MAT 1272 STATISTICSLESSON 1

1.1 STATISTICS AND TYPES OF STATISTICS

WHAT IS STATISTICS?

STATISTICS

STATISTICS IS THE SCIENCE OF

COLLECTING, ANALYZING,

PRESENTING, AND INTERPRETING

DATA, AS WELL AS OF MAKING

DECISIONS BASED ON SUCH

ANALYSES.

READ THE GIVEN INFORMATION BELOW AND THINK OF ONE SIMILAR EXAMPLE OR SEARCH FOR ONE DATA IN THE INTERNET. STATE THE SOURCE AND

THE TIME POSTED.

•The following examples present some statistics:1.During March 2014, a total of 664,000,000 hours were spent by Americans watching March Madness live on TV and/or streaming (Fortune Magazine, March 15, 2015).•2.Approximately 30% of Google's employees were female in July 2014 (USA TODAY, July 24, 2014).•3.According to an estimate, an average family of four living in the United States needs $130,357 a year to to live the American dream (USA TODAY, July 7, 2014).•4.Chicago's O'Hare Airport was the busiest airport in 2014, with a total of 881,933 flight arrivals and departures.•5.In 2013, author James Patterson earned $90 million from the sale of his books (Forbes, September 29, 2014).•6.About 22.8% of U.S. adults do not have a religious affiliation (Time, May 25, 2015).•7.Yahoo CEO Marissa Mayer was the highest paid female CEO in America in 2014, with a total compensation of $42.1 million.

STATISTICAL METHODS HELP US MAKE SCIENTIFIC AND INTELLIGENT DECISIONS

• DECISIONS MADE BY USING STATISTICAL

METHODS ARE CALLED EDUCATED

GUESSES

• DECISIONS MADE WITHOUT USING

STATISTICAL (OR SCIENTIFIC) METHODS

ARE PURE GUESSES AND, HENCE, MAY

PROVE TO BE UNRELIABLE.

• FOR EXAMPLE, OPENING A LARGE STORE IN

AN AREA WITH OR WITHOUT ASSESSING

THE NEED FOR IT MAY AFFECT ITS SUCCESS.

• LIKE ALMOST ALL FIELDS OF STUDY,

STATISTICS HAS TWO ASPECTS:

• THEORETICAL OR MATHEMATICAL

STATISTICS DEALS WITH THE

DEVELOPMENT, DERIVATION, AND PROOF

OF STATISTICAL THEOREMS, FORMULAS,

RULES, AND LAWS.

• APPLIED STATISTICS INVOLVES THE

APPLICATIONS OF THOSE THEOREMS,

FORMULAS, RULES, AND LAWS TO SOLVE

REAL-WORLD PROBLEMS.

• THIS COURSE IS CONCERNED WITH

APPLIED STATISTICS AND NOT WITH

THEORETICAL STATISTICS. (EDUCATED

GUESS, DECISION MAKING DESIGNED FOR

SUCCESS).

The accompanying chart shows the lobbying spending by five selected companies during 2014. Many companies spend millions of dollars to win favors in Washington. According to Fortune Magazine of June 1, 2015, “Comcast has remained one of the biggest corporate lobbyists in the country.” In 2014, Comcast spent $17 million, Google spent $16.8 million, AT&T spent $14.2 million, Verizon spent $13.3 million, and Time Warner Cable spent $7.8 million on lobbying. These numbers simply describe the total amounts spent by these companies on lobbying. We are not drawing any inferences, decisions, or predictions from these data. Hence, this data set and its presentation is an example of descriptive statistics.

Case study 1-1

TYPES OF STATISTICS(DESCRIPTIVE & INFERENTIAL STATISTICS)

DESCRIPTIVE STATISTICS CASE STUDY 1-1

DESCRIPTIVE STATISTICS CHARACTERISTICS

• DESCRIPTIVE STATISTICS CONSISTS OF METHODS FOR ORGANIZING, DISPLAYING, AND DESCRIBING DATA BY USING TABLES, GRAPHS, AND SUMMARY MEASURES.

• CHAPTERS 2 AND 3 DISCUSS DESCRIPTIVE STATISTICAL METHODS. IN CHAPTER 2, WE LEARN HOW TO CONSTRUCT TABLES AND HOW TO GRAPH DATA. IN CHAPTER 3, WE LEARN HOW TO CALCULATE NUMERICAL SUMMARY MEASURES, SUCH AS AVERAGES.

• CASE STUDY 1-1 PRESENTS AN EXAMPLE OF DESCRIPTIVE STATISTICS.

