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Chapter 4: Problem 106

Classroom Games Exercise 4.62 describes a situation in which game theory students are randomly assigned to play either Game 1 or Game 2 , and then are given an exam containing questions on both games. Two one-tailed tests were conducted: one testing whether students who played Game 1 did better than students who played Game 2 on the question about Game \(1,\) and one testing whether students who played Game 2 did better than students who played Game 1 on the question about Game \(2 .\) The p-values were 0.762 and 0.549 , respectively. The p-values greater than 0.5 mean that, in the sample, the students who played the opposite game did better on each question. What does this study tell us about possible effects of actually playing a game and answering a theoretical question about it? Explain.

Short Answer

Expert verified
The study does not provide a statistically significant result to conclude that playing a game improves one's ability to answer theoretical questions about it. The p-values (0.762 for Game 1 and 0.549 for Game 2) are both greater than 0.05, indicating that the differences observed in the exam results could likely be due to chance. As such, this study doesn't provide strong evidence that playing a game has an appreciable effect on answering theoretical questions about it.

Step by step solution

01

Understanding P-value

The p-value is a statistical metric used to assess the strength of the evidence against the null hypothesis. When the p-value is low (below 0.05), it suggests that the likelihood of the observed results happening, assuming the null hypothesis is true, is low and hence, the null hypothesis is unlikely. Conversely, when the p-value is high (above 0.05), it suggests the likelihood of the observed results happening, assuming the null hypothesis is true, is high and hence, the null hypothesis is plausible.
02

Study results

Since the p-values of 0.762 and 0.549 for Game 1 and Game 2 respectively are higher than 0.05, this suggests that the performance differences seen in the sample are likely due to chance, rather than the actual variable being investigated (which game was played). Therefore, it would seem that playing a particular game did not have a significant effect on the test results for questions about that game.
03

Interpretation

The study suggests the result is not statistically significant and hence, the performance of students on questions related to a game is not necessarily affected by whether they played that game or not. The results could have been due to random chance. Therefore, there is no clear evidence from this study of the effect of playing a game on one's ability to answer theoretical questions about it.

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Key Concepts

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

Understanding P-value
The p-value is a vital concept in statistics, often discussed in relation to hypothesis testing. It represents the probability of observing results as extreme, or more extreme, than the results observed, assuming the null hypothesis holds true. Think of it as a way to measure the strength of evidence against the null hypothesis.

Here鈥檚 a simpler way to understand it:

  • If the p-value is low (typically below 0.05), it indicates strong evidence against the null hypothesis, suggesting it's unlikely to be true.
  • If the p-value is high (above 0.05), the data isn鈥檛 surprising if the null hypothesis is correct, suggesting it could be true.
For instance, in the game theory exercise, the p-values were 0.762 and 0.549, both well above 0.05. This means there is not enough evidence to reject the null hypothesis. In practical terms, the different game assignments did not provide significant evidence to suggest they influenced exam performance.
Exploring the Null Hypothesis
The null hypothesis is a fundamental part of statistical hypothesis testing. It is essentially a statement that there is no effect or no difference, and it serves as the default assumption to be tested. We test this assumption to explore if there is statistical evidence to support an alternative hypothesis, which suggests there is an effect.

In the context of the classroom games exercise, the null hypothesis could be stated as: "There is no difference in exam performance between students who played Game 1 and those who played Game 2."

  • The exercise aimed to test whether playing a particular game affected students' understanding as reflected in their test scores.
  • With p-values greater than 0.05, as seen in this study, the null hypothesis stands firm, meaning any observed differences are likely due to random variability rather than the type of game played.
Understanding and testing the null hypothesis helps to discern whether observed patterns in data are noteworthy or if they might simply be due to fluctuations or random chance.
Interpreting Statistical Significance
Statistical significance plays a crucial role in determining if the results of a study are likely due to chance or if they reflect a real-world effect. When results are statistically significant, it means the observed patterns are unlikely to have occurred under the null hypothesis. This decision is often based on the size of the p-value.

Let鈥檚 break it down:

  • A result is considered statistically significant if the p-value is below the conventional threshold of 0.05.
  • However, in the classroom games study, both tests revealed large p-values (0.762 and 0.549), suggesting no statistically significant difference in outcomes based on which game the students played.
The lack of statistical significance here indicates that, based on the sample, playing a game does not notably impact students' ability to answer related theoretical questions. Hence, the study suggests that other factors, not captured by the gaming experience, might be influencing test performance.

