Math 227 Elementary Math 227 Elementary StatisticsStatistics

Bluman 6th edition

CHAPTER 1CHAPTER 1

The Nature of Probability and Statistics

ObjectivesObjectives

• Demonstrate knowledge of statistical terms.

• Differentiate between the two branches of statistics.

• Identify types of data.

• Identify the measurement level for each variable.

Objectives (cont.)Objectives (cont.)

• Identify the five basic sampling techniques.

• Explain the difference between an observational and an experimental study.

• Explain how statistics can be used and misused.

• Explain the importance of computers and calculators in statistics.

Section 1.1 Descriptive and Section 1.1 Descriptive and Inferential StatisticsInferential Statistics

StatisticsStatistics is the science of conducting studies to collect, organize, summarize, analyze, and draw conclusions from data.

Variables and Types Variables and Types of Dataof Data

• In order to gain knowledge about seemingly random events, statisticians collect information for variables that describe the events.

• A variablevariable is a characteristic or attribute that can assume different values.

• DataData are the values that variables can assume.

• A data set is a collection of data values.

• Random variables have values that are determined by chance.

• Descriptive statistics consists of the collection, organization, summarization, and presentation of data.

• Inferential statistics consists of generalizing from samples to populations, performing estimations and hypothesis tests, determining relationships among variables, and making predictions.

Example: Determine whether the results given are Example: Determine whether the results given are examples of descriptive or inferential statistics. examples of descriptive or inferential statistics.

a) In the 1996 presidential election, voters in Massachusetts cast 1,571,763 votes for Bill Clinton, 718,107 for Bob Dole, and 227,217 for H. Ross Perot.

Answer: Descriptive Statistics

b) Allergy therapy makes bees go away.

Answer: Inferential Statistics

Basic Vocabulary Basic Vocabulary

• A population consists of all subjects that are being studied.

• A sample is a group of subjects selected from a population.

• A parameter is a characteristic or measure obtained by using all the data values for a specific population

• A statistic is a characteristic or measure obtained by using the data values from a sample.

Example: Example: Consider the problem of estimating the average grade Consider the problem of estimating the average grade point average (GPA) of the 750 seniors at a college. point average (GPA) of the 750 seniors at a college.

a) What is the population? How many data values are in the population?

Answer: Population – seniors at a college

Data values – 750

b) What is the parameter of interest?

Answer: Their GPA

c) Suppose that a sample of 10 seniors is selected, and their GPAs are 2.72, 2.81, 2.65, 2.69, 3.17, 2.74, 2.57, 2.17, 3.48, 3.10. Calculate a statistic that you would use to estimate the parameter.

d) Suppose that another sample of 10 seniors was selected. Would it be likely that the value of the statistic is the same as in part (c)? Why or why not? Would the value of the parameter remain the same?

Answer: No, because another group of 10 seniors would have different GPA’s.

Answer: Yes, the parameter would be the same because we’re still looking at the GPA of all seniors.

Answer:

1.2 Variables and Types of Data1.2 Variables and Types of Data

Variables can be classified as qualitative (categorical) or quantitative (numerical).

• Qualitative variables can be placed into distinct categories according to some characteristic or attribute.

• Quantitative variables are numerical in nature and can be ordered or ranked.

Classification of VariablesClassification of Variables(Cont.)(Cont.)

Quantitative variables can be further classified into two groups.

• Discrete variables assume values that can be counted. (e.g. # of books, # of desks)

• Continuous variables can assume all values between any two specific values. (e.g. length, time, etc)

Classification of VariablesClassification of Variables

Data

Qualitative Quantitative

Discrete Continuous

Example: Example: Classify each variable as qualitative or quantitative. If the variable Classify each variable as qualitative or quantitative. If the variable is quantitative, further classify it as discrete or continuous.is quantitative, further classify it as discrete or continuous.

a) Number of people in a classroom

Answer: Quantitative – Discrete because # of people can be counted.

b) Weights of new born babies in a hospital

Answer: Quantitative – Continuous because the measurements are within a range.

c) Eye colors of students in Math 227

Answer: Qualitative

Boundaries of a continuous dataBoundaries of a continuous data

Continuous Data must be rounded because of the limits Continuous Data must be rounded because of the limits of the measuring device. Answers are rounded to the of the measuring device. Answers are rounded to the nearest given unit. nearest given unit.

