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Bank employees from a large international bank were recruited with 67 assigned at random to a control group and the remaining 61 assigned to a treatment group. All subjects first completed a short online survey. After answering some general filler questions, 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 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. \({ }^{30}\) 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 four heads). $$ \begin{array}{l|rrrrrrrrrrr} \hline \text { Number of heads } & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\\ \hline \text { Control group } & 0 & 0 & 1 & 8 & 16 & 17 & 14 & 6 & 2 & 1 & 2 \\\ \hline \text { Treatment group } & 0 & 0 & 2 & 4 & 8 & 14 & 15 & 7 & 6 & 0 & 5 \\\ \hline \end{array} $$ If a subject is cheating, we would expect them to report doing better than chance, or tossing six or more heads. (a) Find the proportion of subjects in each group that reported tossing six or more heads. (b) Test the hypotheses that the proportions reporting tossing six or more heads in the two groups are the same against the appropriate alternative. Explain your conclusions in the context of the problem, being sure to relate this to the researcher's conjecture.

Step by step solution

01

Count Subjects Reporting Six or More Heads

First, we need to determine how many subjects in each group reported tossing six or more heads. This is the sum of the numbers for 6, 7, 8, 9, and 10 heads. - **Control group:** 14 (for 6 heads) + 6 (7 heads) + 2 (8 heads) + 1 (9 heads) + 2 (10 heads) = 25 subjects. - **Treatment group:** 15 (6 heads) + 7 (7 heads) + 6 (8 heads) + 0 (9 heads) + 5 (10 heads) = 33 subjects.

02

Calculate Proportions of Six or More Heads

Next, calculate the proportion of subjects in each group who reported tossing six or more heads.- **Control group:** The proportion is 25 out of 67 subjects. \[ \text{Proportion}_\text{control} = \frac{25}{67} \approx 0.373 \]- **Treatment group:** The proportion is 33 out of 61 subjects. \[ \text{Proportion}_\text{treatment} = \frac{33}{61} \approx 0.541 \]

03

Formulate Hypotheses for Proportion Test

Define the null hypothesis and alternative hypothesis to compare the proportions.- **Null Hypothesis \(H_0\):** The proportions of subjects who reported six or more heads are the same in both groups. (\( p_\text{control} = p_\text{treatment} \))- **Alternative Hypothesis \(H_a\):** The proportions of subjects who reported six or more heads are different in the treatment group compared to the control group. (\( p_\text{control} eq p_\text{treatment} \))

04

Perform a Proportion Test

Perform a hypothesis test for two proportions.- Calculate the pooled proportion: \[ \hat{p} = \frac{25 + 33}{67 + 61} = \frac{58}{128} \approx 0.453 \]- Use the formula for the test statistic for two independent samples: \[ z = \frac{0.541 - 0.373}{\sqrt{0.453 \times (1 - 0.453) \left( \frac{1}{67} + \frac{1}{61} \right)}} \] After calculating, this gives a z-value (compute with calculator).

05

Draw Conclusion from Test

Based on the calculated z-value and corresponding p-value, compare against a significance level (commonly \(\alpha = 0.05\)). - If \(p\)-value < \(\alpha\), reject the null hypothesis, suggesting a significant difference between groups. If \(p\)-value > \(\alpha\), do not reject the null hypothesis.- This difference in proportions indicates whether identity priming correlates with increased dishonest behavior aligning with the researcher's conjecture.

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Key Concepts

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

Proportion Test

A proportion test is a statistical method used to determine if there are differences among group proportions, especially for categorical data. In the context of our bank study, a proportion test helps us understand if identity priming affects the number of reported heads in a coin toss. Both the control and treatment groups have subjects reporting six or more heads, and we want to test if the proportion of these reports differ significantly between these groups.

This method assumes that the underlying probability for each group is the same unless shown otherwise by evidence. We'll use a test for two independent proportions to make statistical inferences about the effects of identity priming.

Key steps include:

• Calculating the proportion of subjects in each group that report six or more heads.
• Formulating hypotheses to compare proportions.
• Calculating a test statistic to measure any observed difference.

Finally, the result helps us either support or refute the researcher's hypothesis regarding the influence of professional identity on behavior.

Null Hypothesis

The null hypothesis (ully) is a fundamental concept in hypothesis testing. It represents a statement or position that there is no effect or no difference in the context of the study. In our coin toss study, the null hypothesis suggests that there is no difference between the control group and the treatment group regarding the proportion of subjects reporting six or more heads.

To put it simply, the null hypothesis for this scenario can be stated mathematically as \(H_0: p_\text{control} = p_\text{treatment}\). This implies that the probability of reporting a favorable number of heads (six or more) does not change based on whether participants were asked about their professional identity.

Testing the null hypothesis helps us determine if any observed differences in outcomes are due to random chance or if they are statistically significant. If statistical evidence points otherwise, we might have to reject the null hypothesis.

Alternative Hypothesis

Contrary to the null hypothesis, the alternative hypothesis (alternative) offers the position that there is a meaningful effect or difference in the proportions being studied. In the coin toss experiment, the alternative hypothesis posits that identity priming potentially causes a change in the reported results from the tosses.

In the context of the hypothesis test for proportions, the alternative hypothesis is expressed as \(H_a: p_\text{control} eq p_\text{treatment}\). This means that the proportion of subjects reporting six or more heads is different between the treatment group and the control group.

The alternative hypothesis aligns with the researcher's conjecture that reminding employees of their professional identity might induce dishonest behavior. Therefore, rejecting the null hypothesis in favor of the alternative provides evidence supporting the research idea that identity priming can influence behavior.

The strength of the alternative hypothesis is assessed in the hypothesis testing process, often by calculating a p-value and comparing it against a pre-determined significance level.