A poll of 176,903 American adults, aged 18 and older, was conducted January 2 to December 30, 2014, as part of the Gallup-Healthways Well-Being Index survey. Gallup and Healthways have been “tracking Americans' life evaluations daily” since 2008. According to this poll, in 2014, Americans' outlook on life was the best in seven years, as 54.1% “rated their lives highly enough to be considered thriving,” 42.1% said they were struggling, and 3.8% mentioned that they were suffering. As mentioned in the chart, the margin of sampling error was ± 1%. In Chapter 8, we will discuss the concept of margin of error, which can be combined with these percentages when making inferences. As we notice, the results described in the chart are obtained from a poll of 176,903 adults. We will learn in later chapters how to apply these results to the entire population of adults. Such decision making about the population based on sample results is called inferential statistics.

Case study 2

INFERENTIAL STATISTICS

INFERENTIAL STATISTICS

CASE STUDY 1-2

INFERENTIAL STATISTICS

CHARACTERISTICS

• INFERENTIAL STATISTICS CONSISTS OF METHODS THAT USE SAMPLE RESULTS TO HELP MAKE DECISIONS OR PREDICTIONS ABOUT A POPULATION.

• CASE STUDY 1-2 PRESENTS AN EXAMPLE OF INFERENTIAL STATISTICS. IT SHOWS THE RESULTS OF A SURVEY IN WHICH AMERICAN ADULTS WERE ASKED ABOUT THEIR OPINIONS ABOUT THEIR LIVES.

• CHAPTERS 8 THROUGH 15 AND PARTS OF CHAPTER 7 DEAL WITH INFERENTIAL STATISTICS.

PROBABILITY, WHICH GIVES A MEASUREMENT OF THE LIKELIHOOD THAT A CERTAIN OUTCOME WILL OCCUR, ACTS AS A LINK BETWEEN DESCRIPTIVE

AND INFERENTIAL STATISTICS. PROBABILITY IS USED TO MAKE STATEMENTS ABOUT THE OCCURRENCE

OR NONOCCURRENCE OF AN EVENT UNDER UNCERTAIN CONDITIONS. PROBABILITY AND

PROBABILITY DISTRIBUTIONS ARE DISCUSSED IN CHAPTERS 4 THROUGH 6 AND PARTS OF CHAPTER 7.

1.2 BASIC TERMSTHIS SECTION EXPLAINS THE MEANING OF AN

ELEMENT (OR MEMBER), A VARIABLE, AN OBSERVATION, AND A DATA SET.

WORLD'S EIGHT RICHEST PERSONS AS OF MARCH 2015 DESCRIPTION

• EACH PERSON LISTED IN THIS TABLE IS CALLED AN ELEMENT OR A MEMBER OF THIS GROUP (8 ELEMENTS)

• NOTE THAT ELEMENTS ARE ALSO CALLED OBSERVATIONAL UNITS.

• AN ELEMENT OR MEMBER OF A SAMPLE OR POPULATION IS A SPECIFIC SUBJECT OR OBJECT (FOR EXAMPLE, A PERSON, FIRM, ITEM, STATE, OR COUNTRY) ABOUT WHICH THE INFORMATION IS COLLECTED.

• A VARIABLE IS A CHARACTERISTIC UNDER STUDY THAT ASSUMES DIFFERENT VALUES FOR DIFFERENT ELEMENTS. IN CONTRAST TO A VARIABLE, THE VALUE OF A CONSTANT IS FIXED.

• THE VALUE OF A VARIABLE FOR AN ELEMENT IS CALLED AN OBSERVATION OR MEASUREMENT.

• A DATA SET IS A COLLECTION OF OBSERVATIONS ON ONE OR MORE VARIABLES.

APPLICATIONS

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QUANTITATIVE VARIABLES

A VARIABLE THAT CAN BE

MEASURED NUMERICALLY IS

CALLED A QUANTITATIVE

VARIABLE. THE DATA

COLLECTED ON A

QUANTITATIVE VARIABLE

ARE CALLED QUANTITATIVE

DATA.

DISCRETE VARIABLES

DISCRETE VARIABLE

A VARIABLE WHOSE

VALUES ARE COUNTABLE

IS CALLED A DISCRETE

VARIABLE. IN OTHER

WORDS, A DISCRETE

VARIABLE CAN ASSUME

ONLY CERTAIN VALUES

WITH NO INTERMEDIATE

VALUES.

CONTINUOUS VARIABLES

A VARIABLE THAT CAN

ASSUME ANY NUMERICAL

VALUE OVER A CERTAIN

INTERVAL OR INTERVALS

IS CALLED

A CONTINUOUS

VARIABLE.

