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A casino player has grown suspicious about a specific roulette wheel. Specifically, this player believes that the slots for the numbers 0 and 00 , which can lead to larger payoffs, are slightly smaller than the rest of 36 slots, which means that the ball would land in these two slots less often than it would if all of the slots were of the same size. This player watched 430 spins on this roulette wheel, and found that the ball landed in 0 or 00 slot 14 times. a. What is the value of the point estimate of the proportion of all roulette spins on this wheel in which the ball would land in 0 or 00 slot? b. Construct a \(95 \%\) confidence interval for the proportion of all roulette spins on this wheel in which the ball would land in 0 or 00 slot. c. If all of the slots on this wheel are of the same size, the ball should land in 0 or 00 slot \(5.26 \%\) of the time. Based on the confidence interval you calculated in part b, does the player's suspicion seem reasonable?

Step by step solution

01

Calculate the Point Estimate

The point estimate of the proportion is calculated as 'the number of successful outcomes' divided by 'the total number of outcomes'. In this case, successful outcomes are the times the ball landed on 0 or 00, which is 14 times. Total outcomes are the total number of spins, which is 430. Therefore, the point estimate \( p \) is calculated as: \( p = \frac{14}{430} \)

02

Construct a Confidence Interval for the Proportion

A \(95 \%\) confidence interval for a proportion is calculated using the formula: \( p \pm Z \sqrt{\frac{p(1-p)}{n}}\). Here, \( p \) is the point estimate from Step 1, \( n \) is the number of trials which is 430. For a 95% confidence interval, \( Z \) is the z-score from the standard normal distribution for a two-tailed test, which equals 1.96. So plug these values into the formula and calculate.

03

Evaluate the Player’s Suspicion

The expected value if all slots were the same size is specified to be \(5.26 \% of the time. Compare this value to the confidence interval from Step 2. If 5.26% fall within the confidence interval, then the player's suspicion would seem unreasonable because we would fail to reject the null hypothesis that the ball lands in slot 0 or 00 as often as it would if all slots were the same size.

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Key Concepts

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

Point Estimate

The point estimate is a crucial concept when analyzing data in statistics, especially when trying to make inferences about a population based on a sample. It provides a single value estimate of a population parameter. In the context of this roulette wheel exercise, the point estimate helps us determine the probability or proportion of an event happening. For this problem, we're interested in the proportion of times the ball lands in the 0 or 00 slots.
To calculate the point estimate of the proportion, we take the number of times the ball landed in the preferred slots (14 times as per the observation) and divide it by the total number of spins (430 spins). This gives us the point estimate of the proportion as:
\[ p = \frac{14}{430} \approx 0.0326 \]
This means that based on the observed spins, the ball landed in the 0 or 00 slot approximately 3.26% of the time.

Proportion

In statistics, proportion refers to the fraction or the percentage of the whole that exhibits a particular trait or characteristic. In this problem, the proportion is used to describe the percentage of total roulette spins in which the ball lands in slots 0 or 00.
Understanding proportions is vital because it allows us to translate raw counts into more interpretable formats like percentages that are more intuitive, helping us to make comparisons or decisions. Our estimated proportion of 3.26% suggests that, during observation, a smaller percentage of spins landed in these slots compared to what might be expected if the slots were all of equal size, where it should land in 5.26% of spins.
The discrepancy between the observed proportion and the expected proportion underlines the importance of further statistical testing.

Statistics

Statistics is a branch of mathematics that deals with collecting, analyzing, interpreting, presenting, and organizing data. In the context of this problem, statistical tools are employed to understand whether the observed variation in the game could be due to the casino player’s suspicion or just random chance.
Data from the 430 spins is systematically analyzed to come up with meaningful conclusions using statistical measures like point estimates and confidence intervals. By doing so, they provide insights into the underlying probabilities and help to ascertain if these probabilities significantly deviate from expected norms.

• Helps to transform sample data into generalizable conclusions.
• Involves hypothesis testing to evaluate claims or beliefs regarding the population.

Statistics thus serve as the backbone when trying to make legal and reasonable judgments about the fairness of the roulette wheel in this case.

Hypothesis Testing

Hypothesis testing is a method used in statistics to test if there is sufficient evidence to reject a null hypothesis, considering a certain level of confidence. It’s useful in this roulette wheel scenario to assess whether the player’s suspicion is statistically valid.
Here, the null hypothesis (\( H_0 \)) posits that the ball lands in slots 0 or 00 at the same rate as expected, with a probability of 0.0526 (5.26%). The alternative hypothesis (\( H_a \)) would suggest that it lands at a different rate due to some bias.
To evaluate these hypotheses, we look at whether this expected probability falls within the confidence interval calculated from the observed data. By comparing the expected proportion with the confidence interval, we essentially test the player's claim statistically:

• If the expected 5.26% is outside the confidence interval, we have evidence against the null hypothesis, supporting the player's suspicion.
• However, if it falls within, it suggests that any observed difference might just be due to random chance, thus failing to reject the null hypothesis.

This testing process gives credibility to insights derived and provides a functionally rigorous way to evaluate claims objectively.