Probability: Binomial, Normal

Probability

A certain state is contemplating creating a weekly lottery, the revenues from which will be used to fund improvements in the state’s public education system. The commission chartered to develop the guidelines for the proposed lottery envisions using a process whereby six (6) balls will be randomly selected from a single bin containing a total of forty (40) balls, each of which is individually numbered 1 through 40, in order to determine the winning lottery numbers each week. Once a given ball has been randomly selected it will not be placed back in the bin before selecting the next ball. The commission is currently debating whether an individual should be required to pick the winning lottery numbers in a specific order or be allowed to pick them in a random order.

1. Assuming that the order in which the winning lottery numbers are selected is irrelevant (i.e. not important), what is probability of someone correctly selecting the six (6) winning lottery numbers?

2. Assuming that the order in which the winning lottery numbers are selected is relevant (i.e. is important), what is probability of someone correctly selecting the six (6) winning lottery numbers?

33 students in a college Physics 101 course recently took a mid-term exam. 8 students earned an A, 8 students earned a B, 6 students earned a C, 5 students earned a D and 6 students earned an F on the exam. The students were queried regarding the number of hours they had devoted to studying for the exam. 7 of the students who earned an A, 6 of the students who earned a B, 4 of the students who earned a C, 1 of the students who earned a D, and 1 of the students who earned an F reported that they had devoted more than 8 hours to studying for the exam. The remaining students reported that they had devoted no more than 8 hours to studying for the exam.

3. What is the probability of a randomly selected student having earned an F on the exam?

4. What is the probability of a randomly selected student having devoted more than 8 hours to studying for the exam?

5. What is the probability of a randomly selected student having earned an F on the exam given they devoted no more than 8 hours to studying for the exam?

6. What is the probability of a randomly selected student having earned an A or a B on the exam given they devoted more than 8 hours to studying for the exam?

Frequency Distributions

A graduate student is conducting a study involving individuals who claim to have paranormal capabilities. Participants are asked to reach into a bag that contains seven green balls and three red balls and select a red ball using only their paranormal ability to determine which balls are red (i.e., the participants cannot see the balls in the bag when making their selection). Each participant is given 50 attempts to select a correct colored ball. An attempt is coded as a “success” when the participant selects a red ball or a “failure” when the participant selects a green ball (i.e., the study is structured as a binomial experiment).

7. What is the probability that a randomly selected participant will select exactly 15 red balls?

8. What is the probability that a randomly selected participant will select no more than 15 red balls?

9. What is the probability that a randomly selected participant will select more than15 red balls?

The time required to complete a certain type of construction project is normally distributed with a mean of 60 weeks and a standard deviation of 4 weeks.

10. What is the probability of the project completing in no more than 56 weeks?

11. What is the probability of the project completing in more than 64 weeks?

Customers arrive at a supermarket check-out counter following a Poisson distribution with an average arrival rate of 5 customers per hour. Customers are checked out following an exponential distribution with an average service rate of 6 customers per hour.

12. What is the probability of exactly 5 customers arriving at the supermarket check-out counter in a given one hour period?

13. What is the probability the customer service time will be less than or equal to the expected average service time in a given one hour period?

Probability – Binomial, Normal

1. The Arizona State Office of Tourism Development compiles information about the scenic attractions visited by out-of-state vacationers. The office reports that 75% of out-of-state vacationers visit the Grand Canyon, 35% visit the Sunset Crater (Meteor Crater), and 20% visit both. What is the probability that an out-of-state vacationer will visit at least one of these scenic attractions?

2. You are the sales manager of a life insurance agency. All of your sales representatives are successful in selling a policy on 30% of their sales calls. Determine the probability of a sales representative’s selling at least four policies if he (she) makes five sales calls.

3. Crash Airlines flies from Los Angeles to Phoenix. Their mean flying time is 70 minutes with a standard deviation of 6 minutes. Flying time is approximately normally distributed. You are meeting a friend at Sky Harbor Airport, who left Los Angeles at 4:00 PM (Phoenix time). You want to have only a 10% chance of having to wait for him at the arrival gate. What time should you arrive at the arrival gate?

