Ch. 12 - Analysis of Variance

Triola - Elementary Statistics 14th Edition

Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook

Triola 14th Edition

Ch. 12 - Analysis of Variance

Problem 12.CRE.8

Chapter 12, Problem 12.CRE.8

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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Observe the histogram provided. A normal distribution is characterized by a symmetric, bell-shaped curve where most of the data is concentrated around the mean, and the frequencies taper off equally on both sides.

Analyze the shape of the histogram. In this case, the frequencies of the digits are not symmetric. For example, digits like 6 and 7 have very low frequencies compared to others, and there are peaks at digits like 2, 5, 8, and 9.

Conclude that the histogram does not depict a normal distribution because it lacks symmetry and the bell-shaped curve. Instead, the distribution appears irregular with multiple peaks and valleys.

Consider what the shape of the histogram should be. Since the lottery involves random selection with replacement, each digit (0 through 9) should theoretically have an equal probability of being selected. This would result in a uniform distribution where all bars in the histogram are approximately the same height.

Summarize that the observed histogram deviates from the expected uniform distribution, suggesting potential irregularities or biases in the digit selection process.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Normal Distribution

A normal distribution is a symmetric, bell-shaped distribution characterized by its mean and standard deviation. In a normal distribution, most of the data points cluster around the mean, with fewer points appearing as you move away from the mean. This distribution is important in statistics because many statistical tests assume normality in the data.

Histogram

A histogram is a graphical representation of the distribution of numerical data, where the data is divided into bins or intervals. The height of each bar indicates the frequency of data points within that interval. Histograms are useful for visualizing the shape of the data distribution, including its central tendency and variability.

Random Sampling

Random sampling is a technique used to select a subset of individuals from a larger population, where each individual has an equal chance of being chosen. In the context of the lottery, random sampling ensures that each digit has an equal probability of being drawn, which is essential for the fairness and unpredictability of the lottery results.

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Related Practice

Textbook Question

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.

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Textbook Question

Pulse Rates Shown below are pulse rates from Data Set 1 “Body Data” in Appendix B, and the StatCrunch display from two-way analysis of variance of these data. In analyzing these data, what important feature is addressed with two-way analysis of variance that is not addressed with two separate tests of (1) difference between mean pulse rates based on gender, or (2) differences among the mean pulse rates in the different age brackets?

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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.

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

One vs. Two What is the fundamental difference between one-way analysis of variance and two-way analysis of variance?

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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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