回归分析：第二学期统计学进阶

Statistics

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"《A Second Course in Statistics: Regression Analysis》是威廉·门登霍尔（William Mendenhall）和特里·辛希奇（Terry Sincich）合著的第七版统计学教材，适用于已经完成基础统计学课程的本科生或研究生。这本书特别关注回归分析，是针对具有统计学背景或非统计学背景学生的第二学期或第二季度课程的理想选择，也可用于其他领域的硕士或博士生的实用回归分析课程。" 在回归分析这一主题中，作者深入探讨了统计学中的核心概念，包括但不限于： 1. 回归模型基础：介绍线性回归模型的基本形式，即因变量与一个或多个自变量之间的数学关系，以及如何用最小二乘法估计模型参数。 2. 模型假设：讨论回归分析中对数据的假设，如误差项的独立性、同方差性和正态性，以及残差分析的重要性，以检查这些假设是否成立。 3. 多元回归：扩展到两个或更多自变量的情况，探讨多元线性回归模型，以及多重共线性、异方差性和自相关等问题。 4. 假设检验与置信区间：讲解如何通过t检验和F检验评估回归系数的显著性，以及构建预测的置信区间。 5. 预测与解释：阐述如何使用回归模型进行预测，并解释模型结果的意义，包括系数的解释和影响分析。 6. 决定系数和调整决定系数：定义R²和调整R²，作为衡量模型拟合优度的指标，以及如何根据它们来评估模型的解释能力。 7. 偏相关分析和岭回归：介绍偏相关分析以理解自变量间的关联，以及岭回归作为处理多重共线性的方法。 8. 非线性回归：讨论如何处理非线性关系，包括通过转换变量、使用多项式回归和非参数方法。 9. 分类变量与逻辑回归：引入逻辑回归模型，用于处理二分类响应变量，并解释其背后的概率模型。 10. 时间序列回归：当数据具有时间序列特性时，如何应用回归分析，包括自回归模型（AR）、移动平均模型（MA）和自回归移动平均模型（ARMA）。 11. 考虑异常值和缺失数据：探讨如何处理数据集中可能存在的异常值和缺失数据，包括删除、插补和模型稳健性分析。 12. 回归模型的诊断和改进：学习如何识别并解决模型问题，如选择合适的模型复杂度、正则化方法以及模型选择策略，如逐步回归。本书通过实例研究和案例研究，帮助学生将理论应用于实际问题，强化理解和应用技能。此外，还提供了配套的教学资源，如习题和解决方案，以支持教学和学习过程。《A Second Course in Statistics: Regression Analysis》是学习和掌握回归分析理论与实践的宝贵教材，旨在帮助学生深化统计学理解，提升数据分析能力。

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Chapter

AReviewofBasic

Concepts (Optional)

Contents

1.1 Statistics and Data

1.2 Populations, Samples, and Random Sampling

1.3 Describing Qualitative Data

1.4 Describing Quantitative Data Graphically

1.5 Describing Quantitative Data Numerically

1.6 The Normal Probability Distribution

1.7 Sampling Distributions and the Central

Limit Theorem

1.8 Estimating a Population Mean

1.9 Testing a Hypothesis About a Population Mean

1.10 Inferences About the Difference Between

Two Population Means

1.11 Comparing Two Population Variances

Objectives

1. Review some basic concepts of sampling.

2. Review methods for describing both qualitative

and quantitative data.

3. Review inferential statistical methods: conﬁdence

intervals and hypothesis tests.

Although we assume students have had a prerequisite introductory course in

statistics, courses vary somewhat in content and in the manner in which they present

statistical concepts. To be certain that we are starting with a common background, we

use this chapter to review some basic deﬁnitions and concepts. Coverage is optional.

1.1 Statistics and Data

According to The Random House College Dictionary (2001 ed.), statistics is ‘‘the

science that deals with the collection, classiﬁcation, analysis, and interpretation of

numerical facts or data.’’ In short, statistics is the science of data—a science that

will enable you to be proﬁcient data producers and efﬁcient data users.

Deﬁnition 1.1

Statistics is the science of data. This involves collecting, classify-

ing, summarizing, organizing, analyzing, and interpreting data.

Data are obtained by measuring some characteristic or property of the

objects (usually people or things) of interest to us. These objects upon which

the measurements (or observations) are made are called experimental units,

and the properties being measured are called variables (since, in virtually all

studies of interest, the property varies from one observation to another).

Chapter 1 A Review of Basic Concepts (Optional)

Deﬁnition 1.2

An experimental unit is an object (person or thing) upon which

we collect data.

Deﬁnition 1.3

A variable is a characteristic (property) of the experimental unit

with outcomes (data) that vary from one observation to the next.

All data (and consequently, the variables we measure) are either quantitative

or qualitative in nature. Quantitative data are data that can be measured on a

naturally occurring numerical scale. In general, qualitative data take values that are

nonnumerical; they can only be classiﬁed into categories. The statistical tools that

we use to analyze data depend on whether the data are quantitative or qualitative.

