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Does the Wall Street Culture Corrupt Bankers? Bank employees from a large international bank were recruited, and 67 were randomly assigned to a control group and the remaining 61 to a treatment group. All subjects first completed a short online survey. After answering some general filler questions, the members of the treatment group were asked seven questions about their professional background, such as "At which bank are you currently employed?" or "What is your function at this bank?" These are referred to as "identity priming" questions. The members of the control group were asked seven innocuous questions unrelated to their profession, such as "How many hours a week, on average, do you watch television?" After the survey, all subjects performed a coin tossing task that required tossing any coin 10 times and reporting the results online. They were told they would win \(\$ 20\) for each head tossed, for a maximum payoff of \(\$ 200\). Subjects were unobserved during the task, making it impossible to tell if a particular subject cheated. If the banking culture favors dishonest behavior, it was conjectured that it should be possible to trigger this behavior by reminding subjects of their profession. \(-\) Here are the results. The first line gives the possible number of heads on 10 tosses, and the next two lines give the number of subjects that reported tossing this number of heads for the control and treatment groups, respectively (for example, 16 control subjects reported getting 4 heads). a. Suppose that a subject tosses a fair coin and truthfully reports the number of heads. What is the distribution of the number of heads reported? What is the probability of collecting \(\$ 160\) or more? b. For a truthful subject, what would be the probability of doing better than chance-that is, tossing 6 or more heads? What proportion of subjects in the two groups reported tossing 6 or more heads? c. What does your result in (b) suggest about cheating in the two groups? (In Chapter 23 we will return to this example with more formal tools for comparing the groups.)

Step by step solution

01

Define the Distribution

When a fair coin is tossed 10 times, the number of heads reported follows a binomial distribution. Therefore, it is distributed as \(X \sim \text{Binomial}(10, 0.5)\), where 10 is the number of tosses and 0.5 is the probability of getting a head on each toss.

02

Calculate Probability of Collecting $160 or More

To earn \(160 or more, a participant must report 8, 9, or 10 heads (since earning \)20 per head requires 8 heads for $160). Using the binomial distribution, we calculate the probability \(P(X \geq 8) = P(X = 8) + P(X = 9) + P(X = 10)\). By applying the binomial probability formula, for a fair coin toss of 10 times:- \(P(X = 8) = \binom{10}{8} (0.5)^8 (0.5)^2 = 0.0439\)- \(P(X = 9) = \binom{10}{9} (0.5)^9 (0.5)^1 = 0.0098\)- \(P(X = 10) = \binom{10}{10} (0.5)^{10} = 0.00098\)Thus, \(P(X \geq 8) = 0.0439 + 0.0098 + 0.00098 = 0.05468\).

03

Calculate Probability of Tossing 6 or More Heads

To find the probability of tossing 6 or more heads, calculate \(P(X \geq 6)\). This is the sum of the probabilities of getting exactly 6, 7, 8, 9, or 10 heads:- \(P(X = 6) = \binom{10}{6} (0.5)^{10} = 0.205\)- \(P(X = 7) = \binom{10}{7} (0.5)^{10} = 0.117\)- \(P(X = 8) \ldots = 0.0439\) (calculated earlier)- \(P(X = 9) \ldots = 0.0098\) (calculated earlier)- \(P(X = 10) \ldots = 0.00098\) (calculated earlier)Thus, \(P(X \geq 6) = 0.205 + 0.117 + 0.0439 + 0.0098 + 0.00098 = 0.37798\).

04

Analyze Reported Proportions for 6 or More Heads

The exercise states that we need the proportion of subjects reporting 6 or more heads. Use the reported frequencies to calculate the proportions for each group.

05

Interpretation of Cheating Potential

In comparing the truths and the actual proportions reported by the treatment and control groups, noticeable deviation from 0.378 (the calculated expected probability) may suggest dishonest behavior, especially if the treatment group exceeds the control group significantly in reporting 6 or more heads.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hypothesis Testing

Hypothesis testing is a fundamental concept in statistics that helps us make inferences about populations based on sample data. In this study, we aim to determine if reminding bank employees of their profession influences their honesty during a coin-tossing game. By conducting a hypothesis test, we evaluate if there is evidence to support the conjecture that the banking culture might encourage dishonesty when primed by professional identity.

The null hypothesis might state that there is no difference in cheating behavior between the control group (not professionally primed) and the treatment group (professionally primed). The alternative hypothesis would suggest that the treatment group, when primed with questions about their profession, exhibits higher rates of dishonest reporting (more than 6 heads) than the control group.

To evaluate these hypotheses, we perform statistical tests using the proportions of subjects in each group who reported 6 or more heads. By doing this, we determine if the observed differences are statistically significant or could have occured by random chance, which is a key part of hypothesis testing.

Probability Calculation

Probability calculations are essential in this exercise because they provide a measure of how likely certain outcomes are.

In analyzing the coin toss experiment with bankers, we're interested in the binomial distribution. This describes the probability of achieving a fixed number of 'successes' (heads in our case) in a set number of trials (ten coin tosses), where each trial is independent and has two possible outcomes: head (success) or tail (failure).

The binomial distribution is symbolically represented as \(X \sim \text{Binomial}(n, p)\), where \(n\) represents the number of trials, and \(p\) is the probability of success on each trial. For our fair coin, \(p = 0.5\). This formula allows us to calculate the probability of getting exactly \(x\) heads out of 10 tosses. For example, the probability of getting exactly 6 heads is given by \(P(X = 6)\). Using this, we calculated the probabilities for reporting 6 or more heads, considering the binomial probabilities for outcomes from 6 to 10 heads.

Such probabilities guide us in understanding whether the observed outcomes in the experiment (e.g., more than 6 heads) can be reasonably expected under fair conditions, or if they suggest other factors at play.

Cheating Detection

Detecting cheating in an unobserved setting like a coin toss game relies heavily on probability and statistical analysis. In this experiment, because the task was unobserved, there was no straightforward way to verify the honesty of the participants.

Here, cheating detection is approached by comparing expected outcomes based on probability with the actual reported results. If a significantly larger proportion of subjects report 6 or more heads than the expected probability (approximately 37.8%), it might suggest that some subjects inflated their results to increase their payout.

The control group serves as a baseline of probability-based expectations, assuming no professional priming effect. If the treatment group (those reminded of their profession) has a noticeably higher rate of reporting 6 or more heads, it may indicate a tendency to cheat influenced by the identity priming.

This form of cheating detection is indirect but powerful because it considers statistical deviations from established probabilities as potential indicators of dishonest behavior. This analysis only highlights suspicions of cheating; further investigations would require additional or alternative methods for confirmation.