Exam 6: The Normal Distribution and Other Continuous Distributions

A company that sells annuities must base the annual payout on the probability distribution of the length of life of the participants in the plan.Suppose the probability distribution of the lifetimes of the participants is approximately a normal distribution with a mean of 68 years and a standard deviation of 3.5 years.Find the age at which payments have ceased for approximately 86% of the plan participants.

The interval between patients arriving at an outpatient clinic follows an exponential distribution at a rate of 1.5 patients per hour.What is the probability that a randomly chosen arrival to be less than 10 minutes?

Suppose Z has a standard normal distribution with a mean of 0 and standard deviation of 1.The probability that Z is less than -2.20 is .

A company that receives most of its orders by telephone conducted a study to determine how long customers were willing to wait on hold before ordering a product.The length of waiting time was found to be a variable best approximated by an exponential distribution with a mean length of waiting time equal to 3 minutes .What proportion of customers having to hold more than 4.5 minutes will hang up before placing an order?

The interval between patients arriving at an outpatient clinic follows an exponential distribution with mean 15 minutes.What is the probability that a randomly chosen arrival to be more than 18 minutes?

You were told that the amount of time elapsed between consecutive trades on a foreign stock exchange market followed a normal distribution with a mean of 15 seconds.You were also told that the probability that the time elapsed between two consecutive trades to fall between 16 to 17 seconds was 13%.The probability that the time elapsed between two consecutive trades would fall below 13 seconds was 7%.What is the probability that the time elapsed between two consecutive trades will be between 15 and 16 seconds?

The amount of time necessary for assembly line workers to complete a product is a normal variable with a mean of 15 minutes and a standard deviation of 2 minutes.The probability is that a product is assembled in between 10 and 12 minutes.

The interval between patients arriving at an outpatient clinic follows an exponential distribution at a rate of 1 patient per hour.What is the probability that a randomly chosen arrival to be less than 20 minutes?

The value of the cumulative standardized normal distribution at Z is 0.8770.The value of Z is

Let X represent the amount of time until the next student will arrive in the library parking lot at the university.If we know that the distribution of arrival time can be modeled using an exponential distribution with a mean of 4 minutes (i.e.the mean number of arrivals is 1/4 per minute),find the probability that it will take more than 10 minutes for the next student to arrive at the library parking lot.

You were told that the amount of time elapsed between consecutive trades on a foreign stock exchange market followed a normal distribution with a mean of 15 seconds.You were also told that the probability that the time elapsed between two consecutive trades to fall between 16 to 17 seconds was 13%.The probability that the time elapsed between two consecutive trades would fall below 13 seconds was 7%.The probability is 80% that the time elapsed will be longer than how many seconds?

SCENARIO 6-2 John has two jobs.For daytime work at a jewelry store he is paid $15,000 per month,plus a commission.His monthly commission is normally distributed with mean $10,000 and standard deviation $2000.At night he works occasionally as a waiter,for which his monthly income is normally distributed with mean $1,000 and standard deviation $300.John's income levels from these two sources are independent of each other. -Referring to Scenario 6-2,for a given month,what is the probability that John's commission from the jewelry store is between $11,000 and $12,000?

SCENARIO 6-5 A company producing orange juice buys all its oranges from a large orange orchard.The amount of juice that can be squeezed from each of these oranges is approximately normally distributed with a mean of 4.7 ounces and some unknown standard deviation.The company's production manager knows that the probability is 30.85% that a randomly selected orange will contain less than 4.5 ounces of juice.Also,the probability is 10.56% that a randomly selected orange will contain more than 5.2 ounces of juice.Answer the following questions without the help of a calculator,statistical software or statistical table. -Referring to Scenario 6-5,what is the probability that a randomly selected orange will contain more than 4.2 ounces of juices?

The "middle spread," that is the middle 50% of the normal distribution,is equal to one standard deviation.

SCENARIO 6-5 A company producing orange juice buys all its oranges from a large orange orchard.The amount of juice that can be squeezed from each of these oranges is approximately normally distributed with a mean of 4.7 ounces and some unknown standard deviation.The company's production manager knows that the probability is 30.85% that a randomly selected orange will contain less than 4.5 ounces of juice.Also,the probability is 10.56% that a randomly selected orange will contain more than 5.2 ounces of juice.Answer the following questions without the help of a calculator,statistical software or statistical table. -Referring to Scenario 6-5,what is the probability that a randomly selected orange will contain at least 4.9 ounces of juices?

Suppose Z has a standard normal distribution with a mean of 0 and standard deviation of 1.The probability that Z is between -2.89 and -1.03 is .

Suppose Z has a standard normal distribution with a mean of 0 and standard deviation of 1.The probability that Z values are larger than is 0.3483.

If there is an order for 60,000 ball bearings and it states the bearing diameters must be between 5.96 and 6.04 mm,how many should the manager manufacture?

You were told that the amount of time elapsed between consecutive trades on a foreign stock exchange market followed a normal distribution with a mean of 15 seconds.You were also told that the probability that the time elapsed between two consecutive trades to fall between 16 to 17 seconds was 13%.The probability that the time elapsed between two consecutive trades would fall below 13 seconds was 7%.The middle 86% of the time elapsed will fall between which two numbers?

For some value of Z,the value of the cumulative standardized normal distribution is 0.2090.The value of Z is