Thus, it is important to be able to distinguish between the two types of data.

Deﬁnition 1.4 Quantitative data are observations measured on a naturally

occurring numerical scale.

Deﬁnition 1.5

Nonnumerical data that can only be classiﬁed into one of a

group of categories are said to be qualitative data.

Example

1.1

Chemical and manufacturing plants often discharge toxic waste materials such as

DDT into nearby rivers and streams. These toxins can adversely affect the plants and

animals inhabiting the river and the riverbank. The U.S. Army Corps of Engineers

conducted a study of ﬁsh in the Tennessee River (in Alabama) and its three tributary

creeks: Flint Creek, Limestone Creek, and Spring Creek. A total of 144 ﬁsh were

captured, and the following variables were measured for each:

1. River/creek where each ﬁsh was captured

2. Number of miles upstream where the ﬁsh was captured

3. Species (channel catﬁsh, largemouth bass, or smallmouth buffaloﬁsh)

4. Length (centimeters)

5. Weight (grams)

6. DDT concentration (parts per million)

The data are saved in the FISHDDT ﬁle. Data for 10 of the 144 captured ﬁsh are

shown in Table 1.1.

(a) Identify the experimental units.

(b) Classify each of the ﬁve variables measured as quantitative or qualitative.

Solution

(a) Because the measurements are made for each ﬁsh captured in the Tennessee

River and its tributaries, the experimental units are the 144 captured ﬁsh.

(b) The variables upstream that capture location, length, weight, and DDT con-

centration are quantitative because each is measured on a natural numerical

scale: upstream in miles from the mouth of the river, length in centimeters,

weight in grams, and DDT in parts per million. In contrast, river/creek and

species cannot be measured quantitatively; they can only be classiﬁed into

categories (e.g., channel catﬁsh, largemouth bass, and smallmouth buffaloﬁsh

for species). Consequently, data on river/creek and species are qualitative.

Statistics and Data

FISHDDT

Table 1.1

Data collected by U.S. Army Corps of Engineers (selected

observations)

River/Creek Upstream Species Length Weight DDT

FLINT 5 CHANNELCATFISH 42.5 732 10.00

FLINT 5 CHANNELCATFISH 44.0 795 16.00

SPRING 1 CHANNELCATFISH 44.5 1133 2.60

TENNESSEE 275 CHANNELCATFISH 48.0 986 8.40

TENNESSEE 275 CHANNELCATFISH 45.0 1023 15.00

TENNESSEE 280 SMALLMOUTHBUFF 49.0 1763 4.50

TENNESSEE 280 SMALLMOUTHBUFF 46.0 1459 4.20

TENNESSEE 285 LARGEMOUTHBASS 25.0 544 0.11

TENNESSEE 285 LARGEMOUTHBASS 23.0 393 0.22

TENNESSEE 285 LARGEMOUTHBASS 28.0 733 0.80

1.1 Exercises

1.1

College application data. Colleges and universi-

ties are requiring an increasing amount of informa-

tion about applicants before making acceptance

and ﬁnancial aid decisions. Classify each of the

following types of data required on a college appli-

cation as quantitative or qualitative.

(a) High school GPA

(b) Country of citizenship

(d) Gender of applicant

(e) Parents’ income

(f) Age of applicant

1.2

Fuel Economy Guide. The data in the accompa-

nying table were obtained from the Model Year

2009 Fuel Economy Guide for new automobiles.

(a) Identify the experimental units.

(b) State whether each of the variables measured

is quantitative or qualitative.

MODEL TRANSMISSION ENGINE SIZE NUMBER OF EST. CITY EST. HIGHWAY

NAME MFG TYPE (LITERS) CYLINDERS MILEAGE (MPG) MILEAGE (MPG)

TSX Acura Automatic 2.4 4 21 30

Jetta VW Automatic 2.0 4 29 40

528i BMW Manual 3.0 6 18 28

Fusion Ford Automatic 3.0 6 17 25

Camry Toyota Manual 2.4 4 21 31

Escalade Cadillac Automatic 6.2 8 12 19

Source: Model Year 2009 Fuel Economy Guide,U.S.Dept.ofEnergy,U.S.EnvironmentalProtection

Agency (www.fueleconomy.gov).

1.3

Ground motion of earthquakes. In the Journal of

Earthquake Engineering (November 2004), a team

of civil and environmental engineers studied the

ground motion characteristics of 15 earthquakes

that occurred around the world between 1940 and

1995. Three (of many) variables measured on each

earthquake were the type of ground motion (short,

long, or forward directive), earthquake magnitude

(Richter scale), and peak ground acceleration (feet

per second). One of the goals of the study was to

estimate the inelastic spectra of any ground motion

cycle.

(a) Identify the experimental units for this study.

(b) Identify the variables measured as quantita-

tive or qualitative.

1.4

Use of herbal medicines. The American Asso-

ciation of Nurse Anesthetists Journal (February

2000) published the results of a study on the use

of herbal medicines before surgery. Each of 500

Chapter 1 A Review of Basic Concepts (Optional)

surgical patients was asked whether they used

herbal or alternative medicines (e.g., garlic, ginkgo,

kava, ﬁsh oil) against their doctor’s advice before

surgery. Surprisingly, 51% answered ‘‘yes.’’

