Chapter 1

Chapter 1

Introduction to Statistics

1Larson/Farber 4th ed.

Chapter Outline

• 1.1 What is Statistics?

• 1.2 Random Samples

• 1.3 Experimental Design

Larson/Farber 4th ed. 2

Section 1.1

What is Statistics?


Section 1.1 Objectives

• Define statistics• Define individual/observational unit• Distinguish between a population and a sample• Distinguish between a parameter and a statistic• Distinguish between descriptive statistics and

inferential statistics• Distinguish between levels of measurement


What is Data?


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What is Data?

Data

Consist of information coming from observations, counts, measurements, or responses. Statements are often based on data:

• “People who eat three daily servings of whole grains have been shown to reduce their risk of…stroke by 37%.” (Source: Whole Grains Council)

• “Seventy percent of the 1500 U.S. spinal cord injuries to minors result from vehicle accidents, and 68 percent were not wearing a seatbelt.” (Source: UPI)


What is Statistics?


What is Statistics?

Statistics

The science of collecting, organizing, analyzing, and interpreting data in order to make decisions.


Individuals and Variables

Individual / Observational Unit

Variables

Weight: 1.5 poundsSugars: 16 grams

Age: 6 monthsWeight: 16.53 pounds

Individuals and Variables

• Individuals are people or objects included in the study.

• Variables are characteristics of the individual to be measured or observed.

• Exercise 1: We want to do a study about the people who have climbed Mt. Everest. Identify the individuals and the variables.

Types of Data

Qualitative Data

Consists of attributes, labels, or nonnumerical entries.

Major Place of birth

Eye color


Types of Data

Quantitative data

Numerical measurements or counts.

Guided Ex 1: Is the data quantitative or qualitative?

Age Weight of a letter Temperature


Data Sets

Population The collection of all outcomes, responses, measurements, or counts that are of interest.

Sample A subset of the population.


Exercise 2: Identifying Data Sets

In a recent survey, 1708 adults in the United States were asked if they think global warming is a problem that requires immediate government action. Nine hundred thirty-nine of the adults said yes.

1. Identify the population and the sample.

2. Describe the data set. (Adapted from: Pew Research Center)


Solution: Identifying Data Sets

• The population consists of the responses of all adults in the U.S.

• The sample consists of the responses of the 1708 adults in the U.S. in the survey.

• The sample is a subset of the responses of all adults in the U.S.

• The data set consists of 939 yes’s and 769 no’s.

Responses of adults in the U.S. (population)

Responses of adults in survey (sample)


Parameter and Statistic

Parameter

A number that describes a population characteristic.

Average age of all people in the United States

Statistic A number that describes a sample

characteristic.Average age of people from a sample of three states


Example: Distinguish Parameter and Statistic

Decide whether the numerical value describes a population parameter or a sample statistic.

1. A recent survey of a sample of MBAs reported that the average salary for an MBA is more than $82,000. (Source: The Wall Street Journal)

Solution:Sample statistic (the average of $82,000 is based on a subset of the population)


Guided Exercise 2

Decide whether the numerical value describes a population parameter or a sample statistic.

2. Starting salaries for the 667 MBA graduates from the University of Chicago Graduate School of Business increased 8.5% from the previous year.

Solution:Population parameter (the percent increase of 8.5% is based on all 667 graduates’ starting salaries)


Branches of Statistics

Descriptive Statistics Involves organizing, summarizing, and displaying data.

e.g. Tables, charts, averages

Inferential Statistics Involves using sample data to draw conclusions about a population.


Example: Descriptive and Inferential Statistics

Decide which part of the study represents the descriptive branch of statistics. What conclusions might be drawn from the study using inferential statistics?

A large sample of men, aged 48, was studied for 18 years. For unmarried men, approximately 70% were alive at age 65. For married men, 90% were alive at age 65. (Source: The Journal of Family Issues)


Solution: Descriptive and Inferential Statistics

Descriptive statistics involves statements such as “For unmarried men, approximately 70% were alive at age 65” and “For married men, 90% were alive at 65.”

A possible inference drawn from the study is that being married is associated with a longer life for men.


