Ch. 5 - Discrete Probability Distributions

Triola - Elementary Statistics 14th Edition

Triola14th EditionElementary StatisticsISBN: 9780137366446당신이 사용하는 게 아니라요?교과서 변경

Triola 14th Edition

Ch. 5 - Discrete Probability Distributions

문제 5.1.17

5장, 문제 5.1.17

Lottery. In Exercises 15–20, refer to the accompanying table, which describes probabilities for the California Daily 4 lottery. The player selects four digits with repetition allowed, and the random variable x is the number of digits that match those in the same order that they are drawn (for a “straight” bet).

Range Rule of Thumb for Significant Events Use the range rule of thumb to determine whether 4 matches is a significantly high number of matches.

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Step 1: Understand the problem. The goal is to determine whether 4 matches is a significantly high number of matches using the range rule of thumb. The range rule of thumb states that values are considered significantly high if they are greater than the mean plus two standard deviations.

Step 2: Calculate the mean (μ) of the random variable x. Use the formula for the mean of a probability distribution: μ = Σ[x * P(x)], where x represents the number of matching digits and P(x) represents the corresponding probability.

Step 3: Calculate the standard deviation (σ) of the random variable x. Use the formula for the standard deviation of a probability distribution: σ = √Σ[(x - μ)^2 * P(x)], where μ is the mean calculated in Step 2.

Step 4: Apply the range rule of thumb. Compute the threshold for significantly high values using the formula: μ + 2σ. Compare the value of 4 matches to this threshold.

Step 5: Interpret the result. If 4 matches exceed the threshold calculated in Step 4, it is considered a significantly high number of matches. Otherwise, it is not considered significant.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Probability Distribution

A probability distribution describes how the probabilities of a random variable are distributed across its possible values. In this case, the table shows the probabilities of matching digits in the California Daily 4 lottery, where the random variable x represents the number of digits that match. Understanding this distribution is crucial for analyzing the likelihood of different outcomes in the lottery.

Random Variable

A random variable is a numerical outcome of a random phenomenon. In the context of the lottery question, the random variable x represents the number of digits that match the drawn numbers in the same order. Recognizing how random variables function helps in calculating probabilities and understanding the nature of the outcomes in probabilistic scenarios.

Range Rule of Thumb

The Range Rule of Thumb is a statistical guideline used to determine whether a particular outcome is significantly high or low compared to the expected range of values. It suggests that if an outcome is more than two standard deviations away from the mean, it can be considered significant. In this lottery context, applying this rule will help assess whether matching all four digits is an unusually high event.

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교과서 질문

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Find the probability that no more than 3 of the selected consumers believe that cash will be obsolete in the next 20 years.

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