A day at the movies.

Docan, Carol ; Rymsza, Leonard ; Baum, Paul 等

CASE DESCRIPTION

The primary subject matter of this case concerns business law and statistical analysis. Secondary issues examine contract formation, terms of an agreement, breach of contract, misrepresentation and legal remedies, as well as ethical issues related to business conduct affecting consumers and statistical analysis involving hypothesis testing which may lead to alternate business decisions.

The case has a difficulty of level three, appropriate for junior level courses. The case is designed to be taught in three class hours, including a class presentation by student teams. The case is expected to require a minimum of three hours of outside preparation by student teams that present a report.

CASE SYNOPSIS

Draw your students into a scenario that they will identify with quickly. A busy college student rushes to get to the movie theater, on time, to see the latest big movie hit. The student unwittingly becomes part of a captive audience that must sit through twenty minutes of commercial advertisements before the movie actually begins. Instead of complaining about the cost of a movie ticket, the student is fuming because he had to sit through the commercials and wants his money back. When the manager refuses to return the price of the movie ticket, the student considers whether he has a good lawsuit against the theater on behalf of all moviegoers.

The theater receives a letter from the student expressing his dissatisfaction with the showing of the commercials and threatens a class action lawsuit. The theater learns that competitors have received similar complaints. The theater owners prepare to defend a potential lawsuit by forming a consortium.

Your students will embark on a search for answers to a variety of questions. In Case A, students are required to determine whether a contract exists, identify the terms of the agreement, determine whether a breach of contract occurred, and what remedies, if any, are available, and analyze whether the theater made an innocent misrepresentation or acted fraudulently. In addition, students explore the ethical issues that arise from the theater owner's conduct of showing commercials to a captive audience.

In Case B, the consortium decides to conduct a survey to consider potential legal losses. The results of the survey are used to test the hypotheses regarding the percentage of all moviegoers who are unhappy with the commercials. The student must recognize the statistical issue as one of testing hypotheses about a population proportion, must be able to formulate the null and alternative hypothesis, compute the appropriate test statistic, and draw conclusions about whether the consortium should settle or defend the lawsuit.

The case study was developed after the authors became aware of a consumer fraud lawsuit that was filed against a national movie theatre chain on behalf of all moviegoers who sat through unannounced advertisements. The authors recognized a series of additional legal issues that were not presented in the original litigation, which lead to a discussion of the ethical issues presented in the scenario, and to a discussion of how the national chain might solve the threaten litigation through statistical analysis of a consumer survey.

INSTRUCTORS' NOTES

Recommendations for Teaching Approaches

This case is designed to be used in an upper division business course. The purpose of the course is to enable students to utilize knowledge they have gained in their lower division core business courses. In addition, the course also aims to improve a student's communication, written and oral, and teamwork skills. Student teams prepare the answers to questions presented in the case with coaching from faculty. The faculty coaching is intended to provide answers to team questions. One team of students formally presents their case solution to the class. A second team of students acts as a "challenge team" by asking the presenting team for further explanation or clarification of its case solution. Following the challenge, the entire class is welcome to participate in an active question and answer session.

CASE A QUESTIONS--LEGAL AND ETHICAL

1. Does a contract exist between Tommy and Royal Theater? If a contract is present, what are the terms of the agreement and did Royal breach the agreement? If the contract was breached, what damages, if any, may Tommy recover?

Contract Formation

The breach of contract claim appears to be straightforward. A simple contract was entered into between Tommy and Royal Theater, Tommy requesting a ticket and paying for it in cash. Who is the offeror and who is the offeree is really not critical to the case. Whether Tommy offers to purchase the ticket by tendering the cash and Royal Theater accepts the tender, or Royal Theater offers to sell the ticket and Tommy accepts the offer by requesting a ticket and tendering the cash, makes little difference to the conclusion that a contract was entered into by the parties. The contract that results would have only a few very basic terms. Once Tommy's money is accepted, all that remains is Royal Theater's performance. Performance being Royal Theater's promise to begin showing the movie at 1 pm, the time indicated on the ticket. The ticket did not contain any express written statement authorizing the Royal Theater to show advertisements and promotions. Neither did the ticket contain any express written statement indicating that the movie would begin later than 1 pm on account of the showing of advertisements and promotions.

