374

The Voyage of the St. Andrew

Throughout the picturesque valleys of mid-18th-century Germany echoedthe song of the neuländer (newlander). Their song enticed journeymen

who struggled to feed their families with the dream and promise of colo-nial America. Traveling throughout the German countryside, the typi-

cal neuländer sought to sign up several families from a village forimmigration to a particular colony. By registering a group of neigh-

bors, rather than isolated families, the agent increased the likeli-hood that his signees would not stray to the equally enticingproposals 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 (onefreight) from Rotterdam to Boston was 7.5 pistole. Children between theages of four and thirteen were assessed at half the adult rate (one-halffreight). Children under four were not charged. To get a sense of the ex-pense involved, it has been estimated that the adult fare, 7.5 pistoles, isequivalent to approximately $2000 (1998 U.S.)! For a large family, the costcould easily be well beyond their means. Even though many immigrants didnot have the necessary funds to purchase passage, they were determined tomake the crossing. Years of indentured servitude for themselves and otherfamily members was often the currency of last resort.

As a historian studying the influence of these German immigrants on colo-nial America, Hans Langenscheidt is interested in describing various demo-graphic characteristics of these people. Unfortunately, accurate records arerare. In his research, he has discovered a partially reconstructed 1752 passengerlist for a ship, the St. Andrew. This list contains the names of the head of fami-lies, a list of family members traveling with them, their parish of origin, and thenumber of freights each family purchased. Unfortunately, some of the data aremissing for some of the families. Langenscheidt believes that the demographicparameters of this passenger list are likely to be similar to those of the other nu-merous voyages taken from Germany to America during the mid-eighteenthcentury.Assuming that he is correct, he believes that it is appropriate to createa discrete probability distribution for a number of demographic variables forthis population of German immigrants. His distributions are presented below.

Probability Distribution of theNumber of Families per Parish ofGerman Immigrants on Board the1752 Voyage of the St. Andrew

Number of Families per Parish Probability

1 0.7062 0.1763 0.0004 0.0595 0.0006 0.059

CA

SE S

TUD

Y

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1. Using the information above, describe, through histograms and numer-ical summaries such as the mean and standard deviation, each of theprobability distributions.

2. Does it appear that, on average, the neuländers were successful in sign-ing more than one family from a parish? Does it seem likely that mostof the families knew one another prior to undertaking the voyage? Ex-plain your answers for both of the questions.

3. Using the mean number of freights purchased per family, estimate the av-erage 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 themean 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 fami-lies purchased more than four freights. Using the information con-tained in the appropriate probability distribution for the St. Andrew,calculate the probability that at least five of six German immigrantfamilies would have purchased more than four freights. Does it seemlikely that these families came from a population similar to that of theGermans on board the St. Andrew? Explain.

6. Summarize your findings in a report. Discuss any assumptions madethroughout this analysis. What are the consequences to your calcula-tions and conclusions if your assumptions are subsequently determinedto be invalid? 375

Probability Distribution of theKnown Number of FreightsPurchased by the GermanFamilies on Board the 1752 Voyage of the St. Andrew

Number of Freights Probability

1.0 0.0751.5 0.0252.0 0.4252.5 0.1503.0 0.1253.5 0.1004.0 0.0505.0 0.0256.0 0.025

Probability Distribution of theKnown Number of People in aFamily for the Germans on Boardthe 1752 Voyage of the St. Andrew

Number in Family Probability

1 0.3222 0.1863 0.1364 0.1025 0.0516 0.1367 0.0348 0.0179 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 1752passenger list of the St. Andrew presented by Whitaker and Horlacher.

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376

Should We Convict?

In 1964, a woman who was shopping in Los Angeles had her purse stolen bya young, blonde female who was wearing a ponytail. The blonde female gotinto a yellow car that was driven by a black male who had a mustache and abeard.The police located a blonde female named Janet Collins who wore herhair in a ponytail and had a friend who was a black male who had a mustacheand beard and also drove a yellow car.The police arrested the two subjects.

Because there were no eyewitnesses and no real evidence, the prosecu-tion used probability to make its case against the defendants. The followingprobabilities were presented by the prosecution for the known characteristicsof the thieves:

DEC

ISIO

NS

(a) Assuming that the characteristics listed above are independent of eachother, what is the probability that a randomly selected couple wouldhave all these characteristics? That is, what is P (“yellow car” and “manwith 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 ran-domly selected couple would have all six characteristics listed above.Let the random variable X represent the number of couples that haveall the characteristics listed in the table. Assuming that the randomvariable X follows the binomial probability function, we have

Assuming that there were couples in the Los Angelesarea, what is the probability that more than one of them have the char-acteristics listed in the table? Does this result cause you to change yourmind regarding the defendants’ guilt?

(d) Now, let’s look at this case from a different point of view. We will com-pute the probability that more than one couple has the characteristicsdescribed, given that at least one couple has the characteristics:

Conditional Probability Rule

Compute this probability, assuming Compute this prob-ability again, but this time assume that Do you think thatthe 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

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Quality Assurance in Customer Relations

377

The Customer Relations Department at ConsumersUnion (CU) receives thousands of letters and e-mailsfrom customers each month.Some people write in askinghow well a product performed during CU’s testing, somepeople write in sharing their own experiences with theirhousehold products, and the remaining people write infor an array of other reasons. In order to be able to re-spond to each letter and e-mail that is received, Cus-tomer Relations recently upgraded its customer contactdatabase. Although much of the process has been auto-mated, it still requires employees to manually draft theresponses. Given the current size of the department, eachCustomer Relations representative is required to draftapproximately 300 responses each month.

As part of a quality assurance program, the Cus-tomer Relations manager would like to develop a planthat allows him to evaluate the performance of his em-ployees. From past experience, he knows that theprobability a new employee will write an initial draftof a response that contains errors is approximately10%. The manager would like to know how many ofthe 300 responses he should sample in order to have acost effective quality assurance program.

(a) Let X be a discrete random variable that repre-sents the number of the draft responsesthat contain errors. Describe the probability dis-tribution for X. Be sure to include the name of theprobability distribution, possible values for therandom variable X, and values of the parameters.

n = 300

(b) To be effective, suppose the manager would like tohave a 95% probability of finding at least one draftdocument that contains an error.Assuming that theprobability a draft document will have errors isknown to be 10%, determine the appropriate sam-ple size to satisfy the manager’s requirements. Hint:We are required to find the number of draft docu-ments that must be sampled so that the probabilityof finding at least one document containing anerror is 95%. In other words, we need to determinen by solving:

(c) Suppose the error rate is really 20%. What samplesize will the manager need to review to have a95% probability of finding one or more docu-ments containing an error?

(d) Now, let Y be a discrete random variable that rep-resents the number of errors discovered in a singledraft document. (It is possible for a single draft tocontain more than one error.) The manager deter-mined that errors occurred at the rate of 0.3 errorsper document. Describe the probability distribu-tion for Y. Be sure to include the name of theprobability distribution, possible values for therandom variable Y, and values of the parameters.

(e) What is the probability that a document containsno errors? One error? At least two errors?

Note to Readers: In many cases, our test protocol and analyticalmethods are more complicated than described in these examples.The data and discussions have been modified to make the materialmore appropriate for the audience.

P1x Ú 12 = 0.95.

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