(a) Identify the experimental unit for the study.

(b) Identify the variable measured for each exper-

imental unit.

tive?

1.5 Drinking-water quality study. Disasters (Vol. 28,

2004) published a study of the effects of a trop-

ical cyclone on the quality of drinking water on

aremotePaciﬁcisland.Watersamples(size500

milliliters) were collected approximately 4 weeks

after Cyclone Ami hit the island. The following

variables were recorded for each water sample.

Identify each variable as quantitative or qualita-

tive.

(a) Town where sample was collected

(b) Type of water supply (river intake, stream, or

borehole)

(d) Turbidity level (nephalometric turbidity units

[NTUs])

(e) Temperature (degrees Centigrade)

(f) Number of fecal coliforms per 100 milliliters

(g) Free chlorine-residual (milligrams per liter)

(h) Presence of hydrogen sulphide (yes or no)

1.6

Accounting and Machiavellianism. Behavioral

Research in Accounting (January 2008) published

astudyofMachiavelliantraitsinaccountants.

Machiavellian describes negative character traits

that include manipulation, cunning, duplicity,

deception, and bad faith. A questionnaire was

administered to a random sample of 700 account-

ing alumni of a large southwestern university.

Several variables were measured, including age,

gender, level of education, income, job satisfac-

tion score, and Machiavellian (‘‘Mach’’) rating

score. What type of data (quantitative or qual-

itative) is produced by each of the variables

measured?

1.2 Populations, Samples, and Random Sampling

When you examine a data set in the course of your study, you will be doing so

because the data characterize a group of experimental units of interest to you. In

statistics, the data set that is collected for all experimental units of interest is called a

population.Thisdataset,whichistypicallylarge,eitherexistsinfactorispartofan

ongoing operation and hence is conceptual. Some examples of statistical populations

are given in Table 1.2.

Deﬁnition 1.6

A population data set is a collection (or set) of data measured

on all experimental units of interest to you.

Many populations are too large to measure (because of time and cost); others

cannot be measured because they are partly conceptual, such as the set of quality

Table 1.2 Some typical populations

Variable Experimental Units Population Data Set Type

a. Starting salary of a gradu-

ating Ph.D. biologist

All Ph.D. biologists

graduating this year

Set of starting salaries of all

Ph.D. biologists who graduated

this year

Existing

b. Breaking strength of water

pipe in Philadelphia

All water pipe sections

in Philadelphia

Set of breakage rates for all

water pipe sections in Phila-

delphia

Existing

c. Quality of an item

produced on an assem-

bly line

All manufactured

items

Set of quality measurements for

all items manufactured over the

recent past and in the future

Part existing,

part conceptual

d. Sanitation inspection level

of a cruise ship

All cruise ships Set of sanitation inspection lev-

els for all cruise ships

Existing

Populations, Samples, and Random Sampling

measurements (population c in Table 1.2). Thus, we are often required to select a

subset of values from a population and to make inferences about the population

based on information contained in a sample.Thisisoneofthemajorobjectivesof

modern statistics.

Deﬁnition 1.7

A sample is a subset of data selected from a population.

Deﬁnition 1.8

A statistical inference is an estimate, prediction, or some other

generalization about a population based on information contained in a sample.

Example

1.2

According to the research ﬁrm Magnum Global (2008), the average age of viewers

of the major networks’ television news programming is 50 years. Suppose a cable

network executive hypothesizes that the average age of cable TV news viewers is

less than 50. To test her hypothesis, she samples 500 cable TV news viewers and

determines the age of each.

(a) Describe the population.

(b) Describe the variable of interest.

(d) Describe the inference.

Solution

(a) The population is the set of units of interest to the cable executive, which is

the set of all cable TV news viewers.

(b) The age (in years) of each viewer is the variable of interest.

TV viewers selected by the executive.

(d) The inference of interest involves the generalization of the information con-

tained in the sample of 500 viewers to the population of all cable news viewers.

In particular, the executive wants to estimate the average age of the viewers in

order to determine whether it is less than 50 years. She might accomplish this

by calculating the average age in the sample and using the sample average to

estimate the population average.

Whenever we make an inference about a population using sample information,

we introduce an element of uncertainty into our inference. Consequently, it is

important to report the reliability of each inference we make. Typically, this

is accomplished by using a probability statement that gives us a high level of

conﬁdence that the inference is true. In Example 1.2, we could support the inference

about the average age of all cable TV news viewers by stating that the population

average falls within 2 years of the calculated sample average with ‘‘95% conﬁdence.’’

(Throughout the text, we demonstrate how to obtain this measure of reliability—and

its meaning—for each inference we make.)

Deﬁnition 1.9

A measure of reliability is a statement (usually quantiﬁed with

aprobabilityvalue)aboutthedegreeofuncertaintyassociatedwithastatistical

inference.

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