Ch. 12 - Analysis of Variance

Triola - Elementary Statistics 14th Edition

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

Triola 14th Edition

Ch. 12 - Analysis of Variance

문제 12.CR.6b

12장, 문제 12.CR.6b

Quarters Assume that weights of quarters minted after 1964 are normally distributed with a mean of 5.670 g and a standard deviation of 0.062 g (based on U.S. Mint specifications).
b. If 25 quarters are randomly selected, find the probability that their mean weight is greater than 5.675 g.

검증된 단계별 안내

Step 1: Identify the given values in the problem. The population mean (μ) is 5.670 g, the population standard deviation (σ) is 0.062 g, and the sample size (n) is 25. The problem asks for the probability that the sample mean weight is greater than 5.675 g.

Step 2: Calculate the standard error of the mean (SEM). The SEM is given by the formula:

σ / \sqrt{n}

, where σ is the population standard deviation and n is the sample size.

Step 3: Standardize the sample mean using the z-score formula:

z = (X - μ)/ SEM)

. Here, X is the sample mean (5.675 g), μ is the population mean (5.670 g), and SEM is the standard error of the mean calculated in Step 2.

Step 4: Use the z-score obtained in Step 3 to find the corresponding probability. This can be done by looking up the z-score in a standard normal distribution table or using statistical software to find the cumulative probability.

Step 5: Subtract the cumulative probability from 1 to find the probability that the sample mean weight is greater than 5.675 g. This is because the problem asks for the probability in the upper tail of the distribution.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

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주요 개념

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

Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In this context, the weights of quarters follow a normal distribution, which allows us to use statistical methods to calculate probabilities related to their mean.

Central Limit Theorem

The Central Limit Theorem states that the sampling distribution of the sample mean will be normally distributed, regardless of the shape of the population distribution, provided the sample size is sufficiently large (typically n > 30). In this case, with a sample size of 25, we can still apply the theorem to approximate the distribution of the sample mean of quarter weights.

Z-Score

A Z-score measures how many standard deviations an element is from the mean. It is calculated by subtracting the mean from the value and dividing by the standard deviation. In this problem, we will use the Z-score to determine the probability that the mean weight of the selected quarters exceeds 5.675 g by standardizing the sample mean.

관련 실천

교과서 질문

Cola Weights The displayed results from Exercise 1 are from one-way analysis of variance. What is it about this test that characterizes it as one-way analysis of variance instead of two-way analysis of variance?

109

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교과서 질문

Win 4 Lottery Shown below is a histogram of digits selected in California’s Win 4 lottery. Each drawing involves the random selection (with replacement) of four digits between 0 and 9 inclusive.

c. Identify the frequencies, then test the claim that the digits are selected from a population in which the digits are all equally likely. Is there a problem with the lottery?

140

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교과서 질문

Normal Quantile Plot The accompanying normal quantile plot was obtained from the longevity times of presidents. What does this graph tell us?

154

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교과서 질문

Cola Weights For the four samples described in Exercise 1, the sample of regular Coke has a mean weight of 0.81682 lb, the sample of Diet Coke has a mean weight of 0.78479 lb, the sample of regular Pepsi has a mean weight of 0.82410 lb, and the sample of Diet Pepsi has a mean weight of 0.78386 lb. If we use analysis of variance and reach a conclusion to reject equality of the four sample means, can we then conclude that any of the specific samples have means that are significantly different from the others?

100

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교과서 질문

Win 4 Lottery Shown below is a histogram of digits selected in California’s Win 4 lottery. Each drawing involves the random selection (with replacement) of four digits between 0 and 9 inclusive.

b. Does the display depict a normal distribution? Why or why not? What should be the shape of the histogram?

141

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교과서 질문

In Exercises 1–5, refer to the following list of numbers of years that deceased U.S. presidents, popes, and British monarchs lived after their inauguration, election, or coronation, respectively. (As of this writing, the last president is George H. W. Bush, the last pope is John Paul II, and the last British monarch is George VI.) Assume that the data are samples from larger populations.

[Image]

Exploring the Data Include appropriate units in all answers.

e. What is the level of measurement of the data (nominal, ordinal, interval, ratio)?

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Quarters Assume that weights of quarters minted after 1964 are normally distributed with a mean of 5.670 g and a standard deviation of 0.062 g (based on U.S. Mint specifications).b. If 25 quarters are randomly selected, find the probability that their mean weight is greater than 5.675 g.

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Normal Distribution

Central Limit Theorem

Z-Score

Quarters Assume that weights of quarters minted after 1964 are normally distributed with a mean of 5.670 g and a standard deviation of 0.062 g (based on U.S. Mint specifications).
b. If 25 quarters are randomly selected, find the probability that their mean weight is greater than 5.675 g.