Example: Classifying Data by Type

The base prices of several vehicles are shown in the table. Which data are qualitative data and which are quantitative data? (Source Ford Motor Company)


Solution: Classifying Data by Type

Quantitative Data (Base prices of vehicles models are numerical entries)

Qualitative Data (Names of vehicle models are nonnumerical entries)


Levels of Measurement

Nominal level of measurement• Qualitative data only• Categorized using names, labels, or qualities• No mathematical computations can be made• Example:


Which of the following food items do you tend to buy at least once per month? (Please tick)

Okra Palm Oil Milled Rice

Peppers Prawns Almond Milk


Ordinal level of measurement • Qualitative or quantitative data • Data can be arranged in order• Differences between data entries is not meaningful• Example:


Order of preference

Brand of Pesticide

1 Rambo2 R.I.P.3 BugsBeGone4 D.O.A.5 BestBugSpray

Example: Classifying Data by Level

Two data sets are shown. Which data set consists of data at the nominal level? Which data set consists of data at the ordinal level? (Source: Nielsen Media Research)


Solution: Classifying Data by Level

Ordinal level (lists the rank of five TV programs. Data can be ordered. Difference between ranks is not meaningful.)

Nominal level (lists the call letters of each network affiliate. Call letters are names of network affiliates.)



Interval level of measurement• Quantitative data• Data can be ordered• Differences between data entries is meaningful• Zero represents a position on a scale (not an inherent

zero – zero does not imply “none”)

Example: Time of day on a 12-hour clock

Example: Body temperatures in degrees Celsius



Ratio level of measurement• Zero entry is an inherent zero (implies “none”)• A ratio of two data values can be formed • One data value can be expressed as a multiple of

another• Examples: RULER: inches or centimeters ,YEARS

of work experience, INCOME: money earned last year , NUMBER of children, GPA: grade point average, TEMPERATURE: degrees Kelvin

• Person who earns $2K/ week earns twice as much as person who earns $1K / week




Measurement at the interval or ratio level is desirable because we can use the more powerful statistical procedures available for Means and Standard Deviations.

To have this advantage, often ordinal data are treated as though they were interval; for example, subjective ratings scales (1 = terrible, 2= poor, 3 = fair, 4 = good, 5 = excellent).

Example: Classifying Data by Level

Two data sets are shown. Which data set consists of data at the interval level? Which data set consists of data at the ratio level? (Source: Major League Baseball)


Solution: Classifying Data by Level

Interval level (Quantitative data. Can find a difference between two dates, but a ratio does not make sense.)

Ratio level (Can find differences and write ratios.)


Guided Exercise 3


:

State the level of measurement for each of the following:

Measurement LevelThe senator’s name is Sam Wilson. The senator is 58 years old. The senator was elected in 1963, 1969, 1981, and 1994.

His taxable income is $278,314.19 The senator is married. The senator had divorces in 1965 and 1982.

A newspaper ranked the senator 7th for his voting record on public education

Exercise 4: Summary of Four Levels of Measurement

Level ofMeasurement

Put data in

categories

Arrangedata inorder

Subtractdata

values

Determine if one data value is a

multiple of another

Nominal

Ordinal

Interval

Ratio


Summary of Four Levels of Measurement

Level ofMeasurement

Put data in

categories

Arrangedata inorder

Subtractdata

values

Determine if one data value is a

multiple of another

Nominal Yes No No No

Ordinal Yes Yes No No

Interval Yes Yes Yes No

Ratio Yes Yes Yes Yes


Section 1.2

Random Samples



• Explain the importance of random samples• Construct a simple random sample using random

numbers• Simulate a random process• Describe stratified sampling, cluster sampling,

systematic sampling, multi-stage and convenience sampling


Sampling TechniquesSimple Random Sample• Every possible sample of the same size has the same chance of being selected.• Every individual of the population has an equal chance of being selected.

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Simple Random Sample

• Random numbers can be generated by a random number table, a software program or a calculator.

• Assign a number to each member of the population.

• Members of the population that correspond to these numbers become members of the sample.