Contract Breach

The contract was breached, Tommy argues, by Royal Theater's failure to show the movie at the stated time. In addition, Tommy may also contend that the contract was breached by Royal Theater's unilateral decision to show unwanted advertisements and promotions despite the lack of an agreement regarding such showings. On the other side of this question of breach of contract, Royal Theater may simply state that time was not of the "essence." Specifically, that there was no agreement regarding showing the movie at precisely 1 pm. Consequently, Royal Theater would indicate that it would only be obligated to perform (show the movie) at 1 pm or within a reasonable amount of time after 1 pm. Royal Theater would indicate that the screening of the movie began at 1:20 pm, twenty minutes being a reasonable delay.

Contract Damages

If the contract is breached, there remains the issue of damages that Tommy has suffered. Tommy can argue that he suffered damages for wasted time, confusion and delay. How will Tommy translate these damages into a monetary amount. The complaint to the manager and a request for a refund would seem to be the easiest solution. Royal Theater would not only agree but might argue that refund of the cost of the ticket is the only solution for Tommy. Although Tommy may recover very little in the form of monetary damages on the breach of contract claim, this does not mean that a class action suit is not worthwhile or that punitive damages or injunctive relief is unavailable in a fraud action against Royal Theater.

2. What liabilities, if any, does Royal Theater have for innocent misrepresentation or fraud? In answering the question, consider reviewing the case of Lee P. Cao et al v. Huan Nguyen et al., 607 N.W. 2d 528 (2000) and incorporating in your report the analysis of the court.

Misrepresentation vs. Fraud

The preliminary discussion by the students should distinguish between misrepresentation and fraud. A definition of each concept would be a good start. A misrepresentation is an assertion that is not in accord with the truth. Misrepresentations can be either, (1) "innocent"--where the representation is false but not intentionally deceptive, or (2) "fraudulent"--where the representation is made with knowledge of its falsity and with intent to deceive. In either instance, "innocent" or "fraudulent," the representation that is made is false. The distinction is that in a case of "innocent" misrepresentation the person making the representation is not aware that the representation is false (not in accord with the truth). Whereas, in the case of a "fraudulent" misrepresentation the person making the representation knows the representation is false and is making the representation with the specific intent to deceive another.

Students should recognize that the misrepresentation in this case was not innocent, but instead fraudulent. They should note that the statement, the movie would begin at 1 p.m., was false, that Royal was aware the statement was false, and that it was intended to deceive moviegoers. Students would point out that Royal made the statement to create a captive audience that would sit through twenty minutes of commercials before the movie began. This analysis eliminates any argument that Royal's statement was an innocent misrepresentation. Students then proceed to develop arguments to determine whether the elements of fraud exist.

Prima Facie Case for Fraud

The case Lee P. Cao et al v. Huan Nguyen et al., 607 N.W. 2d 528 (2000), provides the elements of fraud that must be established to prevail in a case. Students should recognize that Cao decision gives the prima facie case for fraud. The case states: In order to maintain an action for fraudulent misrepresentation, a plaintiff must allege and prove the following elements: (1) that a representation was made; (2) that the representation was false; (3) that when made, the representation was known to be false or made recklessly without knowledge of its truth and as a positive assertion; (4) that it was made with the intention that the plaintiff should rely upon it; (5) that the plaintiff reasonably did so rely; and (6) that the plaintiff suffered damage as a result.

Students should understand that in order for Tommy to prevail in a fraud claim against Royal Theater, he must allege and prove all six points stated in the Cao case. Students are expected to go through each point of the prima facie case and determine if the point is met or not met. Most importantly, the students should be able to explain the decision that they reach on each of the six points. Some of the points, as will be seen below, will require very little discussion. However, other points will be more complex, present contrary arguments, and require additional discussion.

Students should apply the elements of fraud in chronological order, as found in the Cao case. If Tommy and Royal Theater have conflicting arguments, each should be discussed. The following cites each element of fraud and applies the facts of the case to discuss the arguments and counter arguments that Tommy and Royal Theater might make.

