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1 QUESTION # 1 1. Let the random variable X represent the number of telephone lines in use by the technical support center of a software manufacturer at noon each day. The probability distribution of X is shown in the table below. X p(x) a. Calculate the expected value (the mean) of X. b. Using past records, the staff at the technical support center randomly selected 20 days and found that an average of 1.25 telephone lines were in use at noon on those days. The staff proposes to select another random sample of 1,000 days and compute the average number of telephone lines in use at noon on those days. How do you expect the average from this new sample to compare to that of the first sample? Justify your response. c. The median of a random variable is defined as any value x such that P(X x) 0.5 and P(X x) 0.5. For the probability distribution shown in the table above, determine the median of X. d. In a sentence or two, comment on the relationship between the mean and the median relative to the shape of this distribution.

2 QUESTION # 2 2. Men s shirt sizes are determined by their neck sizes. Suppose that men s neck sizes are approximately normally distributed with mean 15.7 inches and standard deviation 0.7 inch. A retailer sells men s shirts in sizes S, M, L, XL, where the shirt sizes are defined in the table below. (a) Using a sketch of the normal curve, illustrate the proportion of men whose shirt size is M. Calculate this proportion. (b) Because the retailer only stocks the sizes listed above, what proportion of customers will find that the retailer does not carry any shirts in their sizes? Show your work. (c) You can generate 500 random neck sizes in your calculator with the command: randnorm(15.7, 0.7, 500). [randnorm is found under the MATH/PRB menu]. Generate 500 men s neck sizes and store them into L1. What proportion of those values in L1 are size M? How close are the results of your simulation to the theoretical probability you found in part (a)?

3 QUESTION # 3 3. The Better Business Council of a large city has concluded that students in the city s schools are not learning enough about economics to function in the modern world. These findings were based on test results from a random sample of 20 twelfth-grade students who completed a 46-question multiple-choice test on basic economic concepts. The data set below shows the number of questions that each of the 20 students in the sample answered correctly (a) Display these data in a stemplot (b) Use your stemplot from part (a) to describe the main features of this score distribution. (c) Why would it be misleading to report only a measure of center for this score distribution?

4 QUESTION # 4 4. John believes that as he increases his walking speed, his pulse rate will increase. He wants to model this relationship. John records his pulse rate, in beats per minute (bpm), while walking at each of seven different speeds, in miles per hour (mph). A scatterplot and regression output are shown below. (a) Using the regression output, write the equation of the fitted regression line (b) Do your estimates of the slope and intercept parameters have meaningful interpretations in the context of this question? If so, provide interpretations in this context. If not, explain why not.

QUESTION # 5 5. The graph below displays the scores of 32 students on a recent exam. Scores on this exam ranged from 64 to 95 points. (a) Describe the shape of this distribution.

5 QUESTION # 5 5. The graph below displays the scores of 32 students on a recent exam. Scores on this exam ranged from 64 to 95 points. (a) Describe the shape of this distribution. (b) In order to motivate her students, the instructor of the class wants to report that, overall, the class s performance on the exam was high. Which summary statistic, the mean or the median, should the instructor use to report that the overall exam performance was high? Explain (c) The midrange is defined as of center or spread? Explain. ( min+ max) 2. Compute this value using the data above. Is the midrange considered a measure

QUESTION # 6 6. A consumer advocate conducted a test of two popular gasoline additives, A and B. There are claims that the use of either of these additives will increase gasoline mileage in cars.

The distance traveled was recorded for each car. Next, additive A was randomly assigned to 15 of the cars and additive B was randomly assigned to the other 15 cars.

6 QUESTION # 6 6. A consumer advocate conducted a test of two popular gasoline additives, A and B. There are claims that the use of either of these additives will increase gasoline mileage in cars. A random sample of 30 cars was selected. Each car was filled with gasoline and the cars were run under the same driving conditions until the gas tanks were empty. The distance traveled was recorded for each car. Next, additive A was randomly assigned to 15 of the cars and additive B was randomly assigned to the other 15 cars. The gas tank of each car was filled with gasoline and the assigned additive. The cars were again run under the same driving conditions until the tanks were empty. The distance travelled was recorded and the difference in the distance with the additive minus the distance without the additive for each car was calculated. The following table summarizes the calculated differences. Note that negative values indicate less distance was traveled with the additive than without the additive. (a) On the grid below, display parallel boxplots (showing outliers, if any) of the differences of the two additives. (b) Two ways that the effectiveness of a gasoline additive can be evaluated are by looking at either The proportion of cars that have increased gas mileage when the additive is used or The mean increase in gas mileage when the additive is used i. Which additive, A or B, would you recommend if the goal is to increase gas mileage in the highest proportion of cars? Explain your choice. ii. Which additive, A or B, would you recommend if the goal is to have the highest mean increase in gas mileage? Explain your choice.

QUESTION # 7 7. The Earth s Moon has many impact craters that were created when the inner solar system was subjected to heavy bombardment of small celestial bodies.