1.3 Types of Variables 1.3.1 Quantitative Variables 1.3.2 Qualitative or Categorical Variables

1.3.2 QUALITATIVE OR CATEGORICAL VARIABLES

A variable that cannot assume a numerical value but can be classified into two or more nonnumeric categories is called a qualitative or categorical variable. The data collected on such a variable are called qualitative data.

For example, the status of an undergraduate college student is a qualitative variable because a student can fall into any one of four categories: freshman, sophomore, junior, or senior. Other examples of qualitative variables are the gender of a person, the make of a computer, the opinions of people, and the make of a car. (How would you like this power-point presentation?).

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1.10 A survey of families living in a certain city was conducted to collect information on the following variables: age of the oldest person in the family, number of family members, number of males in the family, number of females in the family, whether or not they own a house, income of the family, whether or not the family took vacations during the past one year, whether or not they are happy with their financial situation, and the amount of their monthly mortgage or rent.

1.10 A survey of families living in a certain city was conducted to collect information on the following variables: age of the oldest person in the family, number of family members, number of males in the family, number of females in the family, whether or not they own a house, income of the family, whether or not the family took vacations during the past one year, whether or not they are happy with their financial situation, and the amount of their monthly mortgage or rent.(a) Which of these variables are qualitative variables?(b) Which of these variables are quantitative variables?(c) Which of the quantitative variables of part b are discrete variables?(d) Which of the quantitative variables of part b are continuous variables?

1.5 POPULATION VERSUS SAMPLE

•Suppose a statistician is interested in knowing the following:•1.The percentage of all voters in a city who will vote for a particular candidate in an election•2.Last year's gross sales of all companies in New York City•3.The prices of all homes in California

We will encounter the terms population and sample on almost every page of the text. Consequently, understanding the meaning of each of these two terms and the difference between them is crucial

Population or Target PopulationA population consists of all elements—individuals, items, or objects—whose characteristics are being studied. The population that is being studied is also called the target population.

POPULATION• A POPULATION CONSISTS OF ALL

ELEMENTS—INDIVIDUALS, ITEMS, OR

OBJECTS—WHOSE CHARACTERISTICS

ARE BEING STUDIED. THE POPULATION

THAT IS BEING STUDIED IS ALSO

CALLED THE TARGET POPULATION.

SAMPLE• A PORTION OF THE POPULATION

SELECTED FOR STUDY IS REFERRED TO

AS A SAMPLE

POPULATION• FOR EXAMPLE, THE ELECTION POLLS

CONDUCTED IN THE UNITED STATES TO

ESTIMATE THE PERCENTAGES OF VOTERS

WHO FAVOR VARIOUS CANDIDATES IN ANY

PRESIDENTIAL ELECTION ARE BASED ON

ONLY A FEW HUNDRED OR A FEW THOUSAND

VOTERS SELECTED FROM ACROSS THE

COUNTRY.

• IN THIS CASE, THE POPULATION CONSISTS OF

ALL REGISTERED VOTERS IN THE UNITED

STATES

SAMPLE• THE SAMPLE IS MADE UP OF A FEW

HUNDRED OR FEW THOUSAND VOTERS

WHO ARE INCLUDED IN AN OPINION

POLL. THUS, THE COLLECTION OF A

NUMBER OF ELEMENTS SELECTED

FROM A POPULATION IS CALLED

A SAMPLE.

CENSUS AND SAMPLE SURVEY

A survey that includes every member of the population is called a census. A survey that includes only a portion of the population is called a sample survey.The purpose of conducting a sample survey is to make decisions about the corresponding population. It is important that the results obtained from a sample survey closely match the results that we would obtain by conducting a census.

CENSUS AND SAMPLE SURVEY

When we collect information on all elements of the target population, it is called a census. Often the target population is very large. Hence, in practice, a census is rarely taken because it is expensive and time-consuming.

Usually, to conduct a survey, we select a sample and collect the required information from the elements included in that sample. We then make decisions based on this sample information

REPRESENTATIVE SAMPLEA SAMPLE THAT REPRESENTS THE CHARACTERISTICS

OF THE POPULATION AS CLOSELY AS POSSIBLE IS CALLED A REPRESENTATIVE SAMPLE.

As an example, to find the average income of families living in New York City by conducting a sample survey, the sample must contain families who belong to different income groups in almost the same proportion as they exist in the population. Such a sample is called a representative sample.

