Exam 20: Probability and Combinatorics

The Human Resources Director of a large corporation wishes to develop an employee benefits package and decides to select 500 employees from a list of all (N = 40,000) workers in order to study their preferences for the various components of a potential package. In this study, methods involving the collection, presentation, and characterization of the data are called _______.

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 F test for the block effects implies that the blocking has been advantageous in reducing the experiment error.

Let X represent the amount of time until 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 more than 10 minutes for the next student to arrive at The library parking lot.

SCENARIO 9-15 A drug company is considering marketing a new local anesthetic. The effective time of the anesthetic the drug company is currently producing has a normal distribution with a mean of 7.4 minutes with a standard deviation of 1.2 minutes. The chemistry of the new anesthetic is such that the effective time should be normally distributed with the same standard deviation. The company will market the new local anesthetic as being better if there is evidence that the population mean effective time is greater than the 7.4 minutes of the current local anesthetic. -Referring to Scenario 9-15, if you select a sample of 25 new local anesthetics and are willing to have a level of significance of 0.05, the probability of the company marketing the new local anesthetic when it is better is _____ if the population mean effective time is 8 minutes.

SCENARIO 16-17 Given below are the prices of a basket of four food items from 2008 to 2012. Year Wheat( \/ Bushel) Corn( \/ Bushel) Soybeans( \/ Bushel) Milk( \/ hundredweight) 2008 4.25 3.71 7.41 15.03 2009 3.43 2.7 7.55 13.63 2010 2.63 2.3 6.05 15.18 2011 2.11 1.97 4.68 14.72 2012 2.16 1.9 4.81 12.32 -Referring to Scenario 16-17, what is the Laspeyres price index for the basket of four food items in 2011 that consisted of 50 bushels of wheat, 30 bushels of corn, 40 bushels of soybeans and 80 hundredweight of milk in 2008 using 2008 as the base year?

A debate team of 4 is to be chosen from a class of 35. There are two twin brothers in the class. How many possible ways can the team be formed which will include only one of the twin brothers?

The Paasche price index uses the consumption quantities in the year of interest as the weights.

SCENARIO 5-14 An accounting firm in a college town usually recruits employees from two of the universities in town. This year, there are fifteen graduates from University A and five from University B and the firm decides to hire six new employees from the two universities. -Referring to Scenario 5-14, what is the probability that none of the new employees will be from University B?

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, based on the Tukey-Kramer procedure with an overall level of significance of 0.01, the agronomist would decide that there is a significant difference between the crop yield of Smith and Trevor seeds.

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 relative efficiency means that 1.0144 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.

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

One of the reasons that a correction for continuity adjustment is needed when approximating the binomial distribution with a normal distribution is because the normal distribution is used for a discrete random variable while the binomial distribution is used for a continuous random variable.

SCENARIO 12-21 A filling machine at a local soft drinks company is calibrated to fill the cans at a mean amount of 12 fluid ounces and a standard deviation of 0.5 ounces. The company wants to test whether the standard deviation of the amount filled by the machine is 0.5 ounces. A random sample of 15 cans filled by the machine reveals a standard deviation of 0.67 ounces. -Referring to Scenario 12-21, the parameter of interest in the test is ________.

SCENARIO 4-10 Are whites more likely to claim bias? It was found that 60% of the workers were white, 30% were black and 10% are other races. Given that a worker was white, the probability that the worker had claimed bias was 30%. Given that a worker was black, the probability that the worker had claimed bias was 40%. Given that a worker was other race, the probability that the worker had claimed bias was 0%. -Referring to Scenario 4-10, if a randomly selected worker had claimed bias, what is the probability that the worker is white?

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 is not from southern Texas, the probability the student has not passed the ELM is ________.

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 between 25 and 30 computers that require repair on a given day using a normal approximation.

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

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 randomized block F test is valid only if there is no interaction between the amount of time needed on the 3 brands of software and the 5 employees.

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 you decide to choose Design A for 70% of the production lines and Design B for the remaining production lines, what is the coefficient of variation of your investment?

SCENARIO 8-12 The superintendent of a unified school district of a small town wants to make sure that no more than 5% of the students skip more than 10 days of school in a year. A random sample of 145 students from a population of 800 showed that 12 students skipped more than 10 days of school last year. -Referring to Scenario 8-12, the superintendent can conclude with 95% level of confidence that no more than 5% of the students in the unified school district skipped more than 10 days of school last year.