Probability: Binomial & Normal

Assume a binomial probability distribution with and . (Round all z values to 2 decimal places.)

Compute the following:

(a) The mean and standard deviation of the random variable. (Round your “σ” to 4 decimal places and mean to 1 decimal place.)

(b) The probability that X is 16 or more. (Use the rounded values found above. Round your answer to 4 decimal places.)

Probability

(c) The probability that X is 8 or less. (Use the rounded values found above. Round your answer to 4 decimal places.)

Probability

The closing price of Schnur Sporting Goods, Inc., common stock is uniformly distributed between $18 and 35 per share.Exercises

What is the probability that the stock price will be:

(a) More than $29? (Round your answer to 4 decimal places.)

Probability

(b) Less than or equal to $24? (Round your answer to 4 decimal places.)

Probability

The number of viewers of American Idol has a mean of 31 million with a standard deviation of 4 million. Assume this distribution follows a normal distribution.

What is the probability that next week’s show will:

(a) Have between 33 and 39 million viewers? (Round z-score computation to 2 decimal places and your final answer to 4 decimal places. Round your answer to 4 decimal places.)

Probability

(b) Have at least 24 million viewers? (Round z-score computation to 2 decimal places and your final answer to 4 decimal places. Round your answer to 4 decimal places.)

Probability

(c) Exceed 43 million viewers? (Round z-score computation to 2 decimal places and your final answer to 4 decimal places. Round your answer to 4 decimal places.)

Probability

List the major characteristics of a normal probability distribution. (Select all that apply.)

Skewed.

Symmetrical.

Bell-shaped.

Asymptotic family of curves.

Uniform.

The manufacturer of a laser printer reports the mean number of pages a cartridge will print before it needs replacing is 12,275. The distribution of pages printed per cartridge closely follows the normal probability distribution and the standard deviation is 745 pages. The manufacturer wants to provide guidelines to potential customers as to how long they can expect a cartridge to last.

How many pages should the manufacturer advertise for each cartridge if it wants to be correct 99 percent of the time? (Round z value to 2 decimal places. Round your answer to the nearest whole number.)

Pages

A recent report in BusinessWeek indicated that 34 percent of all employees steal from their company each year. (Round z-score computation to 2 decimal places and your final answers to 4 decimal places.)

If a company employs 50 people, what is the probability that:

(a) fewer than 12 employees steal?

Probability

(b) more than 12 employees steal?

Probability

(c) exactly 12 employees steal?

Probability

(d) more than 12 but fewer than 22 employees steal?

Probability

Dotties Tax Service specializes in federal tax returns for professional clients, such as physicians, dentists, accountants, and lawyers. A recent audit by the IRS of the returns she prepared indicated that an error was made on 12 percent of the returns she prepared last year. Assuming this rate continues into this year and she prepares 68 returns. What is the probability that she makes errors on:

(a) More than 8 returns? (Round z-score computation to 2 decimal places and your final answer to 4 decimal places.)

Probability

(b) At least 8 returns? (Round z-score computation to 2 decimal places and your final answer to 4 decimal places.)

Probability

(c) Exactly 8 returns? (Round z-score computation to 2 decimal places and your final answer to 4 decimal places.)

Probability

A normal population has a mean of 20 and a standard deviation of 4.

(a) Compute the z value associated with 25 (Round your answer to 2 decimal places.)

Z =

(b) What proportion of the population is between 20 and 25? (Round z-score computation to 2 decimal places and your final answer to 4 decimal places. Round your answer to 4 decimal places.)

Proportion

(c) What proportion of the population is less than 18? (Round z-score computation to 2 decimal places and your final answer to 4 decimal places. Round your answer to 4 decimal places.)

Proportion

Customers experiencing technical difficulty with their Internet cable hookup may call an 800 number for technical support. It takes the technician between 150 seconds to 12 minutes to resolve the problem. The distribution of this support time follows the uniform distribution.

(a) What are the values for a and b in minutes? (Do not round your intermediate calculationsRound your answer to 1 decimal place.)

a

b

(b-1) What is the mean time to resolve the problem? (Do not round your intermediate calculations. Round your answer to 2 decimal places.)