REPRESENTATIVE SAMPLEA SAMPLE THAT REPRESENTS THE CHARACTERISTICS OF THE

POPULATION AS CLOSELY AS POSSIBLE IS CALLED A REPRESENTATIVE SAMPLE.

A sample may be selected with or without replacement. In sampling with replacement, each time we select an element from the population, we put it back in the population before we select the next element.As a result, we may select the same item more than once in such a sample

Sampling without replacement occurs when the selected element is not replaced in the population.In this case, each time we select an item, the size of the population is reduced by one element. Thus, we cannot select the same item more than once in this type of sampling. voter is not selected more than once. Therefore, this is an

example of sampling without replacement.

THREE OF THE MAIN REASONS FOR CONDUCTING A SAMPLE SURVEY INSTEAD OF

A CENSUS

TIME

IN MOST CASES, THE

SIZE OF THE

POPULATION IS QUITE

LARGE.

CONSEQUENTLY,

CONDUCTING A

CENSUS TAKES A LONG

TIME, WHEREAS A

SAMPLE SURVEY CAN

BE CONDUCTED VERY

QUICKLY

COST

THE COST OF

COLLECTING

INFORMATION FROM

ALL MEMBERS OF A

POPULATION MAY

EASILY FALL OUTSIDE

THE LIMITED BUDGET OF

MOST, IF NOT ALL,

SURVEYS, CONDUCTING

A SAMPLE SURVEY MAY

BE THE BEST

APPROACH

IMPOSSIBILITY OF CONDUCTING A CENSUS

SOMETIMES IT IS IMPOSSIBLE TO CONDUCT

A CENSUS

FIRST, IT MAY NOT BE POSSIBLE TO IDENTIFY

AND ACCESS EACH MEMBER OF THE

POPULATION (EX. SURVEY ABOUT

HOMELESS PEOPLE).

SECOND, SOMETIMES CONDUCTING A

SURVEY MEANS DESTROYING THE ITEMS

INCLUDED IN THE SURVEY(EX. TO ESTIMATE

THE MEAN LIFE OF LIGHTBULBS WOULD

NECESSITATE BURNING OUT MANY OF

THEM).

1.5.2 RANDOM AND NONRANDOM SAMPLES

A random sample is a sample drawn in such a way that each member of the population has some chance of being selected in the sample. In a nonrandom sample, some members of the population may not have any chance of being selected in the sample.

RANDOM AND NONRANDOM SAMPLES• SUPPOSE WE HAVE A LIST OF

100 STUDENTS AND WE WANT

TO SELECT 10 OF THEM. IF WE

WRITE THE NAMES OF ALL 100

STUDENTS ON PIECES OF

PAPER, PUT THEM IN A HAT,

MIX THEM, AND THEN DRAW

10 NAMES, THE RESULT WILL

BE A RANDOM SAMPLE OF 10

STUDENTS.

• A RANDOM SAMPLE IS

USUALLY A REPRESENTATIVE

SAMPLE

• HOWEVER, IF WE ARRANGE THE NAMES OF THESE 100 STUDENTS ALPHABETICALLY AND PICK THE FIRST 10 NAMES, IT WILL BE A NONRANDOM SAMPLE BECAUSE THE STUDENTS WHO ARE NOT AMONG THE FIRST 10 HAVE NO CHANCE OF BEING SELECTED IN THE SAMPLE.

• TWO TYPES OF NONRANDOM SAMPLES ARE A CONVENIENCE SAMPLE AND A JUDGMENT SAMPLE

• IN A CONVENIENCE SAMPLE, THE MOST ACCESSIBLE MEMBERS OF THE POPULATION ARE SELECTED TO OBTAIN THE RESULTS QUICKLY. FOR EXAMPLE, AN OPINION POLL MAY BE CONDUCTED IN A FEW HOURS BY COLLECTING INFORMATION FROM CERTAIN SHOPPERS AT A SINGLE SHOPPING MALL.

• IN A JUDGMENT SAMPLE, THE MEMBERS ARE SELECTED FROM THE POPULATION BASED ON THE JUDGMENT AND PRIOR KNOWLEDGE OF AN EXPERT

THE SO-CALLED PSEUDO POLLS ARE EXAMPLES OF NONREPRESENTATIVE SAMPLES

EX. a poll conducted by a television station giving two separate telephone numbers for yes and no votes is not based on a representative sample.

