THE VOYAGE AT ST. ANDREW - 93431

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The Voyage of the St. Andrew Throughout the picturesque valleys of mid-18th-century Germany echoed the song of the neuländer (newlander). Their song enticed journeymen who struggled to feed their families with the dream and promise of colonial America. Traveling throughout the German countryside, the typical neuländer sought to sign up several families from a village for immigration to a particular colony. By registering a group of neighbors, rather than isolated families, the agent increased the likelihood that his signees would not stray to the equally enticing proposals of a competitor. Additionally, by signing large groups, the neuländer fattened his purse, to the tune of one to two florins a head. Generally, the Germans who chose to undertake the hardship of a trans-Atlantic voyage were poor, yet the cost of such a voyage was high. Records from a 1753 voyage indicate that the cost of an adult fare (one freight) from Rotterdam to Boston was 7.5 pistole. Children between the ages of four and thirteen were assessed at half the adult rate (one-half freight). Children under four were not charged. To get a sense of the expense involved, it has been estimated that the adult fare, 7.5 pistoles, is equivalent to approximately $2000 (1998 U.S.)! For a large family, the cost could easily be well beyond their means. Even though many immigrants did not have the necessary funds to purchase passage, they were determined to make the crossing.Years of indentured servitude for themselves and other family members was often the currency of last resort. As a historian studying the influence of these German immigrants on colonial America, Hans Langenscheidt is interested in describing various demographic characteristics of these people. Unfortunately, accurate records are rare. In his research, he has discovered a partially reconstructed 1752 passenger list for a ship, the St.Andrew.This list contains the names of the head of families, a list of family members traveling with them, their parish of origin, and the number of freights each family purchased. Unfortunately, some of the data are missing for some of the families. Langenscheidt believes that the demographic parameters of this passenger list are likely to be similar to those of the other numerous voyages taken from Germany to America during the mid-eighteenth century. Assuming that he is correct, he believes that it is appropriate to create a discrete probability distribution for a number of demographic variables for this population of German immigrants. His distributions are presented below. Probability Distribution of the Number of Families per Parish of German Immigrants on Board the 1752 Voyage of the St. Andrew Number of Families per Parish Probability 1 0.706 2 0.176 3 0.000 4 0.059 5 0.000 6 0.059 CASE STUDY SullStatCH06_Fpp324-377 11/20/02 10:51 AM Page 374 1. Using the information above, describe, through histograms and numerical summaries such as the mean and standard deviation, each of the probability distributions. 2. Does it appear that, on average, the neuländers were successful in signing more than one family from a parish? Does it seem likely that most of the families knew one another prior to undertaking the voyage? Explain your answers for both of the questions. 3. Using the mean number of freights purchased per family, estimate the average cost of the crossing for a family in pistoles and in 1998 U.S. dollars. 4. Is it appropriate to estimate the average cost of the voyage from the mean family size? Why or why not? 5. Langenscheidt came across a fragment of another ship’s passenger list. This fragment listed information for six families. Of these six, five families purchased more than four freights. Using the information contained in the appropriate probability distribution for the St. Andrew, calculate the probability that at least five of six German immigrant families would have purchased more than four freights. Does it seem likely that these families came from a population similar to that of the Germans on board the St. Andrew? Explain. 6. Summarize your findings in a report. Discuss any assumptions made throughout this analysis. What are the consequences to your calculations and conclusions if your assumptions are subsequently determined to be invalid? 375 Probability Distribution of the Known Number of Freights Purchased by the German Families on Board the 1752 Voyage of the St. Andrew Number of Freights Probability 1.0 0.075 1.5 0.025 2.0 0.425 2.5 0.150 3.0 0.125 3.5 0.100 4.0 0.050 5.0 0.025 6.0 0.025 Probability Distribution of the Known Number of People in a Family for the Germans on Board the 1752 Voyage of the St. Andrew Number in Family Probability 1 0.322 2 0.186 3 0.136 4 0.102 5 0.051 6 0.136 7 0.034 8 0.017 9 0.016 Data Source: Wilford W. Whitaker and Gary T. Horlacher, Broad Bay Pioneers (Rockport, Maine: Picton Press, 1998), 63–68. Distributions created from the partially reconstructed 1752 passenger list of the St. Andrew presented by Whitaker and Horlacher. SullStatCH06_Fpp324-377 11/20/02 10:51 AM Page 375 376 Should We Convict? In 1964, a woman who was shopping in Los Angeles had her purse stolen by a young, blonde female who was wearing a ponytail. The blonde female got into a yellow car that was driven by a black male who had a mustache and a beard.The police located a blonde female named Janet Collins who wore her hair in a