(1) That a representation was made: Tommy would assert that Royal Theater made representations in newspaper advertisements, on the theater marquee and by the cashier at the ticket window that the movie would begin at 1 p.m.

Royal Theater has no counter argument.

(2) That the representation was false: Tommy would assert that the representations were false because the movie did not begin at 1 p.m. The movie began after twenty minutes of commercial advertisements were shown. To the contrary, Royal Theater may assert that the representation was not false because the posted time of 1 p.m. was the time when the theater lights would dim, thus advising customers to arrive on time while the theater was lighted.

Royal has the weaker argument.

(3) That when made, the representation was known to be false or made recklessly without knowledge of its truth and as a positive assertion: Tommy would assert that Royal Theater made a positive assertion, "The movie begins at 1 p.m.," and knew it was false because twenty minutes of commercial advertisements would be shown before the movie began.

Royal has no counter argument.

(4) That it was made with the intention that the plaintiff should rely upon it: Tommy may make the simple assertion that moviegoers rely on newspaper advertisements and theater marquees to determine which movies will be shown and at what specific times. Tommy would assert that Royal Theater intended to create a captive audience that would be seated at 1 p.m. only to be forced to sit through twenty-minutes of commercials He will also point out that Royal Theater receives revenues from it's advertisers at the expense of the captive audience that it deceives.

Royal Theater has no counter argument.

(5) That the plaintiff reasonably did so rely: The Cao case provides that a party is justified in relying upon a representation made to the party as a positive statement of fact when an investigation would be required to ascertain its falsity. The idea here is that a party's reliance on a positive statement of fact, under most circumstances, is reasonable in the absence of an attempt to verify the truth or falsity of the statement. If one were required to investigate to determine if every positive statement made to them was in fact true, the tort of fraud would be meaningless. What the students should indicate under this point is that the courts generally do not require a party to make an independent investigation as to the accuracy of the statement on which he relies. However, a person does not act justifiably if he relies on an assertion that is obviously false or not to be taken seriously.

Tommy will assert that he reasonably relied on the newspaper ad, the marquee, and the cashier's statement that the movie would begin at 1 p.m. Tommy would indicate that his reliance on the newspaper ad, the marquee, and the cashier's statement, was justified especially since the statements were not obviously false and were to be taken seriously. It was important that Tommy arrive before the lights in the theater dimmed because he could barely see what was happening when it was dark. He left his house and he arrived with fifteen minutes to spare, enough time to buy a drink and snacks. He entered the viewing room two minutes before the movie was to begin.

Royal Theater may assert that Tommy's reliance was not reasonable. To support this argument, Royal would point out that Tommy had not been to the movie for many years. Additionally, if Tommy had asked others moviegoers he could have easily learned that several minutes of commercial announcements are shown before the movie begins. If Tommy had made this simple inquiry, he would have understood that it was only important for him, personally, to arrive by 1 p.m. because the lights would dim and he would have had difficulty finding a seat in the dark. Once he was comfortably in his seat, the movie would begin after the showing of some commercials.

The decision on element (5) could conceivably be in favor of Tommy or Royal.

(6) That the plaintiff suffered damage as a result: Tommy may assert that he lost twenty minutes of his time and lost $9.00 by paying for a movie that was a bust. Some students might also include the cost of the snacks, in his losses. Tommy will have difficulty determining the value of his time since we do not know what he would have done if he had known the movie started twenty minutes later.

Royal Theater may assert that since the value of Tommy's time cannot be calculated, he is not entitled to recover damages for that loss. Regarding the cost of the movie ticket, Royal Theater might assert that Theaters do not guarantee that consumers will enjoy the movies they pay to view. Moviegoers assume the risk that they will not enjoy the movie, yet they have the opportunity to make that judgment when they pay to see the movie. Royal may also assert that Tommy enjoyed the benefits of consuming the snacks, thereby not suffering a loss.

Royal Theater has the better argument on element (6).

Students who find in favor of Tommy on this point may also raise the issue of whether punitive damages should also be awarded. The issue could be raised because the class of plaintiffs is quite large. Students should be able to point out that the awarding of punitive damages generally requires a showing of conduct that is reprehensible or egregious. On this point, it is left up to the students to determine whether Royal Theater's conduct is to be identified as reprehensible or egregious.