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Most popular questions from this chapter

Translating Information to Other Significance Levels Suppose in a two-tailed test of \(H_{0}: \rho=0 \mathrm{vs}\) \(H_{a}: \rho \neq 0,\) we reject \(H_{0}\) when using a \(5 \%\) significance level. Which of the conclusions below (if any) would also definitely be valid for the same data? Explain your reasoning in each case. (a) Reject \(H_{0}: \rho=0\) in favor of \(H_{a}: \rho \neq 0\) at a \(1 \%\) significance level. (b) Reject \(H_{0}: \rho=0\) in favor of \(H_{a}: \rho \neq 0\) at a \(10 \%\) significance level. (c) Reject \(H_{0}: \rho=0\) in favor of the one-tail alternative, \(H_{a}: \rho>0,\) at a \(5 \%\) significance level, assuming the sample correlation is positive.

Election Poll In October before the 2008 US presidential election, \(A B C\) News and the Washington Post jointly conducted a poll of "a random national sample" and asked people who they intended to vote for in the 2008 presidential election. \(^{37}\) Of the 1057 sampled people who answered either Barack Obama or John McCain, \(55.2 \%\) indicated that they would vote for Obama while \(44.8 \%\) indicated that they would vote for MeCain. While we now know the outcome of the election, at the time of the poll many people were very curious as to whether this significantly predicts a winner for the election. (While a candidate needs a majority of the electoral college vote to win an election, we'll simplify things and simply test whether the percentage of the popular vote for Obama is greater than \(50 \% .\) ) (a) State the null and alternative hypotheses for testing whether more people would vote for Obama than MeCain. (Hint: This is a test for a single proportion since there is a single variable with two possible outcomes.) (b) Describe in detail how you could create a randomization distribution to test this (if you had many more hours to do this homework and no access to technology).

In Exercises 4.150 to \(4.152,\) a confidence interval for a sample is given, followed by several hypotheses to test using that sample. In each case, use the confidence interval to give a conclusion of the test (if possible) and also state the significance level you are using. A \(90 \%\) confidence interval for \(p_{1}-p_{2}: 0.07\) to 0.18 (a) \(H_{0}: p_{1}=p_{2}\) vs \(H_{a}: p_{1} \neq p_{2}\) (b) \(H_{0}: p_{1}=p_{2}\) vs \(H_{a}: p_{1}>p_{2}\) (c) \(H_{0}: p_{1}=p_{2}\) vs \(H_{a}: p_{1}

Paul the Octopus In the 2010 World Cup, Paul the Octopus (in a German aquarium) became famous for being correct in all eight of the predictions it made, including predicting Spain over Germany in a semifinal match. Before each game, two containers of food (mussels) were lowered into the octopus's tank. The containers were identical, except for country flags of the opposing teams, one on each container. Whichever container Paul opened was deemed his predicted winner. \(^{32}\) Does Paul have psychic powers? In other words, is an 8 -for-8 record significantly better than just guessing? (a) State the null and alternative hypotheses. (b) Simulate one point in the randomization distribution by flipping a coin eight times and counting the number of heads. Do this five times. Did you get any results as extreme as Paul the Octopus? (c) Why is flipping a coin consistent with assuming the null hypothesis is true?

Flaxseed and Omega-3 Exercise 4.29 on page 234 describes a company that advertises that its milled flaxseed contains, on average, at least \(3800 \mathrm{mg}\) of ALNA, the primary omega-3 fatty acid in flaxseed, per tablespoon. In each case below, which of the standard significance levels, \(1 \%\) or \(5 \%\) or \(10 \%,\) makes the most sense for that situation? (a) The company plans to conduct a test just to double-check that its claim is correct. The company is eager to find evidence that the average amount per tablespoon is greater than 3800 (their alternative hypothesis) and is not really worried about making a mistake. The test is internal to the company and there are unlikely to be any real consequences either way. (b) Suppose, instead, that a consumer organization plans to conduct a test to see if there is evidence against the claim that the product contains at least \(3800 \mathrm{mg}\) per tablespoon. If the organization finds evidence that the advertising claim is false, it will file a lawsuit against the flaxseed company. The organization wants to be very sure that the evidence is strong, since there could be very serious consequences if the company is sued incorrectly.
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