ponytail and had a friend who was a black male who had a mustache and beard and also drove a yellow car.The police arrested the two subjects. Because there were no eyewitnesses and no real evidence, the prosecution used probability to make its case against the defendants. The following probabilities were presented by the prosecution for the known characteristics of the thieves: DECISIONS (a) Assuming that the characteristics listed above are independent of each other, what is the probability that a randomly selected couple would have all these characteristics? That is, what is P (“yellow car” and “man with a mustache” and and “interracial couple in a car”)? (b) Would you convict the defendants based on this probability? Why? (c) Now let n represent the number of couples in the Los Angeles area that could have committed the crime. Let p represent the probability a randomly selected couple would have all six characteristics listed above. Let the random variable X represent the number of couples that have all the characteristics listed in the table. Assuming that the random variable X follows the binomial probability function, we have Assuming that there were couples in the Los Angeles area, what is the probability that more than one of them have the characteristics listed in the table? Does this result cause you to change your mind regarding the defendants’ guilt? (d) Now, let’s look at this case from a different point of view.We will compute the probability that more than one couple has the characteristics described, given that at least one couple has the characteristics: Conditional Probability Rule Compute this probability, assuming Compute this probability again, but this time assume that Do you think that the couple should be convicted “beyond all reasonable doubt”? Why? n = 2,000,000. n = 1,000,000. = P1X 7 12 P1X Ú 12 P1X 7 1|X Ú 12 = P1X 7 1 and X Ú 12 P1X Ú 12 n = 1,000,000 P1X = x2 = nCx # px11 - p2n-x, x = 0, 1, 2, Á , n. Á Characteristic Probability Yellow car 1/10 Man with a mustache Woman with a ponytail 1/10 Woman with blonde hair 1/3 Black man with beard 1/10 Interracial couple in car 1/1000 14 SullStatCH06_Fpp324-377 12/10/02 11:53 AM Page 376 Quality Assurance in Customer Relations 377 The Customer Relations Department at Consumers Union (CU) receives thousands of letters and e-mails from customers each month. Some people write in asking how well a product performed during CU’s testing, some people write in sharing their own experiences with their household products, and the remaining people write in for an array of other reasons. In order to be able to respond to each letter and e-mail that is received, Customer Relations recently upgraded its customer contact database. Although much of the process has been automated, it still requires employees to manually draft the responses. Given the current size of the department, each Customer Relations representative is required to draft approximately 300 responses each month. As part of a quality assurance program, the Customer Relations manager would like to develop a plan that allows him to evaluate the performance of his employees. From past experience, he knows that the probability a new employee will write an initial draft of a response that contains errors is approximately 10%. The manager would like to know how many of the 300 responses he should sample in order to have a cost effective quality assurance program. (a) Let X be a discrete random variable that represents the number of the draft responses that contain errors. Describe the probability distribution for X. Be sure to include the name of the probability distribution, possible values for the random variable X, and values of the parameters. n = 300 (b) To be effective, suppose the manager would like to have a 95% probability of finding at least one draft document that contains an error.Assuming that the probability a draft document will have errors is known to be 10%, determine the appropriate sample size to satisfy the manager’s requirements. Hint: We are required to find the number of draft documents that must be sampled so that the probability of finding at least one document containing an error is 95%. In other words, we need to determine n by solving: (c) Suppose the error rate is really 20%.What sample size will the manager need to review to have a 95% probability of finding one or more documents containing an error? (d) Now, let Y be a discrete random variable that represents the number of errors discovered in a single draft document. (It is possible for a single draft to contain more than one error.) The manager determined that errors occurred at the rate of 0.3 errors per document. Describe the probability distribution for Y. Be sure to include the name of the probability distribution, possible values for the random variable Y, and values of the parameters. (e) What is the probability that a document contains no errors? One error? At least two errors? Note to Readers: In many cases, our test protocol and analytical methods are more complicated than described in these examples. The data and discussions have been modified to make the material more appropriate for the audience. P1x Ú 12 = 0.95. SullStatCH06_Fpp324-377 11/20/02 10:51 AM Page 377
Solution Description

Question 1

HISTOGRAM SHOWING PROBABILITY DISTRIBUTION OF THE NUMBER OF FAMILIES PER PARISH OFGERMAN IMMIGRANTS.

Text Box: PROBABILITY

                      

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