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A two-way table is a powerful tool in organizing and visualizing data, particularly when you want to analyze the relationships between two categorical variables. In the context of the smoking cessation experiment, one way to understand this is by categorizing our data into two groups. The first category is the number of smokers invited to participate, and the second category is the number of those who actually agreed to participate.
For our experiment setup, the Two-Way Table allows us to clearly see the distribution of participants across the Reward and Deposit groups. In the Reward group, we see 1017 smokers were invited, and 914 agreed to participate. In the Deposit group, 1053 were invited, but only 146 agreed.
These figures help provide a clear snapshot of how many individuals were part of each group and how willing they were to partake in the program based on the incentives offered. The two-way table simplifies complex data, allowing researchers to draw quick insights into participation behavior.

Random assignment plays a crucial role in the integrity of experimental studies. In this smoking cessation program, random assignment ensured that the characteristics of participants were evenly distributed across the different treatment groups, the Reward and Deposit groups.
The idea is to eliminate selection bias, where the traits influencing a smoker to quit, such as motivation, are equally present in both groups preventing one group from having a naturally higher success rate due to such biases.
Random assignment allows researchers to be more confident that differences in outcomes are due to the effects of the financial incentives rather than pre-existing differences among participants. This foundational step strengthens the reliability of the experiment's results, making it easier to attribute any changes in participation rates directly to the type of incentive offered.

Participation rates give us a quantitative measure of success regarding how many invited individuals agree to participate in an experiment. It's a way to gauge interest or willingness from those involved with different incentives. For this experiment, participation rates were determined by dividing the number of individuals who agreed to participate by the total number invited, then multiplying by 100 to convert this into a percentage.
The computation for the Reward group results in a participation rate of \( \frac{914}{1017} \times 100 \approx 89.9\% \). For the Deposit group, it results in \( \frac{146}{1053} \times 100 \approx 13.9\% \).
These rates highlight a significant difference in the willingness of smokers to participate based on financial incentive types, clearly showing that direct cash rewards yielded a higher participation rate compared to those requiring a personal financial risk. Understanding participation rates helps researchers and stakeholders refine intervention strategies by identifying what types of incentives are most effective.