Exam 20: Probability and Combinatorics

The amount of time between successive TV watching by first graders follows an exponential distribution with a mean of 10 hours. The probability that a given first grader spends less than 20 hours between successive TV watching is ______.

SCENARIO 11-11 An agronomist wants to compare the crop yield of 3 varieties of chickpea seeds. She plants all 3 varieties of the seeds on each of 5 different patches of fields. She then measures the crop yield in bushels per acre. Treating this as a randomized block design, the results are presented in the table that follows. 1 11.1 19.0 14.6 2 13.5 18.0 15.7 3 15.3 19.8 16.8 4 14.6 19.6 16.7 5 9.8 16.6 15.2 -Referring to Scenario 11-11, the decision made at a 0.01 level of significance on the randomized block F test for the difference in means implies that all 3 means are significantly different.

SCENARIO 5-12 Two different designs on a new line of winter jackets for the coming winter are available for your manufacturing plants. Your profit (in thousands of dollars) will depend on the taste of the consumers when winter arrives. The probability of the three possible different tastes of the consumers and the corresponding profits are presented in the following table. Probability Taste Design A Design B 0.2 more conservative 180 520 0.5 no change 230 310 0.3 more liberal 350 270 -Referring to Scenario 5-12, what is the total standard deviation of the profit if you increase the shift of your production lines and choose to produce both designs?

A 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 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 (i.e. the mean number of calls answered in a minute is 1/3). Find the waiting time at which only 10% of the customers will continue to hold.

Business analytics combine "traditional" statistical methods with methods and techniques from management science and information systems to form an interdisciplinary tool that supports fact-based management decision making.

SCENARIO 16-16 Given below are the average prices for three types of energy products for five consecutive years. Year Electricity Natural Gas Fuel Oil 1 43.205 25.893 0.892 2 16.959 28.749 0.969 3 47.202 28.933 1.034 4 48.874 29.872 0.913 5 48.693 28.384 0.983 -Referring to Scenario 16-16, what are the simple price indices for electricity, natural gas and fuel oil, respectively, in year 2 using year 5 as the base year?

SCENARIO 5-12 Two different designs on a new line of winter jackets for the coming winter are available for your manufacturing plants. Your profit (in thousands of dollars) will depend on the taste of the consumers when winter arrives. The probability of the three possible different tastes of the consumers and the corresponding profits are presented in the following table. Probability Taste Design A Design B 0.2 more conservative 180 520 0.5 no change 230 310 0.3 more liberal 350 270 -Referring to Scenario 5-12, if your investment preference is to maximize your expected profit while exposing yourself to the minimal amount of risk, will you choose a production mix that will consist of 10%, 30%, 50%, 70%, or 90% of your production lines for Design A and the remaining for Design B?

SCENARIO 8-13 The president of a university is concerned that illicit drug use on campus is higher than the 5% targeted level. A random sample of 250 students from a population of 2,000 revealed that 7 of them had used illicit drugs during the last 12 months. -Referring to Scenario 8-13, what is the critical value for the 90% one-sided confidence interval for the proportion of students who had used illicit drugs during the last 12 months?

The director of a training program wanted to know if a one week orientation would change the perception of potential clients who would perceive the program as being good. He collected Information on the number of clients who would rate the program as being good before and after The orientation. Which of the following tests will be the most appropriate?

SCENARIO 6-8 A company has 125 personal computers. The probability that any one of them will require repair on a given day is 0.15. -Referring to Scenario 6-8 and assuming that the number of computers that requires repair on a given day follows a binomial distribution, compute the probability that there will be more than 25 computers that require repair on a given day using a normal approximation.

A 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 waiting Time was found to be a variable best approximated by an exponential distribution with a mean Length of waiting time equal to 2.8 minutes (i.e. the mean number of calls answered in a minute is 1/2)8). What proportion of callers is put on hold longer than 2.8 minutes?

To construct bootstrap confidence interval estimate for the population mean with confidence level ()1−α 100% , you will order the resampled means for the m samples that you have drawn from the initial sample of size n, use the value of the resampled mean that cuts off the largest α/ 2×100% as the lower limit and the value of that cuts off the smallest α/ 2×100% as the upper limit.

SCENARIO 9-13 A manufacturer produces light bulbs that have a mean life of at least 500 hours when the production process is working properly. Based on past experience, the population standard deviation is 50 hours and the light bulb life is normally distributed. The operations manager stops the production process if there is evidence that the population mean light bulb life is below 500 hours. -Referring to Scenario 9-13, if you select a sample of 100 light bulbs and are willing to have a level of significance of 0.01, the probability of the operations manager stopping the process if the population mean bulb life is 490 hours is _____.

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?

SCENARIO 16-16 Given below are the average prices for three types of energy products for five consecutive years. Year Electricity Natural Gas Fuel Oil 1 43.205 25.893 0.892 2 16.959 28.749 0.969 3 47.202 28.933 1.034 4 48.874 29.872 0.913 5 48.693 28.384 0.983 -Referring to Scenario 16-16, what is the Laspeyres price index for the group of three energy items in year 4 for a family that consumed 15 units of electricity, 24 units of natural gas and 200 units of fuel oil in year 1 using year 1 as the base year?

One of the reasons that a correction for continuity adjustment is needed when approximating the binomial distribution with a normal distribution is because the probability of getting a specific value of a random variable is zero with the normal distribution.

Seven passengers are on a waiting list for an overbooked flight. As a result of cancellations, 3 seats become available. How many different ways can those 3 available seats be filled regardless of the order?

SCENARIO 11-13 An important factor in selecting database software is the time required for a user to learn how to use the system. To evaluate three potential brands (A, B and C) of database software, a company designed a test involving five different employees. To reduce variability due to differences among employees, each of the five employees is trained on each of the three different brands. The amount of time (in hours) needed to learn each of the three different brands is given below: Software Operator A B C 1 17 17 23 2 18 17 23 3 14 13 19 4 12 11 18 5 19 17 22 Mean 16 15 21 $\text { Below is the Excel output for the randomized block design: }$ Source of Variation SS df MS F P-value Fcrit Rows 84.66667 4 21.16667 50.8 9.98-06 3.837853 Columns 103.3333 2 51.66667 124 9.54-07 4.45897 Error 3.333333 8 0.416667 Total 191.3333 14 -Referring to Scenario 11-13, what is the critical value for testing the block effects at a 0.05 level of significance?

If the covariance between two investments is zero, the variance of the sum of the two investments will be equal to the sum of the variances of the investments.

SCENARIO 11-13 An important factor in selecting database software is the time required for a user to learn how to use the system. To evaluate three potential brands (A, B and C) of database software, a company designed a test involving five different employees. To reduce variability due to differences among employees, each of the five employees is trained on each of the three different brands. The amount of time (in hours) needed to learn each of the three different brands is given below: Software Operator A B C 1 17 17 23 2 18 17 23 3 14 13 19 4 12 11 18 5 19 17 22 Mean 16 15 21 $\text { Below is the Excel output for the randomized block design: }$ Source of Variation SS df MS F P-value Fcrit Rows 84.66667 4 21.16667 50.8 9.98-06 3.837853 Columns 103.3333 2 51.66667 124 9.54-07 4.45897 Error 3.333333 8 0.416667 Total 191.3333 14 -Referring to Scenario 11-13, the relative efficiency means that 15.23 times as many observations in each brand would be needed in a one-way ANOVA design as compared to the randomized block design in order to obtain the same precision for comparison of the different means.