91Ó°ÊÓ

Karl Pearson devised the chi-squared goodness-of-fit test partly to analyze data from an experiment to analyze whether a particular roulette wheel in Monte Carlo was fair, in the sense that each outcome was equally likely in a spin of the wheel. For a given European roulette wheel with 37 pockets (with numbers \(0,1,2, \ldots, 36\) ), consider the null hypothesis that the wheel is fair. a. For the null hypothesis, what is the probability for each pocket? b. For an experiment with 3,700 spins of the roulette wheel, find the expected number of times each pocket is selected. c. In the experiment, the 0 pocket occurred 110 times. Show the contribution to the \(X^{2}\) statistic of the results for this pocket. d. Comparing the observed and expected counts for all 37 pockets, we get \(X^{2}=34.4\) Specify the df value and indicate whether there is strong evidence that the roulette wheel is not balanced. (Hint: Recall that the \(d f\) value is the mean of the distribution.)

Step by step solution

01

Understanding the Null Hypothesis

For a European roulette wheel with 37 pockets, if the wheel is fair, each pocket should have an equal probability of being selected. This setup implies that every pocket has the same chance of landing after a spin.

02

Calculating the Probability for Each Pocket

Since there are 37 pockets on the wheel, and assuming each is equally likely under the null hypothesis, the probability for each pocket is given by:\[ P(\text{each pocket}) = \frac{1}{37} \approx 0.0270 \]

03

Finding the Expected Count

With 3,700 spins and each pocket having a probability of \(\frac{1}{37}\), the expected number of times each pocket is selected is:\[ E(\text{each pocket}) = 3,700 \times \frac{1}{37} = 100 \]

04

Calculating the Chi-Square Contribution for Pocket 0

For the 0 pocket occurring 110 times (observed frequency), the contribution to the chi-square statistic is calculated using the formula:\[ \chi^2 = \frac{(O - E)^2}{E} = \frac{(110 - 100)^2}{100} = \frac{10^2}{100} = 1.0 \]

05

Determining Degrees of Freedom and Statistical Significance

The degrees of freedom \(df\) for this scenario is the number of categories minus 1, which is:\[ df = 37 - 1 = 36 \]Given \( X^2 = 34.4 \) and \(df = 36\), we consult the chi-square distribution table to check the significance. Since 34.4 is less than the typical critical value for 36 degrees of freedom (close to 50.998 for \(\alpha = 0.05\)), the evidence is not strong enough to reject the null hypothesis at a 5% significance level.

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Key Concepts

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

Null Hypothesis

In statistical hypothesis testing, the null hypothesis is a starting point when analyzing data. It's a statement that there is no effect or difference. For example, in a fair coin toss, the null hypothesis is that both heads and tails are equally likely. In the case of the European roulette wheel, the null hypothesis posits that the wheel is fair.
Under this hypothesis, every pocket has an equal chance of being the result of the spin. This means each of the 37 pockets would appear with a probability of \(\frac{1}{37} \), or approximately 0.0270, in any given spin. This assumption of fairness is pivotal when conducting a chi-squared goodness-of-fit test.
The purpose of this hypothesis is to give us a baseline or benchmark to compare against the observed outcomes from actual spins of the roulette wheel.

Degrees of Freedom

Degrees of freedom (df) is a fundamental concept in statistical tests, including the chi-squared test. It's essentially the number of values that are free to vary in your data set. To calculate the degrees of freedom for a chi-squared test, you take the number of categories or outcomes and subtract one.
In our roulette example, the 37 pockets represent our categories. Thus, the degrees of freedom would be \(df = 37 - 1 = 36\).
Having 36 degrees of freedom means we have 36 different ways our data can vary while still fitting within the constraints of our test.

• Degrees of freedom help us determine the appropriate chi-squared distribution to reference.
• They are crucial for interpreting test results and determining statistical significance.

By understanding degrees of freedom, we can better assess whether our observed data deviate significantly from what we'd expect under the null hypothesis.

Chi-Squared Statistic

The chi-squared statistic is a measure used to determine how well observed data fit expected data under the null hypothesis. This statistical tool helps us decide if deviations from what's expected are due to chance or if they suggest a pattern requiring further investigation.
The chi-squared statistic is calculated using the formula:\[\chi^2 = \sum \frac{(O - E)^2}{E}\]where \(O\) is the observed frequency, and \(E\) is the expected frequency.
In simpler terms, it assesses whether the observed occurrences in each category (like numbers on a roulette wheel) are consistent with the expected frequencies if everything was random.

• A low chi-squared value suggests that the observed data follow the expected distribution closely.
• A high value indicates a discrepancy that could be statistically significant.

In our example, the chi-squared statistic for the 0 pocket was 1.0, which was derived from the observed count (110) and expected count (100). For the entire wheel, \(X^2\) was calculated to be 34.4, which, when compared to critical values, didn't suggest a significant deviation from fairness.

European Roulette Wheel

The European roulette wheel is a classic tool used in casinos, offering a unmistakable sense of excitement and chance. It features 37 pockets, numbered from 0 to 36. Unlike the American roulette, which has an additional "00" pocket, the European version is slightly more favorable for players.
Each spin of the wheel results in a random outcome, ideally with each number having an equal probability of appearing. This forms the basis of the null hypothesis in our analysis.
The fairness of a roulette wheel can be assessed using statistical methods such as the chi-squared goodness-of-fit test. This involves comparing the observed results from many spins against the expected outcomes under the assumption that the wheel is fair.

• The European roulette wheel provides a straightforward and familiar example for demonstrating statistical concepts, especially in probability and hypothesis testing.
• Due to its simplicity and well-known distribution of outcomes, it is often used in educational settings to teach these concepts.