Ex) Heights might be rounded to the nearest inch. *73 inches could mean any measure from 72.5 inches up to but not including 73.5 inches. So the boundaries of 73 inches is given as

72.5 – 73.5 inches.

(All values up to but not including 73.5 inches)

Recorded Values and Recorded Values and BoundariesBoundaries

Variable Recorded Value

Boundaries

Length 15 centimeters (cm)

Temperature 86 Fahrenheit (F)

Time 0.43 second (sec)

Mass 1.6 grams (g)

14.5-15.5 cm

85.5-86.5 F

0.425-0.435 sec

1.55-1.65 g17Bluman Chapter 1

Levels of MeasurementLevels of Measurement::• Variables can also be classified by how they are

categorized, counted, or measured.

• The level of measurement of the data is useful in deciding what procedure to take to apply statistics to real problems.

Ex) Can the data be organized into specific categories, such as area of residence (rural, suburban, or urban)?

Can the data values be ranked, such as first place, second place, etc.?

Are the values obtained from measurement, such as heights, IQs, or temperature?

• Four common types of measurement scales are used to classify variables: nominal, ordinal, interval, and ratio.

Levels of MeasurementLevels of Measurement::• Nominal—classifies data into mutually exclusive

(nonoverlapping), exhausting categories in which no order or ranking can be imposed on the data.

• Ordinal—classifies data into categories that can be ranked; however, precise differences between the ranks do not exist.

• Interval—ranks data, and precise differences between units of measure do exist; however, there is no meaningful zero.

• Ratio—possesses all the characteristics of interval measurement, and there exists a true zero.

Levels of MeasurementLevels of Measurement::

Nominal-level data

Ordinal-level data

Interval-level data

Ratio-level data

Zip code Gender Eye color

Grade Rating Ranking

SAT score IQ Temperature

Height Weight Time

**Table is missing from the handout (book page 8)

Example 1: Classify each as nominal-level, ordinal-Example 1: Classify each as nominal-level, ordinal-level, interval-level, or ratio level data. level, interval-level, or ratio level data.

a) Sizes of cars

Answer: Categorical – ordinal

b) Nationality of each student

Answer: Categorical – nominal

c) IQ of each student

Answer: Numerical – interval

d) Weights of new born babies

Answer: Numerical – ratio

Section 1.3 Data Collection and Section 1.3 Data Collection and Sampling TechniquesSampling Techniques

Surveys are the most common method of collecting data. Three methods of surveying are:

• Telephone surveys

• Mailed questionnaire surveys

• Personal interviews

Methods to obtain samples:Methods to obtain samples:

RandomRandom samples are selected using chance methods or random methods.

e.g. Lottery

Random Sampling - selection so that each has an equalequal chancechance of being selected

Methods to obtain samples Methods to obtain samples (cont.):(cont.):

• Systematic samples are obtained by numbering each subject of the population and then selecting every kth number.

e.g. A quality control engineer selects every 200th TV remote control from an assembly line and conducts a test of qualities.

• Systematic Sampling

– Select some starting point and then select every Kth element in the population

Methods to obtain samples Methods to obtain samples (cont.):(cont.):

• Stratified samples are obtained by dividing the population into groups according to some characteristic that is important to the study, then sampling from each group.

– e.g. A General Motors researcher has partitioned all registered cars into categories of subcompact, compact, mid-size, and full-size. He is surveying 200 car owners from each category.

• Stratified Sampling - subdivide the population into at least two different subgroups that share the same characteristics, then draw a sample from each subgroup (or stratum)

Methods to obtain samples Methods to obtain samples (cont.):(cont.):

• Cluster samples are obtained by using intact groups called clusters.

– e.g. Two of the nine colleges in the L.A. district are randomly selected, then all faculty from the two selected college are interviewed.

• Cluster Sampling - divide the population into sections (or clusters); randomly select some of those clusters; choose all members from selected clusters

• Convenience Samples are obtained due to the ease of getting

– e.g. An NBC television news reporter gets a reaction to a breaking story by polling people as they pass the front of his studio.

