Question

Write the digits 1,2,3,4 in order on an index card. Bring this card to a busy place (e.g., dining hall, library, university union) and ask at least 30 people to look at the card and select one of the digits at random in their head. Record their responses. (a) If people can think "randomly," about what fraction of the people should respond with the digit \(1 ? 2 ? 3 ? 4 ?\) (b) What fraction of those surveyed responded with the digit \(1 ? 2 ? 3 ? 4 ?\) (c) Do the results suggest anything about people's ability to choose randomly?

Short answer

The expected fraction of people to choose each digit (1, 2, 3, 4) is 1/4 if selection is random. The actual fractions observed from the survey responses will indicate people's ability to choose randomly. By comparing the actual fractions to the expected fractions, one can assess the randomness of the selections.

Step-by-step solution

Understanding Random Selection

If each person makes a random selection of a digit from 1 to 4, each digit should be chosen about an equal number of times. Theoretically, this means that each digit should be chosen by around 1/4 (or 25%) of the participants since there are 4 digits and participants are selecting one digit at random.

Collecting Data

Go to a busy location with the index card that has the numbers 1, 2, 3, and 4 written on it. Ask at least 30 people to select one of the digits at random in their head and record their response. Tally each response for the digits 1, 2, 3, and 4.

Tabulating Results for Each Digit

After collecting data from at least 30 participants, count the number of times each digit was selected. Divide the count for each digit by the total number of responses to find the fraction of responses for each digit. This fraction is the empirical probability based on the survey.

Analyzing the Results

Compare the fraction of responses for each digit with the expected fraction, which is 1/4. If the fraction of responses for each digit is close to 1/4, it would suggest that people are good at choosing randomly. Significant deviations from 1/4 may suggest biases or patterns in the way people choose numbers.

Key concepts

Random Sampling

Random sampling is a critical concept in statistics that ensures each individual in a population has an equal chance of being selected for a survey or experiment. The goal is to obtain a sample that is representative of the overall population. In our exercise, 30 or more participants were asked to choose a digit at random. Ensuring randomness in participant selection is vital; for example, different times of day or different locations could have captured a more diverse pool of participants, reflecting a more accurate cross-section of the general population.

When conducting such an experiment, it's important to avoid any bias in the selection process. Bias can occur if, for example, you only approach people who appear more approachable. This could skew the results as certain types of people might be more inclined to choose certain numbers. Instead, a truly random sample requires that each potential participant – whether shy, busy, or distracted – has an equal opportunity to be part of the study.

Empirical Probability

Empirical probability, also known as experimental probability, is calculated by dividing the number of times an event occurs by the total number of trials. This concept plays a central role in the textbook exercise, where the empirical probability for each digit being chosen is found by dividing the count of each digit by the total number of responses.

In this context, the empirical probability can inform us about human behavior regarding random selection. While in theory, each digit (1, 2, 3, 4) should have an equal probability (1/4) of being chosen if the selection is truly random, empirical data might show deviations due to various factors, such as preferences for certain numbers, intentional or unintentional patterns in choice, or misunderstanding the task of choosing 'at random'.

Survey Methodology

Survey methodology encompasses a range of techniques used to collect data from a sample of individuals. It combines understanding of sampling methods with practical considerations such as survey design, question phrasing, data collection strategies, and analysis techniques. Our exercise illustrates the importance of clear instructions – asking participants to select 'at random' may yield different interpretations and biases. For instance, some may think 'random' means not choosing the number they first saw or avoiding the number they think others would pick.

Using consistent wording when presenting the task to each participant can reduce such biases. Furthermore, the methodology would be improved by randomizing the order of the numbers on the index card to help mitigate any order effects – the tendency for items presented first or last to be selected more frequently than those in the middle.

Expected Distribution

The expected distribution in a random sampling scenario is the theoretical distribution of outcomes if all conditions are ideal and truly random selection occurs, which in our case is equally likely outcomes. For this exercise with four digits, the expected distribution would be 25% for each number – that is, if we had an infinite number of participants each choosing truly at random.

By comparing the empirical data collected from the survey participants to the expected distribution, we can understand whether people tend to choose certain numbers over others when asked to make a random choice. Significant discrepancies can hint towards psychological biases or a misunderstanding of randomness. If people consistently choose one number over another, the distribution would be uneven, which could be very telling about the nature of 'random' choices humans make.