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Chapter 4: Problem 130

Euchre One of the authors and some statistician friends have an ongoing series of Euchre games that will stop when one of the two teams is deemed to be statistically significantly better than the other team. Euchre is a card game and each game results in a win for one team and a loss for the other. Only two teams are competing in this series, which we'll call team A and team B. (a) Define the parameter(s) of interest. (b) What are the null and alternative hypotheses if the goal is to determine if either team is statistically significantly better than the other at winning Euchre? (c) What sample statistic(s) would they need to measure as the games go on? (d) Could the winner be determined after one or two games? Why or why not? (e) Which significance level, \(5 \%\) or \(1 \%,\) will make the game last longer?

Short Answer

Expert verified
The parameter is the success proportions of teams A and B. The null hypothesis is that the teams have an equal chance of winning (\(p_A = p_B\)), and the alternative hypothesis is that the chances are not equal (\(p_A \neq p_B\)). The winning proportions of each team should be measured as games go on. A winner could not be determined after one or two games due to insufficient sample size. Using a 1% significance level would make the game series last longer than a 5% significance level due to the need for stronger evidence to reject the null hypothesis.

Step by step solution

01

Defining the Parameter

Here, the parameter of interest is the true success proportions for teams A and B, with success being defined as winning a game of Euchre. Denote \(p_A\) as the probability of team A winning, and \(p_B\) for team B
02

Formulating the Hypotheses

The null hypothesis (\(H_0\)) would be that both teams have an equal chance of winning the game, that is \(p_A = p_B\). The alternative hypothesis (\(H_A\)) is that the teams do not have an equal chance of winning, hence \(p_A \neq p_B\)
03

Sample Statistics Measurement

As games proceed, they would need to measure the proportion of games won by each team, which will serve as estimates for \(p_A\) and \(p_B\)
04

Number of Games for Winner Determination

The number of games required would be dependent on the difference in team performance. If the difference in proportions is large, fewer games would be required to conclude whether there is a statistically significant difference. However, deciding a winner after one or two games would likely not provide enough evidence to reject the null hypothesis due to a lack of sufficient sample size. So, the winner could not be decided after one or two games.
05

Deciding Significance Level

A lower significance level makes a hypothesis test more conservative and requires more evidence to reject the null hypothesis. Therefore, choosing a significance level of 1% would make the game series last longer than if a 5% significance level was chosen.

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Key Concepts

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

Euchre
Euchre is a classic trick-taking card game, typically played with four players divided into two teams of two. The game is played with a deck of 24, 28, or 32 cards, depending on regional variations. Each team aims to win more tricks than the other, with a trick being a round of play where each player plays one card. Winning a trick can sometimes require both strategy and luck.
In the context of this Euchre game series, determining which team is better involves analyzing the results of multiple games. Each game results in a win or a loss for the teams competing, denoted here as team A and team B. Only one team can win a particular game, making it a clear and straightforward measurement for statistical analysis.
  • Each game of Euchre in this series contributes data points necessary to understand the overall performance of the teams involved.
  • By collecting data from multiple games, players can employ statistical methods to determine if there is a significant difference between the capabilities of the two teams.
Statistical Significance
Statistical significance is a way of determining if the results of an experiment or study are likely to be due to something other than chance. In the context of this Euchre series, statistical significance helps decide which team is genuinely better by analyzing the game outcomes.
Here's how it works:
  • Each win by team A or team B contributes information towards understanding if one team is statistically superior.
  • The degree of statistical significance depends on the observed difference in performance and the margin of error in these observations.
  • A specific "significance level" (such as 5% or 1%) is chosen to determine how strong the evidence needs to be before concluding that one team is better than the other.

By establishing a significance level, players can systematically assess if observed differences in performances are indeed reflective of real differences in abilities or just random variations.
Probability
Probability in this context relates to the team's chances of winning a Euchre game. It provides a quantitative measure of how likely an event is to occur鈥攊n this case, either team winning a game.
Let's break it down further:
  • To decide which team is better, one calculates the probability of winning for each team, denoted as \(p_A\) for team A and \(p_B\) for team B.
  • These probabilities are estimated from the proportion of games won by each team as more games are played.
  • A significant difference in these probabilities can suggest one team has a stronger card-playing ability.

Understanding probabilities in this setting helps clarify which team, A or B, has a higher likelihood of winning future games based on past performance. It serves as a foundation for statistical analysis involved in hypothesis testing.
Null Hypothesis
The null hypothesis is a fundamental concept in hypothesis testing, essentially stating that there is no effect or difference. In the Euchre game series, the null hypothesis posits there is no difference between the two teams' probabilities of winning.
This plays out as follows:
  • The null hypothesis \((H_0)\) claims that team A's probability of winning \((p_A)\) is equal to team B's \((p_B)\), or \(p_A = p_B\).
  • It's a starting point for analysis, meaning it's assumed true until sufficient evidence suggests otherwise.
  • The alternative hypothesis \((H_A)\) states that \(p_A eq p_B\), indicating that one team may be statistically better.