In conclusion, students should conclude that while Tommy will probably be able to establish fraud elements 1-4, he might have some difficulty in establishing elements 5 and 6. If Tommy cannot establish all six elements, he will not prevail in a fraud case against Royal.

3. What ethical issues might be involved in showing the commercials to a captive audience of moviegoers who have paid to see a movie? In answering this question, please read an article entitled, "Only the Ethical Survive." For a copy of the article see: http://www.scu.edu/ethics/publications/iie/ v10n2/ethical-surv.html. Also, search the Internet for other sources that will help you develop your answer.

Students are referred to an article entitled, "Only the Ethical Survive" found at http://www.scu.edu/ethics/publications/iie/v10n2/ethical-surv.html. The basic premise of the article is that, in the long run, it is "good business" to act ethically. Students are encouraged to do some independent research on the question of ethics in business. A search of the Internet, using Google for example and searching "ethics in business" or "business ethics," will produce a considerable volume of material. It is up to each instructor to decide what they would like their class to do in answering this question.

Cost--Benefit Analysis

Here are some possible topics that students can raise. One theory discussed in the literature is a Cost-Benefit analysis. With this ethical theory, a company weighs the costs and benefits of a business decision. If the costs to the company would outweigh the benefits that the company would receive from the decision, one might conclude that the decision is unethical.

Students should identify the "benefits" to Royal Theater from showing advertisements. The obvious benefit to Royal Theater would be revenues received from advertisers for showing the commercials.

On the other hand, what are the "costs" to Royal Theater? Here students might want to look at Royal Theater stakeholders. Stakeholders are entities that are affected by Royal Theater or that have an affect on Royal Theater. Who are the stakeholders in this case--shareholders (in a corporation), employees, suppliers, customers, media, and the local community? The students should be able to identify the stakeholders and explain how the stakeholders might be affected by Royal Theater's decision to show the advertisements in the theater and what affect the stakeholders might have on Royal Theater because of the showing of advertisements.

Justice or Fairness

Another ethical theory is one of "fairness" or "justice." The idea here would be that a decision is ethical if everyone, who is affected by the decision, is treated fairly. In this circumstance the question that students might address would revolve around the question of is it fair to moviegoers to become a "captive audience" with Royal Theater reaping the benefits of commercial revenues.

CASE B QUESTIONS--STATISTICAL

4. In light of this result, should the consortium consider settling or contesting Tommy's lawsuit if it is filed?

The answer to this question involves testing a hypothesis regarding a population proportion. This is a standard statistical test that is covered in an elementary statistics course. The consortium suspects that the proportion of moviegoers who resent the showing of the ads is small, less than 10%. This is the belief or opinion to be tested. This belief or opinion is stated as the alternative hypothesis. The null hypothesis represents all other possibilities regarding the (unknown) population proportion (or percentage), 10% or more in our case.

To conduct the test, let

P = (unknown) proportion of all movie patrons who resent ads.

We then formulate the null (H0) and the alternative (H1) hypotheses, as shown below.

Hypotheses

[H.sub.O]: P [greater than or equal to] 0.10 (Null hypothesis--Avoid lawsuit and negotiate settlement.)

[H.sub.1]: P < 0.10 (Alternative hypothesis- Go to trial and fight the lawsuit.)

Students generally find formulating the hypotheses to be one of the most difficult parts of the problem. They tend to forget that the belief, opinion, suspicion or claim to be tested is stated as the alternative hypothesis while the null hypothesis represents all other possibilities. Additionally, students may also incorrectly formulate the hypotheses as a two-tailed test. In our problem, the correct test is a one-tail test, using the left tail as the rejection region because we would be inclined to reject the null hypothesis only for relatively small values of the sample proportion (much smaller than 10%).

