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Ch. 12 - Analysis of Variance
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoTriola 14th EditionCh. 12 - Analysis of VarianceProblema 12.CRE.8
Capitolo 12, Problema 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?


Histogram showing the frequency of digits drawn in California's Win 4 lottery, with varying heights for each digit from 0 to 9.

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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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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.
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Finding Standard Normal Probabilities using z-Table

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.
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Intro to Histograms

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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Sampling Distribution of Sample Proportion
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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.

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

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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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