation of X? (Round your answer to 3 decimal places.)
Standard deviation X
b. Compute P(X > 0.20). (Round intermediate calculations to 4 decimal places and final answer to 4 decimal places.)
P(X > 0.20)
c. Compute P(0.10 ≤ X ≤ 0.20). (Round intermediate calculations to 4 decimal places and final answer to 4 decimal places.)
P(0.10 ≤ X ≤ 0.20)

10. When crossing the Golden Gate Bridge, traveling into San Francisco, all drivers must pay a toll. Suppose the amount of time (in minutes) drivers wait in line to pay the toll follows an exponential distribution with a probability density function of f(x) = 0.2e−.2x.

a. What is the mean waiting time that drivers face when entering San Francisco via the Golden Gate Bridge?
Mean waiting time

b. What is the probability that a driver spends more than the average time to pay the toll? (Round intermediate calculations to 4 decimal places and final answer to 4 decimal places.)
Probability

c. What is the probability that a driver spends more than 10 minutes to pay the toll? (Round intermediate calculations to 4 decimal places and final answer to 4 decimal places.)
Probability

d. What is the probability that a driver spends between 4 and 6 minutes to pay the toll? (Round intermediate calculations to 4 decimal places and final answer to 4 decimal places.)
Probability

11. Lisa Mendes and Brad Lee work in the sales department of an AT&T Wireless Store. Lisa has been signing in an average of 48 new cell phone customers every month with a standard deviation of 22, while Brad signs in an average of 56 new customers with a standard deviation of 17. The store manager offers both Lisa and Brad a \$100 incentive bonus if they can sign in more than 100 new customers in a month. Assume a normal distribution to answer the following questions. Use Table 1.

a. What is the probability that Lisa will earn the \$100 incentive bonus? (Round “z” value to 2 decimal places and final answer to 4 decimal places.)
Probability

b. What is the probability that Brad will earn the \$100 incentive bonus? (Round “z” value to 2 decimal places and final answer to 4 decimal places.)
Probability

12.  Entrance to a prestigious MBA program in India is determined by a national test where only the top 10% of the examinees are admitted to the program. Suppose it is known that the scores on this test are normally distributed with a mean of 420 and a standard deviation of 80. Parul Monga is trying desperately to get into this program. What is the minimum score that she must earn to get admitted? Use Table 1. (Round “z” value to 2 decimal places and final answer to 1 decimal place.)

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