The following examples might be presented in class and discussed prior to assigning this case. They would help clarify these issues and guide the student to the correct formulation of the hypotheses for this case. A test is either a one-tail or two-tail test, depending on the values of the sample result (usually, the sample proportion or sample mean) for which we reject the null hypothesis. If we reject the null hypothesis for both very large and very small values of the sample result, the test is a two-tailed test. For example, in testing the effectiveness of a new insulin pump, we would reject the null hypothesis (the pump is effective in that it produces the desired amount of insulin on average) for both very large or very small values of the average amount of insulin released-that is, if the average amount of insulin released is too high or too small. The correct hypotheses would be:

[H.sub.0]: Average level of insulin= Desired level of insulin

[H.sub.1]: Average level of insulin [not equal to] Desired level of insulin

On the other hand, if we reject the null hypothesis for only very large values or only very small values of the sample result, the test is a one-tail test. The rejection region where the null hypothesis is rejected) of a one-tail test can either be in the right tail or in left tail. For example, if we are testing the breaking strength of a paper bag, the appropriate test is a one-tail test, using the lower or left tail as the rejection region because we would reject the null hypothesis (that the breaking strength of the bag is equal to or greater than the desired level) for very small values of the test statistic. These values would provide evidence that the breaking strength of the bag is far below the desired level.

[H.sub.0]: Average breaking strength of the bag = Desired breaking strength of the bag

[H.sub.1]: Average breaking strength of the bag < Desired breaking strength of the bag

Lastly, the rejection region of a one-tail test could be in the right or upper tail. This would be the case if we rejected the null hypothesis for very large values of the test statistic. To illustrate, suppose we are testing the noise level of a new lawn mower. We don't care if the noise level is very low; our concern is that it is not too high. Therefore, the appropriate test is a one-tail test, using the right or upper tail as the rejection region. In this case,

[H.sub.0]: Average noise level = Desired noise level

[H.sub.1]: Average noise level > Desired noise level

If the test statistic does not fall in the rejection region, students will often incorrectly conclude that the null hypothesis is true and therefore should be accepted. Instructors should remind their students that no statistical test will prove that the null hypothesis is true. The test can only provide evidence that it is false and hence should be rejected. To get this point across, the best analogy to present to students is from our criminal system. In a criminal trail, the null hypothesis is that "the defendant is innocent." (In our legal system this is the presumption--"innocent until proven guilty.") The alternative hypothesis is that the defendant is "guilty." (The alternative is what the prosecution [the state] believes; otherwise the state would not be prosecuting the defendant.) The jury, after reviewing the evidence, may render a verdict of "not guilty." We therefore would state that the null hypothesis should not be rejected. Not rejecting the null hypothesis means that there was not sufficient evidence to reach a guilty verdict. A "not guilty" verdict does not imply that we accept the null hypothesis and conclude that the defendant is innocent. Thus, we can never prove that a defendant is innocent, but we can, with enough evidence, prove that the defendant is guilty.

The next step is to test the hypotheses. We begin by assuming that the null hypothesis is true and then determine if the evidence (sample data) supports or conflicts with this assumption. We use the sample evidence and the value of the population proportion to compute a value called z, or the test statistic, by applying the following formula:

Test Statistic, z

z = [bar.P] - [P.sub.0] / [square root of,(P(1 - p)/n)]

where

n = sample size = 100

X = number of patrons in the sample who resent the ads = 6

[bar.p] = X/n = proportion of patrons in the sample who resent the ads = 6/100 = 0.06

[p.sub.0] = proportion of patrons under the null hypothesis (hypothesized value of p) = 0.10. The decision to reject or not reject the null hypothesis depends on how large the computed z value is relative to the critical z value. The critical z value is a percentile of the standard normal distribution and is obtained from a standard normal table. The percentile depends on the risk we are willing to take of incorrectly rejecting the null hypothesis-that is, rejecting the null hypothesis when it is true. This risk is called the Type I error and is specified prior to conducting the test. For example, if we set the Type error at 5%, then, if we repeated the test a very large number of times, 5% of the time we would incorrectly reject the null hypothesis. With a Type I error of 5%, we state that the test is conducted at the 5% level of significance. The level of significance is often denoted as". (Thus," = 0.05 means that the test is conducted at the 5% level of significance.)