Guided Ex 1: Simple Random Sample

There are 731 students currently enrolled in statistics at your school. You wish to form a sample of eight students to answer some survey questions. Select the students who will belong to the simple random sample.

• Assign numbers 1 to 731 to each student taking statistics.

• On the table of random numbers, choose a starting place at random (suppose you start in the third row, second column.)


Solution: Simple Random Sample

• Read the digits in groups of three• Ignore numbers greater than 731

The students assigned numbers 719, 662, 650, 4, 53, 589, 403, and 129 would make up the sample.


What is a Simulation?


A simulation is a numerical facsimile or representation of a real-world phenomenon

Exercise 1: Simple Random Samples


Use a random number table to simulate each of the following:a. Choose the numbers for the next lottery. That is,

randomly choose six numbers from 1 to 52b. The outcomes of tossing a die 20 times.

Exercise 2: TI-83 Random Number Generation


Exercise 3: Coin Toss Simulation using TI-83

Other Sampling Techniques

Stratified Sample• Divide a population into groups (strata) and select a

random sample from each group.

• To collect a stratified sample of the number of people who live in West Ridge County households, you could divide the households into socioeconomic levels and then randomly select households from each level.



Stratified Sample

• We stratify to ensure that our sample represents different groups in the population

• Result: reduced sample variability• Samples taken within a stratum vary less, so our

estimates can be more precise• Example: If we stratify by sex, we can create the

sample so that the proportions of men and women within our sample match the proportions in the population



Cluster Sample• Divide the population into groups (clusters) and

select all of the members in one or more, but not all, of the clusters.

• In the West Ridge County example you could divide the households into clusters according to zip codes, then select all the households in one or more, but not all, zip codes.



Cluster Sample• If each cluster represents the population fairly,

cluster sampling will be unbiased.• Clusters are internally heterogeneous, each

resembling the overall population.• We select clusters to make sampling more

practical or affordable.


Non-Random Sampling Technique

Convenience Sample• One of the main types of non-probability

sampling methods• Made up of people who are easy to reach • Usually not representative of population• Example: A pollster interviews shoppers at a local

mall. If the mall was chosen because it was a convenient site from which to solicit survey participants, this would be a convenience sample.



Systematic Sample• Choose a starting value at random. Then choose

every kth member of the population.

• In the West Ridge County example you could assign a different number to each household, randomly choose a starting number, then select every 100th household.


Exercise 4: Identifying Sampling Techniques

You are doing a study to determine the opinion of students at your school regarding stem cell research. Identify the sampling technique used:

You divide the student population with respect to majors and randomly select and question some students in each major.

Solution:Stratified sampling (the students are divided into strata (majors) and a sample is selected from each major)


Sampling Terminology

Sampling Frame: • a list of individuals from which a sample is actually

selected• ideally, should match the population

Example: When doing a phone survey, the sampling frame might be the phone book


Sampling Undercoverage

Undercoverage: the condition resulting from omitting population members from the sample frame

Example: The phone book might not be representative of all residents of a community


Population

Sampling Frame

Sampling Terminology

Sampling Error is the mismatch between • measurements taken from samples• corresponding measurements taken from the

respective population• sampling errors do not represent mistakes!

Nonsampling Error results from- poor sample design- sloppy data collection techniques- bias in questions- nonsampling errors are inadvertent errors


Multi-Stage Sampling


Multi-stage sampling involves selecting a sample in at least two stages

• Successively smaller groups are created at each stage

• Most surveys conducted by professional polling organizations use some combination of stratified and cluster sampling as well as simple random samples

Example: Multi-Stage Sampling

• Suppose that college freshmen are housed in separate freshman dorms

• Within a freshman dorm, men and women are housed on alternating floors

• You wish to sample their attitudes about the campus food by going to dorms at random, but you are still concerned about possible differences between men and women

• Can you design a suitable sampling plan?


