91Ó°ÊÓ

Rock-paper-scissors is a hand game played by two or more people where players choose to sign either rock, paper, or scissors with their hands. For your statistics class project, you want to evaluate whether players choose between these three options randomly, or if certain options are favored above others. You ask two friends to play rock-paper-scissors and count the times each option is played. The following table summarizes the data: $$ \begin{array}{ccc} \text { Rock } & \text { Paper } & \text { Scissors } \\ \hline 43 & 21 & 35 \end{array} $$ Use these data to evaluate whether players choose between these three options randomly, or if certain options are favored above others. Make sure to clearly outline each step of your analysis, and interpret your results in context of the data and the research question.

Step by step solution

01

Define the Hypotheses

First, define the null and alternative hypotheses. The null hypothesis \((H_0)\) is that players choose rock, paper, and scissors with equal probability. The alternative hypothesis \((H_a)\) states that players do not choose the options equally.

02

Collect Observed Frequencies

From the data, we have the following observed frequencies: Rock \(O_1 = 43\), Paper \(O_2 = 21\), and Scissors \(O_3 = 35\).

03

Calculate Expected Frequencies

Assuming random choice, each option should be equally likely. Total number of plays is \(43 + 21 + 35 = 99\). Thus, the expected frequency for each option is \(E = \frac{99}{3} = 33\).

04

Use the Chi-Square Test

Calculate the chi-square statistic using the formula: \[\chi^2 = \sum \frac{(O_i - E)^2}{E}\]where \(O_i\) are observed frequencies and \(E\) are expected frequencies.

05

Perform Calculations

Substitute the values into the chi-square formula:\[\chi^2 = \frac{(43-33)^2}{33} + \frac{(21-33)^2}{33} + \frac{(35-33)^2}{33}\]Calculate each term: \(\frac{(43-33)^2}{33} = \frac{100}{33} \approx 3.03\), \(\frac{(21-33)^2}{33} = \frac{144}{33} \approx 4.36\), \(\frac{(35-33)^2}{33} = \frac{4}{33} \approx 0.12\).Finally, compute \(\chi^2\), which equals approximately \(3.03 + 4.36 + 0.12 = 7.51\).

06

Determine the Critical Value

Find the critical chi-square value from the chi-square distribution table for \(df = 2\) (degrees of freedom = number of outcomes - 1) at a significance level \(\alpha\) (commonly 0.05). The critical value \(\chi^2_{0.05, 2} \approx 5.991\).

07

Interpret Results

Compare the calculated chi-square statistic \(7.51\) to the critical value \(5.991\). Since \(7.51 > 5.991\), we reject the null hypothesis. This suggests that players do not choose rock, paper, or scissors equally.

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Key Concepts

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

Null Hypothesis

In hypothesis testing, the null hypothesis, denoted as \(H_0\), is a critical concept. It represents a statement of no effect or no difference, implying no change or preference in the case under examination. For our rock-paper-scissors game scenario, the null hypothesis states that the players choose rock, paper, and scissors with equal probability.

In simpler terms, \(H_0\) presupposes that each choice occurs purely by chance, so there is no tendency toward any specific option.

This idea forms the cornerstone of statistical testing, as it provides a baseline or unexciting assumption against which we can test a new theory or observation.

Alternative Hypothesis

The alternative hypothesis, often represented as \(H_a\), is the statement you want to test against the null hypothesis. In the context of our exercise, the alternative hypothesis suggests that players do not choose rock, paper, and scissors with equal probability.

This implies that there is a noticeable difference, meaning players favor certain options over others.

The alternative hypothesis is essential because rejecting the null hypothesis implies acceptance of the alternative, thus revealing new insights or trends in the data. It drives the investigation by suggesting an outcome where behavior is not random.

Observed Frequencies

Observed frequencies (\(O_i\)) represent the actual data collected during an experiment or observation. For the rock-paper-scissors game, the observed frequencies are the counts of each choice made by the players: Rock \(O_1 = 43\), Paper \(O_2 = 21\), and Scissors \(O_3 = 35\).

These figures act as the raw data used to assess whether any significant deviation from randomness exists.

• They reflect the real-world behavior shown by the participants.
• They are crucial for comparing against expected frequencies in statistical analysis.

Understanding observed frequencies is fundamental, as they are the starting point in your comparison to assess if players act randomly or prefer certain moves more than others.

Expected Frequencies

Expected frequencies (\(E\)) are the theoretical counts we'd anticipate if the players' choices were random. Given that there are three options—Rock, Paper, Scissors—with a total of 99 plays, and assuming each choice is equally likely, we calculate the expected frequency for each option as follows: \(E = \frac{99}{3} = 33\).

Expected frequencies provide a benchmark or a point of reference in chi-square testing:

• They help determine how far off the actual observations are from the assumption of random choice.
• They are essential in calculating the chi-square statistic, which measures the discrepancy between observed and expected values.

By comparing observed frequencies to expected frequencies, we can assess the likelihood that any deviations are due to chance, or if there's potentially a preference in choices.