Critical z value

What is the critical z value? The answer depends on whether a one-tail or a two-tail test is being conducted. If the test is a one-tail test using the lower or left tail, the critical z value is the 5th percentile of the standard normal distribution, -1.645. (This value is obtained from the [cumulative] standard normal distribution table.) If the computed z value is less than this value, we reject the null hypothesis. That is, we reject the null hypothesis if the computed z value is more than 1.645 standard deviations below the hypothesized value of the proportion. Conversely, if the computed z value is greater than -1.645, we do not reject the null hypothesis. If the test is a one-tail test using the left or lower tail, the critical z value is the same, expect that we drop the minus sign and use 1.645. (Lastly, if the test is a two-tailed test, we split the risk of a Type I error among the lower and upper tails, with 2.5% in each tail as opposed to 5% in one tail. From the standard normal table, we get two critical z values, namely, -1.96 and 1.96.)

Students often find it difficult to interpret the meaning of the z value. The following explanation should help. The z value is simply the number of standard deviations (or, more precisely, standard errors) the sample proportion is away from the hypothesized value of the proportion. The numerator of the z value is the difference between the sample proportion and the hypothesized value of the proportion. The denominator is the standard error of the sample proportion, or more simply, the sampling error. The sampling error depends on the sample size, n, and the hypothesized value of p. The larger the sample size, the smaller the sampling error and the bigger the z value. The larger the z value, the more likely it is that the null hypothesis will be rejected. Conversely, the smaller the sample size, the larger the sampling error and the smaller the value of z. Small z values tend to support the null hypothesis; consequently, the less likely it is that the null hypothesis will be rejected.

Decision Rule (at 5% level of significance)

[Z.sub.0.05] = critical z value at the 5% level of significance (5th percentile of standard normal distribution)

= -1.645

We would reject the null hypothesis for z values that are less than the critical value. Thus, we may state the decision rule as:

If z < -1.645, reject the null hypothesis at the 5% level of significance; otherwise, do not reject.

Test Result

Since

[bar.p] = X/n = 6/100 = .06

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Students often find it difficult to interpret the meaning of this result. It simply states that the sample proportion (0.06) is 1.33 standard deviations below the hypothesized value of the proportion (0.10). Since the computed z value (-1.33) is greater than the critical z value (-1.645) we cannot reject the null hypothesis. Thus, the difference between sample proportion and the hypothesized value of the proportion is due to sampling error.

Students tend to also have difficulty interpreting the magnitude of negative numbers. They should be reminded that -3, for example, is bigger than -5, but -3 is smaller than -2.

Since z = -1.33 > -1.645, we do not reject the null hypothesis. Thus, there is not sufficient evidence, at the 5% level of significance, to conclude that the proportion of patrons who resent the showing of the ads is less than 0.10 (10%). Thus, there is insufficient evidence to justify going to trail. The consortium should, therefore, negotiate a settlement.

[GRAPHIC OMITTED]

These results are illustrated below in the form of a graph.

The graph is the standard normal distribution. This is a graph of all possible z values. Recall that the z values are the number of standard deviations the sample proportion is away from the hypothesized value of the (unknown) population proportion, [p.sub.0] = 0.10. The mean of the z distribution is 0 because the hypothesized value of the population proportion is zero standard deviations away from itself. The computed z value for the sample proportion, p = 0.06, is -1.33, meaning that the sample proportion is 1.33 standard deviations below the hypothesized value of the population proportion.

The critical value of z corresponding to the 5% level of significance is -1.645. The critical value of z is the value of z that cuts off 5% of the area in the lower tail of the distribution. (More precisely, it is the 5th percentile of the standard normal distribution.) We reject the null hypothesis H0 if the computed z value is less than the critical z value. (A computed value of z that is less than the critical value of z, -1.645, represents the "rejection" region of the test.) On the other hand, we do not reject the null hypothesis if the computed z value is greater than the critical z value. (A computed value of z that is greater than the critical value of z represents the "do not reject" region of the test.)

Students will also have difficulty in interpreting the results of the test. They should first state whether the null hypothesis is rejected or not rejected. If it is rejected, they should conclude that there is sufficient evidence to support the alternative hypothesis. If it is not rejected, the correct conclusion is that there is insufficient evidence to support the alternative hypothesis.