Possible Solution: Multi-Stage Sampling

• Plan: Stratify sample dorms by sex• Stage 1: Select some freshman dorms at random • Stage 2: Within each dorm, select floors (e.g.

sexes) at random • Treat each floor as a cluster and interview

everyone on that floor


Exercise 4: Sampling Strategy

In order to try to gauge freshman opinion about the food served on campus, Food Services suggests that you just stand outside the school cafeteria at lunchtime and stop people to ask them questions.

Critical Thinking:

1. What’s wrong with this sampling strategy?

2. Suggest some possible ways to improve the quality of the data.


Section 1.2 Summary

• Describe simple random samples• Construct a simple random sample using random

numbers• Simulate a random process• Describe stratified sampling, cluster sampling,

systematic sampling, multi-stage and convenience sampling


Section 1.3

Experimental Design



• Discuss what it means to take a census• Describe simulations, observational studies and

experiments• Identify control groups, placebo effects, completely

randomized experiments• Discuss potential pitfalls that might your data

unreliable


Designing a Statistical Study


In a statistical study, the researcher will• Take measurements, or survey the population of

interest• Observe or manipulate the samples in some manner

Exercise 1: Designing a Statistical Study


A group of students is interested in knowing if the number of times they can sink a basketball is related to the color of the basketball. The students shoot a series of baskets and record their success using a regulation colored basketball. They then switch to a blue colored basketball and shoot the same series of baskets. A statistical analysis is performed.

What steps do you think might be needed in order to carry out such a statistical study?

Designing a Statistical Study

3. Collect the data.

4. Describe the data using descriptive statistics techniques.

5. Interpret the data and make decisions about the population using inferential statistics.

6. Identify any possible errors.

1. Identify the variable(s) of interest (the focus) and the population of the study.

2. Develop a detailed plan for collecting data. If you use a sample, make sure the sample is representative of the population.


Ways to Produce Data

• Census- measurements/ observations of entire population

• Sampling- Measurements/observations from representative

part of population• Simulation• Experiment

- impose treatment, then measure variable of interest• Observational Study• Survey


Data Collection

Observational study • A researcher observes and measures characteristics of

interest of part of a population.• Example:

Researchers observed and recorded the mouthing behavior on nonfood objects of children up to three years old. (Source: Pediatric Magazine)


Data Collection

Experiment• A treatment is applied to part of a population and

responses are observed.• Example:

An experiment was performed in which diabetics took cinnamon extract daily while a control group took none. After 40 days, the diabetics who had the cinnamon reduced their risk of heart disease while the control group experienced no change. (Source: Diabetes Care)


Data Collection

Simulation • Uses a mathematical or physical model to reproduce

the conditions of a situation or process.• Computers are often used• Examples:

Automobile manufacturers use simulations with dummies to study the effects of crashes on humans.

Airlines use simulators to train pilots on different models of aircraft


Example: Methods of Data Collection

Consider the following statistical studies. Which method of data collection would you use to collect data for each study?

1. A study of the effect of changing flight patterns on the number of airplane accidents.

Solution:Simulation (It is impractical to create this situation)



2. A study of the effect of eating oatmeal on lowering blood pressure.

Solution:Experiment (Measure the effect of a treatment – eating oatmeal)



Solution:Observational study (observe and measure certain characteristics of part of a population)

3. A study of how fourth grade students solve a puzzle.



Solution:Survey (Ask “Do you approve of the way the president is handling his job?”)

4. A study of U.S. residents’ approval rating of the U.S. president.


Data Collection

Survey• An investigation of one or more characteristics of a

population.• Commonly done by interview, mail, or telephone.• Example: A survey is conducted on a sample of

female physicians to determine whether the primary reason for their career choice is financial stability.


Surveys: Potential Pitfalls

• Non-response• Truthfulness of Response• Hidden Bias• Vague Wording

e.g. “often”, “seldom”, “occasionally”•Interviewer Influence

Tone of voice, body language, dress• Voluntary / Non-voluntary response


Exercise 2: Elements of Experimental Design

Identify some activities that must be carried out in order to conduct an experiment:

• Randomly assign subjects to treatments• Manipulate treatment factors (e.g. amount of

medication dispensed)• Compare responses of subject groups across different

treatments


Key Elements of Experimental Design: Control

• Control Group Used to account for the influence of other known or

unknown variables that might be the cause of an underlying response in an experimental group

Control groups frequently receive a dummy treatment


Key Elements of Experimental Design:

Confounding variables Occurs when an experimenter cannot tell the

difference between the effects of different factors on a variable.

