Exam 6: The Normal Distribution and Other Continuous Distributions

Scientists in the Amazon are trying to find a cure for a deadly disease that is attacking the rain forests there. One of the variables that the scientists have been measuring involves the diameter of the trunk of the trees that have been affected by the disease. Scientists have calculated that the average diameter of the diseased trees is 42 centimeters. They also know that approximately 95% of the diameters fall between 32 and 52 centimeters and almost all of the diseased trees have diameters between 27 and 57 centimeters. When modeling the diameters of diseased trees, which distribution should the scientists use?

The amount of tea leaves in a can from a particular production line is normally distributed with μ = 110 grams and σ = 25 grams. What is the probability that a randomly selected can will contain between 100 and 110 grams of tea leaves?

Patients arriving at an outpatient clinic follow an exponential distribution at a rate of 1 patient per hour. What is the probability that a randomly chosen arrival to be more than 2.5 hours?

TABLE 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 $2,000. At night he works 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 Table 6-2, for a given month, what is the probability that John's income as a waiter is less than $1,300?

If we know that the length of time it takes a college student to find a parking spot in the library parking lot follows a normal distribution with a mean of 3.5 minutes and a standard deviation of 1 minute, find the probability that a randomly selected college student will find a parking spot in the library parking lot in less than 3 minutes.

You were told that the mean score on a statistics exam is 75 with the scores normally distributed. In addition, you know the probability of a score between 55 and 60 is 4.41% and that the probability of a score greater than 90 is 6.68%. What is the probability of a score between 75 and 90?

TABLE 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 $2,000. At night he works 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 Table 6-2, for a given month, what is the probability that John's commission from the jewelry store is at least $12,000?

The amount of time necessary for assembly line workers to complete a product is a normal random variable with a mean of 15 minutes and a standard deviation of 2 minutes. The probability is ________ that a product is assembled in between 16 and 21 minutes.

A catalog company that receives the majority 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 time was found to be a random variable best approximated by an exponential distribution with a mean equal to 2.8 minutes. What proportion of callers is put on hold longer than 2.8 minutes?

TABLE 6-4 The interval between consecutive hits at a web site is assumed to follow an exponential distribution with an average of 40 hits per minute. -Referring to Table 6-4, what is the probability that the next hit at the web site will occur within 10 seconds after just being hit by a visitor?

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

The probability that a standard normal random variable, Z, is between 1.50 and 2.10 is the same as the probability Z is between -2.10 and -1.50.

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. What proportion of the plan recipients die before they reach the standard retirement age of 65?

TABLE 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 $2,000. At night he works 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 Table 6-2, for a given month, what is the probability that John's commission from the jewelry store is between $9,000 and $11,000?

The owner of a fish market determined that the average weight for a catfish is 3.2 pounds with a standard deviation of 0.8 pound. Assuming the weights of catfish are normally distributed, the probability that a randomly selected catfish will weigh between 3 and 5 pounds is ________?

The owner of a fish market determined that the average weight for a catfish is 3.2 pounds with a standard deviation of 0.8 pound. Assuming the weights of catfish are normally distributed, the probability that a randomly selected catfish will weigh less than 2.2 pounds is ________?

Any set of normally distributed data can be transformed to its standardized form.

TABLE 6-3 Suppose the time interval between two consecutive defective light bulbs from a production line has a uniform distribution over an interval from 0 to 90 minutes. -Referring to Table 6-3, what is the standard deviation of the time interval?

TABLE 6-1 The number of column inches of classified advertisements appearing on Mondays in a certain daily newspaper is normally distributed with population mean 320 and population standard deviation 20 inches. -Referring to Table 6-1, a single Monday is chosen at random. State in which of the following ranges the number of column inches of classified advertisement is most likely to be:

The amount of time necessary for assembly line workers to complete a product is a normal random variable with a mean of 15 minutes and a standard deviation of 2 minutes. So, 90% of the products require more than ________ minutes for assembly.