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Exam 20: Probability and Combinatorics

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Question 71

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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, using the 90% one-sided confidence interval, the president can be 95% confident that no more than 5% of the students at the university had used illicit drugs during the last 12 months.

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Question 72

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The oranges grown in corporate farms in an agricultural state were damaged by some unknown fungi a few years ago. Suppose the manager of a large farm wanted to study the impact of the fungi on the orange crops on a daily basis over a 6-week period. On each day a random sample of orange trees was selected from within a random sample of acres. The daily average number of damaged oranges per tree and the proportion of trees having damaged oranges were calculated. In this study, drawing conclusions on any one day about the true population characteristics based on information obtained from the sample is called _______.

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Question 73

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The D in the DCOVA framework stands for "data".

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. $\begin{array}{rrrr}\underline {\text { Fields }} &\underline {\text { Smith}}& \underline{\text {Walsh}}& \underline{\text {Trevor }}\\1&11.1 & 19.0 & 14.6 \\2&13.5 & 18.0 & 15.7 \\3&15.3 & 19.8 & 16.8 \\4&14.6 & 19.6 & 16.7 \\5&9.8 & 16.6 & 15.2\end{array}$ -Referring to Scenario 11-11, what are the degrees of freedom of the F test statistic for testing the block effects?

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 0.10, the probability of making a Type II error is _____ if the mean power consumption of all such microwave ovens is in fact 248 W.

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. She should reject the null hypothesis using a 5% level of significance.

SCENARIO 16-16 Given below are the average prices for three types of energy products for five consecutive years. $\begin{array} { r r r r } \text { Year } & \text { Electricity } & \text { Natural Gas } & \text { Fuel Oil } \\1 & 43.205 & 25.893 & 0.892 \\2 & 16.959 & 28.749 & 0.969 \\3 & 47.202 & 28.933 & 1.034 \\4 & 48.874 & 29.872 & 0.913 \\5 & 48.693 & 28.384 & 0.983\end{array}$ -Referring to Scenario 16-16, what is the Paasche price index for the group of three energy items in year 4 for a family that consumed 13 units of electricity, 26 units of natural gas and 235 units of fuel oil in year 4 using year 1 as the base year?

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, what is the p-value of the F test statistic for testing the block effects?

To construct a bootstrap confidence interval estimate for the population mean, you will first select a random sample of size n with replacement from a population of size N as the initial sample.

At the International Pancakes Hut, there are 4 different ways to have an egg cooked, 7 different choices of pancakes, 5 different types of syrups and 8 different beverages. How many different ways are there to order an egg, a pancake with a choice of syrup and a beverage?

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 failing to market the new local anesthetic when its population mean effective time is greater than the 7.4 minutes is _____ if the population mean effective time is 8 minutes.

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 decision is to reject the null hypothesis when using a 5% level of significance.

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 does not live in the dormitory, what is the probability that the student is a junior or a senior?

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 not a freshman?

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 248 W.

The oranges grown in corporate farms in an agricultural state were damaged by some unknown fungi a few years ago. Suppose the manager of a large farm wanted to study the impact of the fungi on the orange crops on a daily basis over a 6-week period. On each day a random sample of orange trees was selected from within a random sample of acres. The daily average number of damaged oranges per tree and the proportion of trees having damaged oranges were calculated. In this study, the presentation and characterization of the two main measures calculated each day (i.e., average number of damaged oranges per tree and proportion of trees having damaged oranges) is called _______ .

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: $\begin{array}{ | r | r | r | r | } \hline & & \text { Software } & \\\hline \text { Operator } & \text { A } & \text { B } & \text { C } \\\hline 1 & 17 & 17 & 23 \\\hline 2 & 18 & 17 & 23 \\\hline 3 & 14 & 13 & 19 \\\hline 4 & 12 & 11 & 18 \\\hline 5 & 19 & 17 & 22 \\\hline \text { Mean } & 16 & 15 & 21 \\\hline\end{array}$ $\text { Below is the Excel output for the randomized block design: }$ $\begin{array}{lrrrrrr}\hline \text { Source of Variation } & S S & d f & \text { MS } & {\text { F }} & \text { P-value } & \text { Fcrit } \\\hline \text { Rows } & 84.66667 & 4 & 21.16667 & 50.8 & 9.98 \mathrm{E}-06 & 3.837853 \\\text { Columns } & 103.3333 & 2 & 51.66667 & 124 & 9.54 \mathrm{E}-07 & 4.45897 \\\text { Error } & 3.333333 & 8 & 0.416667 & & & \\\text { Total } & 191.3333 & 14 & & & & \\\hline\end{array}$ -Referring to Scenario 11-13, the among-block variation or SSBL is __________.

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 your investment preference is to maximize your expected return while exposing yourself to the minimal amount of risk, will you choose a portfolio that will consist of 10%, 30%, 50%, 70%, or 90% of your money on the house in neighborhood A and the remaining on the house in neighborhood B?

The consumer price index is a Paasche price index.

A debate team of 4 members for a high school will be chosen randomly from a potential group of 15 students. Ten of the 15 students have no prior competition experience while the others have some degree of experience. What is the probability that no more than 1 of the members chosen for the team have some prior competition experience?