Rejecting or failing to reject the null hypothesis after analyzing several games' outcomes can guide decisions on whether a team is significantly better based on the rules of statistical inference.

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Most popular questions from this chapter

By some accounts, the first formal hypothesis test to use statistics involved the claim of a lady tasting tea. \({ }^{11}\) In the 1920 's Muriel Bristol- Roach, a British biological scientist, was at a tea party where she claimed to be able to tell whether milk was poured into a cup before or after the tea. R.A. Fisher, an eminent statistician, was also attending the party. As a natural skeptic, Fisher assumed that Muriel had no ability to distinguish whether the milk or tea was poured first, and decided to test her claim. An experiment was designed in which Muriel would be presented with some cups of tea with the milk poured first, and some cups with the tea poured first. (a) In plain English (no symbols), describe the null and alternative hypotheses for this scenario. (b) Let \(p\) be the true proportion of times Muriel can guess correctly. State the null and alternative hypothesis in terms of \(p\).

In a test to see whether males, on average, have bigger noses than females, the study indicates that " \(p<0.01\)."

Scientists studying lion attacks on humans in Tanzania \(^{32}\) found that 95 lion attacks happened between \(6 \mathrm{pm}\) and \(10 \mathrm{pm}\) within either five days before a full moon or five days after a full moon. Of these, 71 happened during the five days after the full moon while the other 24 happened during the five days before the full moon. Does this sample of lion attacks provide evidence that attacks are more likely after a full moon? In other words, is there evidence that attacks are not equally split between the two five-day periods? Use StatKey or other technology to find the p-value, and be sure to show all details of the test. (Note that this is a test for a single proportion since the data come from one sample.)

The consumption of caffeine to benefit alertness is a common activity practiced by \(90 \%\) of adults in North America. Often caffeine is used in order to replace the need for sleep. One study \(^{24}\) compares students' ability to recall memorized information after either the consumption of caffeine or a brief sleep. A random sample of 35 adults (between the ages of 18 and 39 ) were randomly divided into three groups and verbally given a list of 24 words to memorize. During a break, one of the groups takes a nap for an hour and a half, another group is kept awake and then given a caffeine pill an hour prior to testing, and the third group is given a placebo. The response variable of interest is the number of words participants are able to recall following the break. The summary statistics for the three groups are in Table 4.9. We are interested in testing whether there is evidence of a difference in average recall ability between any two of the treatments. Thus we have three possible tests between different pairs of groups: Sleep vs Caffeine, Sleep vs Placebo, and Caffeine vs Placebo. (a) In the test comparing the sleep group to the caffeine group, the p-value is \(0.003 .\) What is the conclusion of the test? In the sample, which group had better recall ability? According to the test results, do you think sleep is really better than caffeine for recall ability? (b) In the test comparing the sleep group to the placebo group, the p-value is 0.06 . What is the conclusion of the test using a \(5 \%\) significance level? If we use a \(10 \%\) significance level? How strong is the evidence of a difference in mean recall ability between these two treatments? (c) In the test comparing the caffeine group to the placebo group, the p-value is 0.22 . What is the conclusion of the test? In the sample, which group had better recall ability? According to the test results, would we be justified in concluding that caffeine impairs recall ability? (d) According to this study, what should you do before an exam that asks you to recall information?

How influenced are consumers by price and marketing? If something costs more, do our expectations lead us to believe it is better? Because expectations play such a large role in reality, can a product that costs more (but is in reality identical) actually be more effective? Baba Shiv, a neuroeconomist at Stanford, conducted a study \(^{25}\) involving 204 undergraduates. In the study, all students consumed a popular energy drink which claims on its packaging to increase mental acuity. The students were then asked to solve a series of puzzles. The students were charged either regular price ( \(\$ 1.89\) ) for the drink or a discount price \((\$ 0.89)\). The students receiving the discount price were told that they were able to buy the drink at a discount since the drinks had been purchased in bulk. The authors of the study describe the results: "the number of puzzles solved was lower in the reduced-price condition \((M=4.2)\) than in the regular-price condition \((M=5.8) \ldots p<.0001 . "\) (a) What can you conclude from the study? How strong is the evidence for the conclusion? (b) These results have been replicated in many similar studies. As Jonah Lehrer tells us: "According to Shiv, a kind of placebo effect is at work. Since we expect cheaper goods to be less effective, they generally are less effective, even if they are identical to more expensive products. This is why brand-name aspirin works better than generic aspirin and why Coke tastes better than cheaper colas, even if most consumers can't tell the difference in blind taste tests."26 Discuss the implications of this research in marketing and pricing.
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