Exam 20: Probability and Combinatorics

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). What proportion of customers having to hold more than 1.5 minutes will hang up before Placing an order?

SCENARIO 4-15 At a Texas college, 60% of the students are from the southern part of the state, 30% are from the northern part of the state, and the remaining 10% are from out-of-state. All students must take and pass an Entry Level Math (ELM) test. 60% of the southerners have passed the ELM, 70% of the northerners have passed the ELM, and 90% of the out-of-staters have passed the ELM. -Referring to Scenario 4-15, if a randomly selected student has passed the ELM, the probability the student is from out-of-state is ________.

To determine the probability of getting more than 3 events of interest in a binomial distribution, you will find the area under the normal curve for X = 3.5 and above.

SCENARIO 8-14 The president of a university is concerned that the percentage of students who have cheated on an exam is higher than the 1% acceptable level. A confidential random sample of 1,000 students from a population of 7,000 revealed that 6 of them said that they had cheated on an exam during the last semester. -Referring to Scenario 8-14, the president can be 90% confident that no more than 1% of the students at the university had cheated on an exam during the last semester.

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 no more than 8 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 3 minutes (i.e. the mean number of calls answered in a minute is 1/3). What proportion of customers having to hold more than 4.5 minutes will hang up before Placing an order?

To determine the probability of getting between 3 and 4 events of interest in a binomial distribution, you will find the area under the normal curve between X = 3.5 and 4.5.

The interval between patients arriving at an outpatient clinic follows an exponential distribution at a rate of 15 patients per hour. What is the probability that a randomly chosen arrival to be less than 15 minutes?

SCENARIO 9-11 An appliance manufacturer claims to have developed a compact microwave oven that consumes a population mean of no more than 250 W. From previous studies, it is believed that power consumption for microwave ovens is normally distributed with a population standard deviation of 15 W. If there is evidence that the population mean consumption is greater than 250 W, the manufacturer will be unable to make the claim. -Referring to Scenario 9-11, if you select a sample of 20 compact microwave ovens and are willing to have a level of significance of 0.05, the probability of making a Type II error is _____ if the mean power consumption of all such microwave ovens is in fact 257.3 W.

SCENARIO 11-12 A student team in a business statistics course designed an experiment to investigate whether the brand of bubblegum used affected the size of bubbles they could blow. To reduce the person-to-person variability, the students decided to use a randomized block design using themselves as blocks. Four brands of bubblegum were tested. A student chewed two pieces of a brand of gum and then blew a bubble, attempting to make it as big as possible. Another student measured the diameter of the bubble at its biggest point. The following table gives the diameters of the bubbles (in inches) for the 16 observations. -Referring to Scenario 11-12, the randomized block F test is valid only if the population of diameters is normally distributed for the 4 brands.

SCENARIO 5-11 There are two houses with almost identical characteristics available for investment in two different neighborhoods with drastically different demographic composition. The anticipated gain in value when the houses are sold in 10 years has the following probability distribution: -Referring to Scenario 5-11, if you can invest 90% of your money on the house in neighborhood A and the remaining on the house in neighborhood B, what is the portfolio expected return of your investment?

SCENARIO 5-11 There are two houses with almost identical characteristics available for investment in two different neighborhoods with drastically different demographic composition. The anticipated gain in value when the houses are sold in 10 years has the following probability distribution: -Referring to Scenario 5-11, if you can invest half of your money on the house in neighborhood A and the remaining on the house in neighborhood B, what is the portfolio risk of your investment?

SCENARIO 5-13 From an inventory of 48 new cars being shipped to local dealerships, corporate reports indicate that 12 have defective radios installed. -Referring to Scenario 5-13, what is the probability out of the 8 new cars it just received that, when each is tested, none of the cars have defective radios?

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 less than 25 or more than 30 computers that require repair on a given day using a normal approximation.

The Commissioner of Health in New York State wanted to study malpractice litigation in New York. A sample of 31 thousand medical records was drawn from a population of 2.7 million patients who were discharged during 2010. Using the information obtained from the sample to predict population characteristics with respect to malpractice litigation is an example of _______.

SCENARIO 9-11 An appliance manufacturer claims to have developed a compact microwave oven that consumes a population mean of no more than 250 W. From previous studies, it is believed that power consumption for microwave ovens is normally distributed with a population standard deviation of 15 W. If there is evidence that the population mean consumption is greater than 250 W, the manufacturer will be unable to make the claim. -Referring to Scenario 9-11, if you select a sample of 20 compact microwave ovens and are willing to have a level of significance of 0.01, the probability of making a Type II error is _____ if the mean power consumption of all such microwave ovens is in fact 257.3 W.

To determine the probability of getting between 2 and 4 events of interest in a binomial distribution, you will find the area under the normal curve between X = 1.5 and 4.5.

SCENARIO 5-11 There are two houses with almost identical characteristics available for investment in two different neighborhoods with drastically different demographic composition. The anticipated gain in value when the houses are sold in 10 years has the following probability distribution: -Referring to Scenario 5-11, what is the total standard deviation of value gain if you invest in both houses?

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.

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 agronomist decided to compare the 3 treatment means by using the Tukey multiple comparison procedure with an overall level of significance of 0.01. How many pairwise comparisons can be made?