Exam 6: The Normal Distribution and Other Continuous Distributions

Suppose the probability of finding a bark beetles-infested pine tree is the same anywhere over a piece of 100-acre national forest land. Which of the following distributions would you use to determine the probability of finding a bark beetles-infested pine tree in a piece of 10-acre national forest land?

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, John's income as a waiter will be between what two values symmetrically distributed around the population mean 90% of the time?

Let X represent the amount of time till 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 between 2 and 12 minutes for the next student to arrive at the library parking lot.

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 probability that the time interval between two consecutive defective light bulbs will be between 10 and 35 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 "middle spread," that is the middle 50% of the normal distribution, is equal to one standard deviation.

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?

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 82 and 100 grams of tea leaves?

The interval between patients arriving at an outpatient clinic follows an exponential distribution with mean 15 minutes. What is the mean number of arrivals per minute?

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, 15% of the products require more than ________ minutes for assembly.

TABLE 6-6 According to Investment Digest, the arithmetic mean of the annual return for common stocks from 1926-2010 was 9.5% but the value of the variance was not mentioned. Also 25% of the annual returns were below 8% while 65% of the annual returns were between 8% and 11.5%. The article claimed that the distribution of annual return for common stocks was bell-shaped and approximately symmetric. Assume that this distribution is normal with the mean given above. Answer the following questions without the help of a calculator, statistical software or statistical table. -Referring to Table 6-6, find the probability that the annual return of a random year will be more than 11.5%.

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 55 and 90?

Which of the following about the normal distribution is not true?

The true length of boards cut at a mill with a listed length of 10 feet is normally distributed with a mean of 123 inches and a standard deviation of 1 inch. What proportion of the boards will be less than 124 inches?

The probability that a standard normal variable Z is positive is ________.

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 more than 2.5 hours?

Suppose the probability of a car accident taking place anywhere on a stretch of a 20 miles highway is the same. Which of the following distributions would you use to determine the probability that a car accident will occur somewhere between the 5-mile and 15-mile posts of the highway?

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 probability that the time interval between two consecutive defective light bulbs will be at least 80 minutes?

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

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%. The middle 86.64% of the students will score between which two scores?