determine how many data values one would expect to fall within the intervals and . Compare your results with the actual number of data values that fall in these intervals. Also, using technology, verify the assumption that the observations come from a population having a bell shaped probability distribution.37 The following data give the number of defective parts received in the last 15 shipments at a manufacturing plant:8101211139151410161812141613Find the mean of these data.Find the standard deviation of these data.Find the coefficient of variation for these data.
38 The owner of the facility in Problem 37 has another plant where the shipments received are much larger than at the first plant. The quality engineer at this facility also decides to collect the data on defectives received in each shipment. The last 15 shipments provided the following data:213038475839351559604347393041Find the mean and the standard deviation of these data.Find the coefficient of variation of these data, compare it with the one obtained in Problem 37, and comment on which facility receives more stable shipments.
39 Prepare box plots for the data in Problems 37 and 38. Comment on the shape of the distribution of these two data sets.
40 The following data give the test scores of 40 students in a statistics class:68789280877974858688919771728186604076772099807989878780839592988786959676757980Find the sample mean and the sample standard deviation S for these data.Prepare a frequency distribution table for these data.Use the grouped data in part (b) to determine the grouped mean and the grouped standard deviation .Compare the values of and with the values of and the standard deviation . Notice that the grouped mean and grouped standard deviations are only the approximate values of the actual mean and standard deviations of the original (i.e., ungrouped) sample.
41 The following two data sets give the number of defective ball bearings found in 20 boxes randomly selected from two shipments:Shipment I6065797167687356596366597277796971706055Shipment II4555565059604838424137575549433945515355Find the quartiles for each of these two sets.Prepare the box plots for each of the two data sets and display them side by side on one sheet of graph paper.Use part (b) to compare the two shipments. Which shipment in your opinion is of better quality?
42 The following data give the number of flights that left late at a large airport over the past 30 days:5059633012055494743514751576258503953504543465259483651334232Prepare a complete frequency distribution table for these data.Prepare a box plot for these data to comment on the shape of the distribution of these data. Does the set contain any outliers?Find the mean and the standard deviation for these data.
43 The following data gives the inflation rate and interest rates in the United States over 10 consecutive periods. Determine the correlation coefficient between the inflation rate and the interest rates in the United States. Interpret the value of the correlation coefficient you determined.Period12345678910Inflation rate2.342.542.222.671.983.222.512.572.752.67Interest rate4.554.654.754.824.464.854.354.254.554.35
44 The following data gives the heights (in.) and weights (lb) of eight individuals. Determine the correlation coefficient between the heights and weights. Interpret the value of the correlation coefficient you have determined.Individuals12345678Heights 77 72 73 76 72 73 77 72Weights156172195181158164164191
45 It is generally believed that students' performance on a test is related to number of hours of sleep they have the night before the test. To verify this belief, 12 students were asked how many hours they slept on the night before the test. The following data shows the number of hours of sleep on the night before the test and the test scores of each of the 12 students. Determine the correlation coefficient between the hours of sleep and test scores. Interpret the value of the correlation coefficient you have determined.Student123456789101112Hours of sleep 8 8 6 5 8 8 7 6 7 5 4 6Test scores898488858797939087908672
Notes
1 1 Source: Reproduced with permission of ASQ, Jacobson (1998).
2 2 Source: Based on data from The Engineering Statistics Handbook, National Institute of Standards and Technology (NIST).
Chapter 3 Elements of Probability
The focus of this chapter is the study of basic concepts of probability.
Topics Covered
Random experiments and sample spaces
Representations of sample spaces and events using Venn diagrams
Basic concepts of probability
Additive and multiplicative rules of probability
Techniques of counting sample points: permutations, combinations, and tree diagrams
Conditional probability and Bayes's theorem
Introducing random variables
Learning Outcomes
After studying this chapter, the reader will be able to
Handle basic questions about probability using the definitions and appropriate counting techniques.
Understand various characteristics and rules of probability.
Determine probability of events and identify them as independent or dependent.
Calculate conditional probabilities and apply Bayes's theorem for appropriate experiments.
Understand the concept of random variable defined over a sample space.
3.1 Introduction
In day‐to‐day activities and decisions, we often confront two scenarios: one where we are certain about the outcome of our action and the other where we are uncertain or at a loss. For example, in making a decision about outcomes, an engineer knows that a computer motherboard requires four RAM chips and plans to manufacture 100 motherboards. On the one hand, the engineer is certain that he will need 400 RAM chips. On the other hand, the manufacturing process of the RAM chips produces both nondefective and defective chips. Thus, the engineer has to focus on how many defective chips could be produced at the end of a given shift and so she is dealing with uncertainty.
Probability is a measure of chance. Chance, in this context, means there is a possibility that some sort of event will occur or will not occur. For example, the manager needs to determine the probability that the manufacturing process of RAM chips will produce 10 defective chips in a given shift. In other words, one would like to measure the chance that in reality, the manufacturing process of RAM chips does produce 10 defective chips in a given shift. This small example shows that the theory of probability plays a fundamental role in dealing with problems where there is any kind of uncertainty.
3.2 Random Experiments, Sample Spaces, and Events
3.2.1 Random Experiments and Sample Spaces
Inherent in any situation where the theory of probability is applicable is the notion of performing a repetitive operation, that is, performing a trial or experiment that is capable of being repeated over and over “under essentially
