Textbook Question
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?
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Ch. 12 - Analysis of Variance
Triola14th EditionElementary StatisticsISBN: 9780137366446
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Table of contents
- Ch. 1 - Introduction to Statistics88
- Ch. 2 - Exploring Data with Tables and Graphs70
- Ch. 3 - Describing, Exploring, and Comparing Data66
- Ch. 4 - Probability112
- Ch. 5 - Discrete Probability Distributions109
- Ch. 6 - Normal Probability Distributions122
- Ch. 7 - Estimating Parameters and Determining Sample Sizes107
- Ch. 8 - Hypothesis Testing75
- Ch. 9 - Inferences from Two Samples94
- Ch. 10 - Correlation and Regression80
- Ch. 11 - Goodness-of-Fit and Contingency Tables29
- Ch. 12 - Analysis of Variance42
Chapter 12, Problem 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.
Verified step by step guidance1
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: , where σ is the population standard deviation and n is the sample size.
Step 3: Standardize the sample mean using the z-score formula: . 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.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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09:47
Finding Standard Normal Probabilities using z-Table
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.
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04:52Calculating the Mean
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.
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06:31Z-Scores From Given Probability - TI-84 (CE) Calculator
Related Practice
Textbook Question
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?
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Textbook Question
Normal Quantile Plot The accompanying normal quantile plot was obtained from the longevity times of presidents. What does this graph tell us?
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Textbook Question
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?
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Textbook Question
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?
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Textbook Question
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.
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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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