Exam 20: Probability and Combinatorics

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 unweighted aggregate price index for the basket of four food items in 2010 using 2008 as the base year?

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 value of MSA is __________, while MSBL is __________.

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 A?

SCENARIO 9-12 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 9-12, if you select a sample of 25 new local anesthetics and are willing to have a level of significance of 0.1, the confidence coefficient of the test is _____ if the population mean effective time is 8 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 5 for a family that consumed 15 units of electricity, 24 units of natural gas and 200 units of fuel oil in year 1 using hear 1 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 you decide to choose Design A for 90% of the production lines and Design B for the remaining production lines, what is the coefficient of variation of your investment?

As a general rule, one can use the normal distribution to approximate a binomial distribution whenever the sample size is at least 30.

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, exactly two of the cars have non-defective radios?

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, in order to perform the test, you need to assume that the amount filled by the machine follows a normal distribution.

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 value of the F test statistic for testing the block effects?

SCENARIO 4-17 Jake woke up late in the morning on the day that he has to go to school to take an important test. He can either take the shuttle bus which is usually running late 20% of the time or ride his unreliable motorcycle which breaks down 40% of the time. He decides to toss a fair coin to make his choice. -Referring to Scenario 4-17, if Jake is late to the test, what is the probability that he rode his bike?

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, exactly half of the cars have defective radios?

SCENARIO 6-7 The interval between consecutive hits at a web site is assumed to follow an exponential distribution with a mean of 40 hits per minute. -Referring to Scenario 6-7, what is the mean time between consecutive hits?

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 0.05, the probability of the operations manager stopping the process when the process is not working properly is _____ if the population mean bulb life is 490 hours.

SCENARIO 4-16 According to the record of the registrar's office at a state university, 35% of the students are freshman, 25% are sophomore, 16% are junior and the rest are senior. Among the freshmen, sophomores, juniors and seniors, the portion of students who live in the dormitory are, respectively, 80%, 60%, 30% and 20%. -Referring to Scenario 4-16, if a randomly selected student lives in the dormitory, what is the probability that the student is a freshman?

As a general rule, one can use the normal distribution to approximate a binomial distribution whenever the sample size is at least 15.

SCENARIO 9-12 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 9-12, if you select a sample of 25 new local anesthetics and are willing to have a level of significance of 0.01, the probability of the company not marketing the new local anesthetic when its population mean effective time is not greater than the 7.4 minutes is _____.

SCENARIO 12-19 The director of transportation of a large company is interested in the usage of the company's van pool program. She surveyed 129 of her employees on the usage of the program before and after a campaign to convince her employees to use the service and obtained the following: She will use this information to perform test using a level of significance of 0.05. -Referring to Scenario 12-19, the director now wants to know if the proportion of employees who use the service before the campaign and the proportion of employees who use the service after the Campaign are the same. Which test should she use?

The covariance

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, no more than 2 of the cars have defective radios?