Book cover for Stats Data and Models

Stats Data and Models

Richard D. De Veaux, Paul F. Velleman, David E. Bock

ISBN #9780321986498

4th Edition

1,406 Questions

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Summary

Learning Objectives

Key Concepts

Example Problems

Explanations

Common Mistakes

Summary

Re-expression is a critical tool in data analysis that transforms variables to better meet the assumptions required for advanced statistical techniques. By using methods from the Ladder of Powers and guidelines such as Tukey’s Rule of Thumb, analysts can make skewed data more symmetric, straighten curved relationships, and stabilize variances. Not only does this process improve the fit and interpretation of linear models, but it also enables meaningful comparisons across groups. However, care must be taken in choosing and reversing the transformation to avoid over-correction and misinterpretation.

Learning Objectives

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Key Concepts

CONCEPT

DEFINITION

Analysis of Variance (ANOVA)

A collection of statistical techniques used to compare the means of three or more groups by partitioning total variability into components due to differences among groups (between-group) and within groups.

Example Problems

Example 1

Every year, US News and World Report publishes a special issue on many U.S. colleges and universities. The scatterplots below have Student/Faculty Ratio (number of students per faculty member) for the colleges and universities on the $y$ -axes plotted against 4 other variables. The correct correlations for these scatterplots appear in this list. Match them. $\begin{array}{llllll}-0.98 & -0.71 & -0.51 & 0.09 & 0.23 & 0.69\end{array}$

Example 2

Are good grades in high school associated with family togetherness? A random sample of 142 high school students was asked how many meals per week their families ate together. Their responses produced a mean of 3.78 meals per week, with a standard deviation of $2.2 .$ Researchers then matched these responses against the students' grade point averages (GPAs). The scatterplot appeared to be reasonably linear, so they created a line of regression. No apparent pattern emerged in the residuals plot. The equation of the line was $\widehat{G P A}=2.73+0.11$ Meals. a) Interpret the $y$ -intercept in this context. b) Interpret the slope in this context. c) What was the mean GPA for these students? d) If a student in this study had a negative residual, what did that mean? e) Upon hearing of this study, a counselor recommended that parents who want to improve the grades their children get should get the family to eat together more often. Do you agree with this interpretation? Explain.

Example 3

Here are the scatterplot and regression analysis for Case Prices of 36 wines from vineyards in the Finger Lakes region of New York State and the Ages of the vineyards. (Data in Vineyards full): a) Does it appear that vineyards in business longer get higher prices for their wines? Explain. b) What does this analysis tell us about vineyards in the rest of the world? c) Write the regression equation. d) Explain why that equation is essentially useless.

Example 4

Instead of Age, perhaps the Size of the vineyard (in acres) is associated with the price of the wines. Look at the scatterplot (Data in Vineyards full): a) Do you see any evidence of an association? b) What concern do you have about this scatterplot? c) If the red "+" data point is removed, would the correlation become stronger or weaker? Explain. d) If the red " $+$ " data point is removed, would the slope of the line increase or decrease? Explain.

Example 5

In January $2012,$ the New York Times published a story called Twin Births in the U.S., Like Never Before, in which they reported a 76 percent increase in the rate of twin births from 1980 to $2009 .$ Here are the number of twins born each year (per 1000 live births). (www.cdc.gov/nchs/births.htm) a) Using the data only up to 2009 (as in the NYTimes article), find the equation of the regression line for predicting the number of twin births by Years since 1980 . b) Explain in this context what the slope means. c) Predict the number of twin births in the United States for the year 2014 . Comment on your faith in that prediction. d) Comment on the residuals plot and the appropriateness of the model. Look at the past three years' rates.

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Common Mistakes

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