Students may also wonder why we have to perform a statistical test in the first place. After all, they might reason that since the sample revealed that 6 out of 100 moviegoers, or 6%, resent the ads and that 6% is less than 10%, they can conclude, without going through all the effort of conducting a statistical test, that the consortium should fight the case. What is the underlying flaw in this argument? It ignores the sampling error and looks only at how far away the sample proportion is from the hypothesized value. While each of these results is important, neither is sufficient by itself. The formula for the z value allows us to account for both results. The numerator of z is the difference between the sample proportion and the hypothesized value. The denominator is the sampling error. The larger the sample size, the smaller the sampling error. If, for example, the difference between the sample proportion and the hypothesized value is very large, it would be tempting to argue that the null hypothesis should be rejected. However, if the sampling size is very small, the sampling error (the denominator) will be quite large. If it is very large relative to the difference between the sample proportion and the hypothesized value of the proportion (the numerator), the z value will be very small and, consequently, the null hypothesis would not be rejected. (Small z values tend to support the null hypothesis while large z values provide evidence that the null hypothesis should be rejected.) At the other extreme, if the difference between the sample proportion and the hypothesized value of the proportion is very small, students might be inclined to not reject the null hypothesis. But if the sampling error, due, for example, to a large sample size, is very small, a large z value could be obtained, thereby rejecting the null hypothesis.

The point is that any result that is based on sample data is subject to sampling error. The sampling error must be accounted for when judging the difference between the sample result and the result expected if the null hypothesis were true. For example, if a coin is claimed to be fair is tossed a very large number of times, it would be expected that heads would come up 50% of the time. If, however, the coin is tossed 100 times, but heads come up 48 times, could it be argued that the coin is biased in favor of getting more tails? Probably not, since the difference between the sample result (48 heads) and the expected result (50 heads) would be attributed to random variation or sampling error. The larger the sample size, the smaller the sampling error and the more statistically reliable is the sample outcome. The computed z value allows us to account for both the difference between the sample result and the expected result (if the null hypothesis were true) as well as the sampling error.

5. When would the consortium make a Type I error? A Type II error?

Solution

A Type I error is made if we reject the null hypothesis if it is true. This would mean going to trial when the consortium should try to settle the case.

A Type II error is made if we do not reject the null hypothesis if it is false. This would mean avoiding going to trail by trying to negotiating a settlement when the consortium should actually fight the case in court.

6. Would your answer to Question 4 change if 300 patrons were randomly surveyed and 18 out of the 300 patrons agreed with Tommy and resented the ads? Explain.

Solution

Let

n = sample size = 300,

X = number of patrons in the sample who resent the ads = 18,

[bar.p] = X/n = proportion of patrons in the sample who resent the ads = 18/100 = 0.06,

[p.sub.0] = hypothesized proportion of patrons who resent the ads = 0.10.

We now compute z as before, but inserting the new values above:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since z =-2.31 < -1.645, we reject the null hypothesis. Thus, there is sufficient evidence, at the 5% level of significance, to conclude that the proportion of patron who resent the ads is less than 0.10. Hence, there the consortium would be justified in going to trial.

Students might wonder why the null hypothesis is rejected with a sample size of 300, but is not rejected with a sample size of 100, particularly in view of the fact that the sample proportion (0.06) is the same in both cases? The answer is straightforward: with the sample size is this question (300) being three times as large as the sample size in the previous question (100), the sampling error is smaller, resulting in a larger z value. As we stated earlier, larger z values provide weaker evidence in support of the null hypothesis and, consequently, lead us to reject the null hypothesis.

It is also helpful to point out that if, for example, the sample size is quadrupled, the resulting sampling error would not be one-forth as large as is was before; rather it would be one-half as large. This is just another way of saying that, as the sample size continues to increase, the reduction in the sampling error gets smaller and smaller-that is, the sampling error decreases as the sample size increases, but at a decreasing rate. In the extreme case, if you sampled the entire population, the sampling error would, of course, be zero, the z value would be infinitely large and any difference that exists, whether large or small, would be significant and, hence, lead to the rejection of the null hypothesis.

Carol Docan, California State University, Northridge Leonard Rymsza, California State University, Northridge Paul Baum, California State University, Northridge