Example 1: Identify which of these types of sampling is Example 1: Identify which of these types of sampling is used: random, systematic, convenience, stratified, or used: random, systematic, convenience, stratified, or cluster.cluster.

a) A marketing expert for MTV is planning a survey in which 500 people will be randomly selected from each age group of 10-19, 20-29, and so on.

Answer: Stratified

b) A news reporter stands on a street corner and obtains a sample of city residents by selecting five passing adults about their smoking habits.

Answer: Convenience

c) In a Gallup poll of 1059 adults, the interview subjects were selected by using a computer to randomly generate telephone numbers that were then called.

Answer: Random

Example 1: Identify which of these types of sampling is Example 1: Identify which of these types of sampling is used: random, systematic, convenience, stratified, or used: random, systematic, convenience, stratified, or cluster. cluster. (Cont.)(Cont.)

d) At a police sobriety checkpoint at which every 10th driver was stopped and interviewed.

Answer: Systematic

e) A market researcher randomly selects 10 blocks in the Village of Newport, then asks all adult residents of the selected blocks whether they own a DVD player.

Answer: Cluster

f) Foods plans to conduct a marketing survey of 100 men and 100 women in Orange County.

Answer: Stratified

Example 1: Identify which of these types of sampling is Example 1: Identify which of these types of sampling is used: random, systematic, convenience, stratified, or used: random, systematic, convenience, stratified, or cluster. cluster. (Cont.)(Cont.)

g) CNN is planning an exit poll in which 100 polling stations will be randomly selected and all voters will be interviewed as they leave the premises.

Answer: Cluster

h) An executive mixes all the returned surveys in a bin, then obtains a sample group by pulling 50 of those surveys.

Answer: Random

i) The Dutchess County Commissioner of Jurors obtains a list of 42,763 car owners and constructs a pool of jurors by selecting every 150th name on that list.

Answer: Systematic

Section 1-4 Observational and Section 1-4 Observational and Experimental StudiesExperimental Studies

• Observational Study – The experimenter records the outcomes of an experiment without control.

• Experimental Study – The experimenter intervenes by administering treatment to the subjects in order to study its effects on the subject.

– An Independent Variable – the variable that is being manipulated by the researcher.

– A Dependent Variable – the outcome variable.

– A Treatment Group – the group that is being treated.

– A Controlled Group – the group that is not being treated.

– Confounding Factors – factors other than the treatment that can influence a study.

Example 1: Lipitor is a drug that is supposed to lower the cholesterol level. To test the effectiveness of the drug, 100 patients were randomly selected and 50 were randomly chosen to use Lipitor. The other 50 were given a placebo that contained no drug at all.

a) What is the treatment? Answer: Lipitor

b) Identify the treatment group and the control group.Answer: Treatment group – The group given Lipitor

Control group – The group given a placebo

c) Is this an observational or experimental study?Answer: Experimental

d) What factor could confound the result?Answer: Change eating habits, diet, exercise, smoking, genes.

Section 1.5 Uses and Misuses of Section 1.5 Uses and Misuses of StatisticsStatistics

Statistics can be misused in ways that are deceptive:

1) using samples that are not representative of the population;

2) questionnaire or interview process may be flawed;

3) conclusions are based on samples that are far too small;

4) using graphs that produce a misleading impression; etc.

Section 1.6 Computers and Section 1.6 Computers and CalculatorsCalculators

• In the past, statistical calculations were done with pencil and paper. However, with the advent of calculators, numerical computations became easier.

• Excel, MINITAB, and the TI-83 graphing calculator can be used to perform statistical computations.

• Students should realize that the computer and calculator merely give numerical answers and save time and effort of doing calculations by hand.

SUMMARYSUMMARY• The two major areas of statistics are descriptive and

inferential.

• When the populations to be studied are large, statisticians use subgroups called samples.

• The five basic methods for obtaining samples are: random, systematic, stratified, cluster, convenience.

• Data can be classified as qualitative or quantitative.

• The four basic types of measurement are nominal, ordinal, interval, and ratio.

• The two basic types of statistical studies are observational and experimental.