Mean

(b-2) What is the standard deviation of the time? (Do not round your intermediate calculations. Round your answer to 2 decimal places.)

Standard deviation

(c) What percent of the problems take more than 5 minutes to resolve? (Do not round your intermediate calculations. Round your answer to 2 decimal places. Omit the “%” sign in your response.)

Percent %

(d) Suppose we wish to find the middle 50 percent of the problem-solving times. What are the end points of these two times? (Do not round your intermediate calculations. Round your answers to 3 decimal places.)

End point 1

End point 2

For the most recent year available the mean annual cost to attend a private university in the United States was $20,257. Assume the distribution of annual costs follows a normal probability distribution and the standard deviation is $4,325.

Ninety-nine percent of all students at private universities pay less than what amount? (Round z value to 2 decimal places and your final answer to the nearest whole number. Omit the “$” sign in your response.)

Amount $

A uniform distribution is defined over the interval from 2 to 5.

(a) What are the values for a and b?

a

b

(b) What is the mean of this uniform distribution? (Round your answer to 2 decimal places.)

Mean

(c) What is the standard deviation? (Round your answer to 4 decimal places.)

Standard deviation

(e) Find the probability of a value more than 2.6. (Round your answer to 2 decimal places.)

Probability

(f) Find the probability of a value between 2.9 and 3.7. (Round your answer to 4 decimal places.)

Probability

According to a government study among adults in the 25- to 34-year age group, the mean amount spent per year on reading and entertainment is $2,075. Assume that the distribution of the amounts spent follows the normal distribution with a standard deviation of $500. (Round z-score computation to 2 decimal places and your final answer to 2 decimal places. Omit the “%” sign in your response.)

(a) What percent of the adults spend more than $2,500 per year on reading and entertainment?

Percent %

(b) What percent spend between $2,500 and $3,050 per year on reading and entertainment?

Percent %

(c) What percent spend less than $1,275 per year on reading and entertainment?

Percent %

Among U.S. cities with a population of more than 250,000 the mean one-way commute to work is 24.3 minutes. The longest one-way travel time is New York City, where the mean time is 38.8 minutes. Assume the distribution of travel times in New York City follows the normal probability distribution and the standard deviation is 7.7 minutes.

(a) What percent of the New York City commutes are for less than 27 minutes? (Round your answers to 2 decimal places. Omit the “%” sign in your response.)

Percent %

(b) What percent are between 27 and 35 minutes? (Round your answers to 2 decimal places. Omit the “%” sign in your response.)

Percent %

(c) What percent are between 27 and 43 minutes? (Round your answers to 2 decimal places. Omit the “%” sign in your response.)

Percent

In establishing warranties on HDTV sets, the manufacturer wants to set the limits so that few will need repair at manufacturer expense. On the other hand, the warranty period must be long enough to make the purchase attractive to the buyer. For a new HDTV the mean number of months until repairs are needed is 36.84 with a standard deviation of 3.34 months.

Where should the warranty limits be set so that only 10 percent of the HDTVs need repairs at the manufacturers expense? (Round z-score computation to 2 decimal places. Round your final answers to 2 decimal places.)

warranty limits months

The monthly sales of mufflers in the Richmond, Virginia, area follow the normal distribution with a mean of 1,200 and a standard deviation of 225. The manufacturer would like to establish inventory levels such that there is only a 5 percent chance of running out of stock.

Where should the manufacturer set the inventory levels? (Enter the value without the separators. Round your answer to the nearest whole number.)

The mean of a normal probability distribution is 420; the standard deviation is 12.

(a) About 68 percent of the observations lie between what two values?

Value 1

Value 2

(b) About 95 percent of the observations lie between what two values?

Value 1

Value 2

(c) Practically all of the observations lie between what two values?

Value 1

Value 2

Assume that the mean hourly cost to operate a commercial airplane follows the normal distribution with a mean of $2,250 per hour and a standard deviation of $180.

What is the operating cost for the lowest 10 percent of the airplanes? (Round z value to 2 decimal places. Omit the “$” sign in your response.)

Operating cost $

See the attachment.