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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16,980 Students Helped

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

Scatterplots are essential tools in statistics that allow for the initial exploration of the relationship between two quantitative variables by revealing direction (positive or negative), form (linear or curved), and strength (tight or dispersed grouping of data points). The Pearson correlation coefficient quantifies linear associations and is immune to changes in scale or units, but it requires that the data meet conditions such as linearity and the absence of influential outliers. Nonparametric measures like Kendall’s tau and Spearman’s rho are valuable when the linearity condition is not met. Importantly, correlation does not imply causation, and careful assessment—including a check for lurking variables—is critical for valid interpretation.

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

Association Suppose you were to collect data for each pair of variables. You want to make a scatterplot. Which variable would you use as the explanatory variable and which as the response variable? Why? What would you expect to see in the scatterplot? Discuss the likely direction, form, and strength. a) Apples: weight in grams, weight in ounces b) Apples: circumference (inches), weight (ounces) c) College freshmen: shoe size, grade point average d) Gasoline: number of miles you drove since filling up, gallons remaining in your tank

Example 2

Association II Suppose you were to collect data for each pair of variables. You want to make a scatterplot. Which variable would you use as the explanatory variable and which as the response variable? Why? What would you expect to see in the scatterplot? Discuss the likely direction, form, and strength. a) T-shirts at a store: price each, number sold b) Scuba diving: depth, water pressure c) Scuba diving: depth, visibility d) All elementary school students: weight, score on a reading test

Example 3

Bookstore sales Consider the following data from a small bookstore. $$ \begin{array}{|c|c|} \hline {\begin{array}{c} \text { Number of Sales } \\ \text { People Working } \end{array}} & {\text { Sales (in \$1000) }} \\ \hline 2 & 10 \\ \hline 3 & 11 \\ \hline 7 & 13 \\ \hline 9 & 14 \\ \hline 10 & 18 \\ 10 & 20 \\ 12 & 20 \\ 15 & 22 \\ 16 & 22 \\ \hline 20 & 26 \\ \hline \bar{x}=10.4 & \bar{y}=17.6 \\ S D(x)=5.64 & S D(y)=5.34 \\ \hline \end{array} $$ a) Prepare a scatterplot of Sales against Number of sales people working. b) What can you say about the direction of the association? c) What can you say about the form of the relationship? d) What can you say about the strength of the relationship? e) Does the scatterplot show any outliers?

Example 4

Disk drives 2014 Disk drives have been getting larger. Their capacity is now often given in terabytes (TB) where $1 \mathrm{~TB}=1000$ gigabytes, or about a trillion bytes. A search of prices for external disk drives on Amazon.com in early 2014 found the following data: $$ \begin{array}{|c|c|} \hline \text { Capacity (in IB) } & \text { Price (in \$) } \\ \hline 0.50 & 59.99 \\ 1.0 & 79.99 \\ 2.0 & 111.97 \\ 3.0 & 109.99 \\ 4.0 & 149.99 \\ 6.0 & 423.34 \\ 8.0 & 596.11 \\ 12.0 & 1079.99 \\ 32.0 & 4461.00 \\ \hline \end{array} $$ a) Prepare a scatterplot of Price against Capacity. b) What can you say about the direction of the association? c) What can you say about the form of the relationship? d) What can you say about the strength of the relationship? e) Does the scatterplot show any outliers?

Example 5

Correlation facts If we assume that the conditions for correlation are met, which of the following are true? If false, explain briefly. a) A correlation of -0.98 indicates a strong, negative association. b) Multiplying every value of $x$ by 2 will double the correlation. c) The units of the correlation are the same as the units of $y$.

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Common Mistakes

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