To select such a sample, we divide the target population into different subpopulations based on certain characteristics.As an example of a quota sample, suppose we want to select a sample of 1000 persons from a city whose population has 48% men and 52% women. To select a quota sample, we choose 480 men from the male population and 520 women from the female population. The sample selected in this way will contain exactly 48% men and 52% women

• SAMPLING ERROR

THE SAMPLING ERROR IS THE

DIFFERENCE BETWEEN THE RESULT

OBTAINED FROM A SAMPLE SURVEY

AND THE RESULT THAT WOULD HAVE

BEEN OBTAINED IF THE WHOLE

POPULATION HAD BEEN INCLUDED IN

THE SURVEY.

• NONSAMPLING ERRORS

• THE ERRORS THAT OCCUR IN THE

COLLECTION, RECORDING, AND

TABULATION OF DATA ARE

CALLED NONSAMPLING

ERRORS OR BIASES.

1.5.3 Sampling and Nonsampling Errors

SELECTION ERROR OR BIASTHE LIST OF MEMBERS OF THE TARGET POPULATION THAT IS USED TO SELECT A SAMPLE IS CALLED THE SAMPLING FRAME. THE ERROR THAT OCCURS BECAUSE THE SAMPLING FRAME IS NOT REPRESENTATIVE OF

THE POPULATION IS CALLED THE SELECTION ERROR OR BIAS.Nonresponse Error or BiasThe error that occurs because many of the people included in the sample do not respond to a survey is called the nonresponse error or bias.Response Error or BiasThe response error or bias occurs when people included in the survey do not provide correct answers.Voluntary response error or bias occurs when a survey is not conducted on a randomly selected sample but on a questionnaire published in a magazine or newspaper and people are invited to respond to that questionnaire.

Stratified Random SampleIn a stratified random sample, we first divide the population into subpopulations, which are called strata. Then, one sample is selected from each of these strata. The collection of all samples from all strata gives the stratified random sample.

Suppose we need to select a sample from the population of a city, and we want households with different income levels to be proportionately represented in the sample.For example, we may form three groups of low-, medium-, and high-income households. We will now have three subpopulations, which are usually called strata. We then select one sample from each subpopulation or stratum. The collection of all three samples selected from the three strata gives the required sample, called the stratified random sample.

Cluster SamplingIn cluster sampling, the whole population is first divided into (geographical) groups called clusters. Each cluster is representative of the population. Then a random sample of clusters is selected. Finally, a random sample of elements from each of the selected clusters is selected.divide the population into different geographical groups or clusters and, as a first step, select a random sample of certain clusters from all clusters. We then take a random sample of certain elements from each selected cluster

For example, suppose we are to conduct a survey of households in the state of New York. First, we divide the whole state of New York into, say, 40 regions, which are called clusters or primary units.We make sure that all clusters are similar and, hence, representative of the population.We then select at random, say, 5 clusters from 40. Next, we randomly select certain households from each of these 5 clusters and conduct a survey of these selected households. This is called cluster sampling.

1.13 BRIEFLY EXPLAIN THE TERMS POPULATION, SAMPLE, REPRESENTATIVE SAMPLE, SAMPLING WITH REPLACEMENT, AND SAMPLING WITHOUT

REPLACEMENT.

1.13 BRIEFLY EXPLAIN THE TERMS POPULATION, SAMPLE, REPRESENTATIVE SAMPLE, SAMPLING WITH REPLACEMENT, AND SAMPLING WITHOUT

REPLACEMENT.•A census is a survey that includes every member of the population. A survey based on a portion of the population is called a sample survey. A sample survey is preferred over a census for the following reasons:1.Conducting a census is very expensive because the size of the population is often very large.•2.Conducting a census is very time consuming.•3.In many cases it is impossible to identify each element of the target population.

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1.16 Explain the following.(a) Random sample(b) Nonrandom sample(c) Convenience sample(d) Judgment sample(e) Quota sample

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1.16 Explain the following.(a) Random sample(b) Nonrandom sample(c) Convenience sample(d) Judgment sample(e) Quota sample

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1.17 Explain the following four sampling techniques.(a) Simple random sampling(b) Systematic random sampling(c) Stratified random sampling(d) Cluster sampling

31.17 Explain the following four sampling techniques.(a) Simple random sampling(b) Systematic random sampling (c) Stratified random sampling (d) Cluster sampling a) Simple random sampling (b) Systematic random sampling (c) In a stratified random sample, we first divide the population into subpopulations which are called strata. Then, one sample is selected from each of these strata. The collection of all samples from all strata gives the stratified random sample.In cluster sampling, the whole population is divided into (geographical) groups called clusters. Each cluster is representative of the population. Then, a random sample of clusters is selected. Finally, a random sample of elements of each of the selected clusters is selected.

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1.5.1 WHY SAMPLE?