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Chapter 11: Problem 2

Roulette Casinos are required to verify that their games operate as advertised. American roulette wheels have 38 slots鈥�18 red, 18 black, and 2 green. In one casino, managers record data from a random sample of 200 spins of one of their American roulette wheels. The one-way table below displays the results. \(\begin{array}{llll}{\text { Color: }} & {\text { Red }} & {\text { Black }} & {\text { Green }} \\ \hline \text { Count: } & {85} & {99} & {16} \\\ \hline\end{array}\) (a) State appropriate hypotheses for testing whether these data give convincing evidence that the distribution of outcomes on this wheel is not what it should be. (b) Calculate the expected counts for each color. Show your work.

Short Answer

Expert verified
The hypotheses test if the roulette is fair; expected counts: Red 94.74, Black 94.74, Green 10.53.

Step by step solution

01

Define Hypotheses

To test if the roulette wheel is fair, we set up the following hypotheses:- Null Hypothesis (\(H_0\)): The distribution of outcomes is as expected for an American roulette wheel: 18/38 red, 18/38 black, and 2/38 green.- Alternative Hypothesis (\(H_a\)): The distribution of outcomes is not as expected.
02

Determine Total Outcomes

First, calculate the total number of spins recorded by summing up the counts for red, black, and green: \(85 + 99 + 16 = 200\) spins.
03

Compute Expected Counts

Calculate the expected count for each color using the probabilities from the null hypothesis.- Expected count for red: \(\text{Probability of Red} = \frac{18}{38}\) \[\text{Expected Red} = 200 \times \frac{18}{38} \approx 94.74\]- Expected count for black: \(\text{Probability of Black} = \frac{18}{38}\) \[\text{Expected Black} = 200 \times \frac{18}{38} \approx 94.74\]- Expected count for green: \(\text{Probability of Green} = \frac{2}{38}\) \[\text{Expected Green} = 200 \times \frac{2}{38} \approx 10.53\]

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Key Concepts

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

Expected Counts
When we investigate whether a game like roulette is operating fairly, one of the key concepts we rely on is "Expected Counts." This relates to what we anticipate seeing in a perfectly random and unbiased scenario. Expected counts are calculated by applying the theoretical probabilities of each outcome to the total number of trials. For example, in the context of our roulette wheel:
  • The chance of landing on red is 18 out of 38 slots.
  • The same probability applies to black, as there are also 18 black slots.
  • For green, with just 2 slots, it is 2 out of 38.
To find the expected count for red, you multiply the total spins (200) by the probability of landing on red, which is \( \frac{18}{38} \). This results in about 94.74 expected red outcomes.
You'll apply the same method to calculate for black and green. The expected count is important because it acts as our baseline for comparison, helping us determine if the actual outcomes significantly deviate from what we'd expect by chance.
Null Hypothesis
In hypothesis testing, the Null Hypothesis is your starting assumption. It's the statement you seek to test and potentially reject through your experiment. For roulette, it proposes that the outcome distribution is as theoretically expected. In our scenario:
  • The null hypothesis (\(H_0\)) asserts that each color's probability of appearance is consistent with the given odds: 18/38 for red, 18/38 for black, and 2/38 for green.
The null hypothesis assumes there's no effect or no difference, meaning the roulette wheel is functioning in alignment with expectations.
It is vital to understand that the null hypothesis isn't necessarily what we believe, but it's what we assume to be true unless proven otherwise by data analysis.
Alternative Hypothesis
The Alternative Hypothesis is developed to contrast the null hypothesis. It represents a different, often more exciting proposition that there is some effect or difference. In the case of the roulette wheel:
  • The alternative hypothesis (\(H_a\)) claims that the distribution of outcomes is not as expected.
This essentially means that we suspect the roulette may not be fair. It might be poorly calibrated or manipulated.
The goal of our statistical test is to determine whether there is enough evidence in the data to reject the null hypothesis in favor of the alternative hypothesis. Typically, rejecting the null hypothesis in favor of the alternative suggests that the roulette wheel isn't functioning as per the expected probabilities.

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Most popular questions from this chapter

Aw, nuts! A company claims that each batch of its deluxe mixed nuts contains 52% cashews, 27% almonds, 13% macadamia nuts, and 8% brazil nuts. To test this claim, a quality control inspector takes a random sample of 150 nuts from the latest batch. The one-way table below displays the sample data. (a) State appropriate hypotheses for performing a test of the company鈥檚 claim. (b) Calculate the expected counts for each type of nut. Show your work.

The P-value for a chi-square goodness-of-fit test is 0.0129. The correct conclusion is (a) reject H0 at A 0.05; there is strong evidence that the trees are randomly distributed. (b) reject H0 at A 0.05; there is not strong evidence that the trees are randomly distributed. (c) reject H0 at A 0.05; there is strong evidence that the trees are not randomly distributed. (d) fail to reject H0 at A 0.05; there is not strong evidence that the trees are randomly distributed. (e) fail to reject H0 at A 0.05; there is strong evidence that the trees are randomly distributed.

Yahtzee (5.3, 6.3) In the game of Yahtzee, 5 six-sided dice are rolled simultaneously. To get a Yahtzee, the player must get the same number on all 5 dice. (a) Luis says that the probability of getting a Yahtzee in one roll of the dice is \(\left(\frac{1}{6}\right)^{5} .\) Explain why Luis is wrong. (b) Nassir decides to keep rolling all 5 dice until he gets a Yahtzee. He is surprised when he still hasn鈥檛 gotten a Yahtzee after 25 rolls. Should he be? Calculate an appropriate probability to support your answer.

Mendel and the peas Gregor Mendel (1822鈥�1884), an Austrian monk, is considered the father of genetics. Mendel studied the inheritance of various traits in pea plants. One such trait is whether the pea is smooth or wrinkled. Mendel predicted a ratio of 3 smooth peas for every 1 wrinkled pea. In one experiment, he observed 423 smooth and 133 wrinkled peas. The data were produced in such a way that the Random and Independent conditions are met. Carry out a chi-square goodness-of-fit test based on Mendel鈥檚 prediction. What do you conclude?

Inference recap (8.1 to 11.2) In each of the following settings, say which inference procedure from Chapter 8, 9, 10, or 11 you would use. Be specific. For example, you might say 鈥渢wo-sample z test for the difference between two proportions.鈥� You do not need to carry out any procedures.34 (a) Is there a relationship between attendance at religious services and alcohol consumption? A random sample of 1000 adults was asked whether they regularly attend religious services and whether they drink alcohol daily. (b) Separate random samples of 75 college students and 75 high school students were asked how much time, on average, they spend watching television each week. We want to estimate the difference in the average amount of TV watched by high school and college students.
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