Example: A coffee shop owner remodels her shop at the same time a nearby mall has its grand opening. If business at the coffee shop increases, it cannot be determined whether it is because of the remodeling or the new mall.


Experimental Design: Lurking Variables

MEASURED VARIABLES EXCLUDED

Gasoline (gallons) Commute Time (minutes) Traffic Congestion [ LURKING ]2 32 low3 45 moderate4 55 high


Lurking Variable

• Variable for which no data have been collected• Variable which has impact on other variables in a

study

Key Elements of Experimental Design: Control

• Placebo effect A subject reacts favorably to a placebo when in

fact he or she has been given no medical treatment at all.

Blinding is a technique where the subject does not know whether he or she is receiving a treatment or a placebo.

Double-blind experiment neither the subject nor the experimenter knows if the subject is receiving a treatment or a placebo.


Key Elements of Experimental Design: Randomization

• Randomization is a process of randomly assigning subjects to different treatment groups.

• Completely randomized design All subjects are assigned to different treatment

groups through random selection.• Randomized block design

Divide subjects with similar characteristics into blocks, and then within each block, randomly assign subjects to treatment groups.



Randomized block design• An experimenter testing the effects of a new weight

loss drink may first divide the subjects into age categories. Then within each age group, randomly assign subjects to either the treatment group or control group.



• Matched Pairs Design Subjects are paired up according to a similarity.

One subject in the pair is randomly selected to receive one treatment while the other subject receives a different treatment.

Example: Subjects exposed to the same toxins at a work site might be paired together


Key Elements of Experimental Design: Replication

• Replication is the repetition of an experiment using a large group of subjects.

• Example: To test a vaccine against a strain of influenza, 10,000 people are given the vaccine and another 10,000 people are given a placebo. Because of the sample size, the effectiveness of the vaccine would most likely be observed. We are making two complete re-runs of the experiment!


Example: Experimental Design

A company wants to test the effectiveness of a new gum developed to help people quit smoking. Identify a potential problem with the given experimental design and suggest a way to improve it.

The company identifies one thousand adults who are heavy smokers. The subjects are divided into blocks according to gender. After two months, the female group has a significant number of subjects who have quit smoking.


Solution: Experimental Design

Problem:

The groups are not similar. The new gum may have a greater effect on women than men, or vice versa.

Correction:

The subjects can be divided into blocks according to gender, but then within each block, they must be randomly assigned to be in the treatment group or the control group.


Example: Bad Experimental Design


Goal: Test effectiveness of fertilizer on different plots of land

Issue: There will be no way to know whether the results are attributable to the treatment or soil type

Example: Randomized Block Design


Within each block, use randomness to determine which trees are treated with fertilizer and which trees are not treated

Example: Completely Randomized Design


Use randomness to determine which trees are treated with fertilizer and which are not

Exercise 3: Identify Best Data Gathering Technique


a. Study the effect of stopping the cooling process in a nuclear reactor.

b. Study the amount of time college students taking a full course load would spend watching TV.

c. Study the effect of a calcium supplement given to young girls on bone mass.

d. Study the number of academic clubs that every MATES student participates in

Exercise 4: Analyze a Statistical Study


In 1778 Captain James Cook introduced goats to the Hawaiian Islands. It was later observed that the Silver Sword plant appeared to be less and less common. Botanists suspected the goats to be the cause and conducted a statistical study. They set up stations around the islands with similar climate and soil conditions. Each station consisted of two plots of land, one with a fence around it to keep the goats out.

Identify thea) Treatment b) Experimental Groupc) Control Group

Section 1.3 Summary

• Sampling techniques• Design of an experiment• Data collection techniques• What is a census?• Describe simulations, observational studies and

experiments• Identify control groups, placebo effects, completely

randomized experiments• Discuss potential pitfalls that might your data

unreliable


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