7 QUESTION # 7 7. The Earth s Moon has many impact craters that were created when the inner solar system was subjected to heavy bombardment of small celestial bodies. Scientists studied 11 impact craters on the Moon to determine whether there was any relationship between the age of the craters (based on radioactive dating of lunar rocks) and the impact rate (as deduced from the density of the craters). The data are displayed in the scatterplot below. (a) Describe the nature of the relationship between impact rate and age. Prior to fitting a linear regression model, the researchers transformed both impact rate and age by using logarithms. The following computer output and residual plot were produced. (b) Interpret the value of r 2. (c) Comment on the appropriateness of this linear regression for modeling the relationship between the transformed variables.

QUESTION # 8 8. At a certain university, students who live in the dormitories eat at a common dining hall. Recently, some students have been complaining about the quality of the food served there.

8 QUESTION # 8 8. At a certain university, students who live in the dormitories eat at a common dining hall. Recently, some students have been complaining about the quality of the food served there. The dining hall manager decided to do a study to estimate the proportion of students living in the dormitories who think that the quality of the food should be improved. One evening, the manager asked the first 100 students entering the dining hall to answer the following question. Many students believe that the food served in the dining hall needs improvement. Do you think that the quality of food served here needs improvement, even though that would increase the cost of the meal plan? Yes No No Opinion a. In this setting, explain how bias may have been introduced biased on the way this convenience sample was selected and suggest how the sample could have been selected differently to avoid that bias. b. In this setting, explain how bias may have been introduced based on the way the question was worded and suggest how it could have been worded differently to avoid that bias.

9 QUESTION # 9 9. In search of a mosquito repellent that is safer than the ones that are currently on the market, scientists have developed a new compound that is rated as less toxic than the current compound, thus making a repellent that contains this new compound safer for human use. Scientists also believe that a repellent containing the new compound will be more effective than the ones that contain the current compound. To test the effectiveness of the new compound versus that of the current compound, scientists have randomly selected 100 people from a state. Up to 100 bins, with an equal number of mosquitoes in each bin, are available for use in this study. After a compound is applied to a participant s forearm, the participant will insert his or her forearm into a bin for 1 minute, and the number of mosquito bites on the arm at the end of that time will be determined. a. Suppose this study is to be conducted using a completely randomized design. Describe a randomization process and outline the experiment. b. Suppose this study is to be conducted using a matched-pairs design. Describe a randomization process and outline the experiment. c. Which of the designs, the one in part (a) or the one in part (b), is better for testing the effectiveness of the new compound versus that of the current compound? Justify your answer.

QUESTION # 10 10. A plot of the number of defective items produced during 20 consecutive days at a factory is shown below. a. Draw a histogram that shows the frequencies of the number of defective items.

10 QUESTION # A plot of the number of defective items produced during 20 consecutive days at a factory is shown below. a. Draw a histogram that shows the frequencies of the number of defective items. b. Give one fact that is obvious from the histogram but is not obvious from the scatterplot. c. Give one fact that is obvious from the scatterplot but is not obvious from the histogram.

11 QUESTION # The manager of a cultured pearl farm has received a special order for two pearls between 7 millimeters and 9 millimeters in diameter. From past experience, the manager knows that the pearls found in his oyster bed have diameters that are normally distributed with a mean of 8 millimeters and a standard deviation of 2 millimeters. Assume that every oyster contains one pearl. The manager wants to know how many oysters he should expect to open to find two pearls of the appropriate size for this special order. Complete the following parts to design a simulation to answer the manager's question. a. Determine the probability of finding a pearl of the appropriate size in an oyster selected at random. (Express this probability as a number between 0 and 1. Round this probability to the nearest tenth). b. Describe how you would use a table of random digits to carry out a simulation to determine the number of oysters needed to find two pearls of the appropriate size. Include a description of what each of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 will represent in your simulation. c. Perform your simulation 3 times. (That is, run 3 trials of your simulation) Start at the upper left most digit in the first row of the table and move across. Make your procedure clear so that someone can follow what you did. You must do this by marking directly on or above the table. Report how long it took to find two pearls of the appropriate size in each trial

12 QUESTION # A department supervisor is considering purchasing one of two comparable photocopy machines, A or B. Machine A costs $10,000 and machine B costs $10,500. This department replaces photocopy machines every three years. The repair contract for machine A costs $50 per month and covers an unlimited number of repairs. The repair contract for machine B costs $200 per repair. Based on past performance, the distribution of number of repairs needed over any one-year period for machine B is shown below. Number of Repairs Probability You are asked to give a recommendation based on overall cost as to which machine, A or B along with its repair contract, should be purchased. What would your recommendation be? Give a statistical justification to support your recommendation.

CHAPTER 1. Introduction

CHAPTER 1. Introduction CHAPTER 1 Introduction Engineers and scientists are constantly exposed to collections of facts, or data. The discipline of statistics provides methods for organizing and summarizing data, and for drawing